Package 'generalCorr'

Title: Generalized Correlations, Causal Paths and Portfolio Selection
Description: Function gmcmtx0() computes a more reliable (general) correlation matrix. Since causal paths from data are important for all sciences, the package provides many sophisticated functions. causeSummBlk() and causeSum2Blk() give easy-to-interpret causal paths. Let Z denote control variables and compare two flipped kernel regressions: X=f(Y, Z)+e1 and Y=g(X, Z)+e2. Our criterion Cr1 says that if |e1*Y|>|e2*X| then variation in X is more "exogenous or independent" than in Y, and the causal path is X to Y. Criterion Cr2 requires |e2|<|e1|. These inequalities between many absolute values are quantified by four orders of stochastic dominance. Our third criterion Cr3, for the causal path X to Y, requires new generalized partial correlations to satisfy |r*(x|y,z)|< |r*(y|x,z)|. The function parcorVec() reports generalized partials between the first variable and all others. The package provides several R functions including get0outliers() for outlier detection, bigfp() for numerical integration by the trapezoidal rule, stochdom2() for stochastic dominance, pillar3D() for 3D charts, canonRho() for generalized canonical correlations, depMeas() measures nonlinear dependence, and causeSummary(mtx) reports summary of causal paths among matrix columns. Portfolio selection: decileVote(), momentVote(), dif4mtx(), exactSdMtx() can rank several stocks. Functions whose names begin with 'boot' provide bootstrap statistical inference, including a new bootGcRsq() test for "Granger-causality" allowing nonlinear relations. A new tool for evaluation of out-of-sample portfolio performance is outOFsamp(). Panel data implementation is now included. See eight vignettes of the package for theory, examples, and usage tips. See Vinod (2019) \doi{10.1080/03610918.2015.1122048}.
Authors: Prof. H. D. Vinod, Fordham University, NY.
Maintainer: H. D. Vinod <[email protected]>
License: GPL (>= 2)
Version: 1.2.6
Built: 2025-03-04 04:48:45 UTC
Source: https://github.com/cran/generalCorr

Help Index

Absolute residuals of kernel regression of x on y.

Description

This internal function calls the kern function to implement kernel regression with the option residuals=TRUE and returns absolute residuals.

Usage

abs_res(x, y)

Arguments

x

vector of data on the dependent variable
y

vector of data on the regressor

Details

The first argument is assumed to be the dependent variable. If abs_res(x,y) is used, you are regressing x on y (not the usual y on x)

Value

absolute values of kernel regression residuals are returned.

Note

This function is intended for internal use.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

## Not run: 
set.seed(330)
x=sample(20:50)
y=sample(20:50)
abs_res(x,y)

## End(Not run)

Absolute values of gradients (apd's) of kernel regressions of x on y when both x and y are standardized.

Description

1) standardize the data to force mean zero and variance unity, 2) kernel regress x on y, with the option ‘gradients = TRUE’ and finally 3) compute the absolute values of gradients

Usage

abs_stdapd(x, y)

Arguments

x

vector of data on the dependent variable
y

data on the regressors which can be a matrix

Details

The first argument is assumed to be the dependent variable. If abs_stdapd(x,y) is used, you are regressing x on y (not the usual y on x). The regressors can be a matrix with 2 or more columns. The missing values are suitably ignored by the standardization.

Value

Absolute values of kernel regression gradients are returned after standardizing the data on both sides so that the magnitudes of amorphous partial derivatives (apd's) are comparable between regression of x on y on the one hand and regression of y on x on the other.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

## Not run: 
set.seed(330)
x=sample(20:50)
y=sample(20:50)
abs_stdapd(x,y)

## End(Not run)

Absolute values of gradients (apd's) of kernel regressions of x on y when both x and y are standardized and control variables are present.

Description

1) standardize the data to force mean zero and variance unity, 2) kernel regress x on y and a matrix of control variables, with the option ‘gradients = TRUE’ and finally 3) compute the absolute values of gradients

Usage

abs_stdapdC(x, y, ctrl)

Arguments

x

vector of data on the dependent variable
y

data on the regressors which can be a matrix
ctrl

Data matrix on the control variable(s) beyond causal path issues

Details

The first argument is assumed to be the dependent variable. If abs_stdapdC(x,y) is used, you are regressing x on y (not the usual y on x). The regressors can be a matrix with 2 or more columns. The missing values are suitably ignored by the standardization.

Value

Absolute values of kernel regression gradients are returned after standardizing the data on both sides so that the magnitudes of amorphous partial derivatives (apd's) are comparable between regression of x on y on the one hand and regression of y on x on the other.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

See Also

See abs_stdapd.

Examples

## Not run: 
set.seed(330)
x=sample(20:50)
y=sample(20:50)
z=sample(20:50)
abs_stdapdC(x,y,ctrl=z)

## End(Not run)

Absolute values of residuals of kernel regressions of x on y when both x and y are standardized.

Description

1) Standardize the data to force mean zero and variance unity, 2) kernel regress x on y, with the option ‘residuals = TRUE’ and finally 3) compute the absolute values of residuals.

Usage

abs_stdres(x, y)

Arguments

x

vector of data on the dependent variable
y

data on the regressors which can be a matrix

Details

The first argument is assumed to be the dependent variable. If abs_stdres(x,y) is used, you are regressing x on y (not the usual y on x). The regressors can be a matrix with 2 or more columns. The missing values are suitably ignored by the standardization.

Value

Absolute values of kernel regression residuals are returned after standardizing the data on both sides so that the magnitudes of residuals are comparable between regression of x on y on the one hand and regression of y on x on the other.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Examples

## Not run: 
set.seed(330)
x=sample(20:50)
y=sample(20:50)
abs_stdres(x,y)

## End(Not run)

Absolute values of residuals of kernel regressions of x on y when both x and y are standardized and control variables are present (C for control presence).

Description

1) standardize the data to force mean zero and variance unity, 2) kernel regress x on y and a matrix of control variables, with the option ‘residuals = TRUE’ and finally 3) compute the absolute values of residuals.

Usage

abs_stdresC(x, y, ctrl)

Arguments

x

vector of data on the dependent variable
y

data on the regressors which can be a matrix
ctrl

Data matrix on the control variable(s) beyond causal path issues

Details

The first argument is assumed to be the dependent variable. If abs_stdres(x,y) is used, you are regressing x on y (not the usual y on x). The regressors can be a matrix with two or more columns. The missing values are suitably ignored by the standardization.

Value

Absolute values of kernel regression residuals are returned after standardizing the data on both sides so that the magnitudes of residuals are comparable between regression of x on y on the one hand and regression of y on x on the other.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D.'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

See Also

See abs_stdres.

Examples

## Not run: 
set.seed(330)
x=sample(20:50)
y=sample(20:50)
z=sample(21:51)
abs_stdresC(x,y,ctrl=z)

## End(Not run)

Absolute residuals kernel regressions of standardized x on y and control variables, Cr1 has abs(RHS*y) not gradients.

Description

1) standardize the data to force mean zero and variance unity, 2) kernel regress x on y and a matrix of control variables, with the option ‘residuals = TRUE’ and finally 3) compute the absolute values of residuals.

Usage

abs_stdrhserC(x, y, ctrl, ycolumn = 1)

Arguments

x

vector of data on the dependent variable
y

data on the regressors which can be a matrix
ctrl

Data matrix on the control variable(s) beyond causal path issues
ycolumn

if y has more than one column, the column number used when multiplying residuals times this column of y, default=1 or first column of y matrix is used

Details

The first argument is assumed to be the dependent variable. If abs_stdrhserC(x,y) is used, you are regressing x on y (not the usual y on x). The regressors can be a matrix with 2 or more columns. The missing values are suitably ignored by the standardization.

Value

Absolute values of kernel regression residuals are returned after standardizing the data on both sides so that the magnitudes of residuals are comparable between regression of x on y on the one hand and regression of y on x on the other.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

See Also

See abs_stdres.

Examples

## Not run: 
set.seed(330)
x=sample(20:50)
y=sample(20:50)
z=sample(21:51)
abs_stdrhserC(x,y,ctrl=z)

## End(Not run)

Absolute values of Hausman-Wu null in kernel regressions of x on y when both x and y are standardized.

Description

1) standardize the data to force mean zero and variance unity, 2) kernel regress x on y, with the option ‘gradients = TRUE’ and finally 3) compute the absolute values of Hausman-Wu null hypothesis for testing exogeneity, or E(RHS.regressor*error)=0 where error is approximated by kernel regression residuals

Usage

abs_stdrhserr(x, y)

Arguments

x

vector of data on the dependent variable
y

data on the regressors which can be a matrix

Details

The first argument is assumed to be the dependent variable. If abs_stdrhserr(x,y) is used, you are regressing x on y (not the usual y on x). The regressors can be a matrix with 2 or more columns. The missing values are suitably ignored by the standardization.

Value

Absolute values of kernel regression RHS*residuals are returned after standardizing the data on both sides so that the magnitudes of Hausman-Wu null values are comparable between regression of x on y on the one hand and flipped regression of y on x on the other.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

## Not run: 
set.seed(330)
x=sample(20:50)
y=sample(20:50)
abs_stdrhserr(x,y)

## End(Not run)

Block version of abs-stdres Absolute values of residuals of kernel regressions of standardized x on standardized y, no control variables.

Description

1) Standardize the data to force mean zero and variance unity, 2) kernel regress x on y, with the option ‘residuals = TRUE’ and finally 3) compute the absolute values of residuals.

Usage

absBstdres(x, y, blksiz = 10)

Arguments

x

vector of data on the dependent variable
y

data on the regressors which can be a matrix
blksiz

block size, default=10, if chosen blksiz >n, where n=rows in matrix then blksiz=n. That is, no blocking is done

Details

The first argument is assumed to be the dependent variable. If abs_stdres(x,y) is used, you are regressing x on y (not the usual y on x). The regressors can be a matrix with 2 or more columns. The missing values are suitably ignored by the standardization.

Value

Absolute values of kernel regression residuals are returned after standardizing the data on both sides so that the magnitudes of residuals are comparable between regression of x on y on the one hand and regression of y on x on the other.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Examples

## Not run: 
set.seed(330)
x=sample(20:50)
y=sample(20:50)
abs_stdres(x,y)

## End(Not run)

Block version of Absolute values of residuals of kernel regressions of standardized x on standardized y and control variables.

Description

1) standardize the data to force mean zero and variance unity, 2) kernel regress x on y and a matrix of control variables, with the option ‘residuals = TRUE’ and finally 3) compute the absolute values of residuals.

Usage

absBstdresC(x, y, ctrl, blksiz = 10)

Arguments

x

vector of data on the dependent variable
y

data on the regressors which can be a matrix
ctrl

Data matrix on the control variable(s) beyond causal path issues
blksiz

block size, default=10, if chosen blksiz >n, where n=rows in matrix then blksiz=n. That is, no blocking is done

Details

The first argument is assumed to be the dependent variable. If abs_stdres(x,y) is used, you are regressing x on y (not the usual y on x). The regressors can be a matrix with two or more columns. The missing values are suitably ignored by the standardization.

Value

Absolute values of kernel regression residuals are returned after standardizing the data on both sides so that the magnitudes of residuals are comparable between regression of x on y on the one hand and regression of y on x on the other.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D.'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

See Also

See abs_stdres.

Examples

## Not run: 
set.seed(330)
x=sample(20:50)
y=sample(20:50)
z=sample(21:51)
absBstdresC(x,y,ctrl=z)

## End(Not run)

Block version abs_stdrhser Absolute residuals kernel regressions of standardized x on y and control variables, Cr1 has abs(Resid*RHS).

Description

1) standardize the data to force mean zero and variance unity, 2) kernel regress x on y and a matrix of control variables, with the option ‘residuals = TRUE’ and finally 3) compute the absolute values of residuals.

Usage

absBstdrhserC(x, y, ctrl, ycolumn = 1, blksiz = 10)

Arguments

x

vector of data on the dependent variable
y

data on the regressors which can be a matrix
ctrl

Data matrix on the control variable(s) beyond causal path issues
ycolumn

if y has more than one column, the column number used when multiplying residuals times this column of y, default=1 or first column of y matrix is used
blksiz

block size, default=10, if chosen blksiz >n, where n=rows in matrix then blksiz=n. That is, no blocking is done

Details

The first argument is assumed to be the dependent variable. If absBstdrhserC(x,y) is used, you are regressing x on y (not the usual y on x). The regressors can be a matrix with 2 or more columns. The missing values are suitably ignored by the standardization.

Value

Absolute values of kernel regression residuals are returned after standardizing the data on both sides so that the magnitudes of residuals are comparable between regression of x on y on the one hand and regression of y on x on the other.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

See Also

See abs_stdres.

Examples

## Not run: 
set.seed(330)
x=sample(20:50)
y=sample(20:50)
z=sample(21:51)
absBstdrhserC(x,y,ctrl=z)

## End(Not run)

Report causal identification for all pairs of variables in a matrix (deprecated function). It is better to choose a target variable and pair it with all others, instead of considering all possible targets.

Description

This studies all possible (perhaps too many) causal directions in a matrix. It is deprecated because it uses older criterion 1 by caling abs_stdapd I recommend using causeSummary or its block version cuseSummBlk. This uses abs_stdres, comp_portfo2, etc. and returns a matrix with 7 columns having detailed output. Criterion 1 has been revised as described in Vinod (2019) and is known to work better.

Usage

allPairs(mtx, dig = 6, verbo = FALSE, typ = 1, rnam = FALSE)

Arguments

mtx

Input matrix with variable names
dig

Digits of accuracy in reporting (=6, default)
verbo

Logical variable, set to 'TRUE' if printing is desired
typ

Causal direction criterion number (typ=1 is default) Criterion 1 (Cr1) compares kernel regression absolute values of gradients. Criterion 2 (Cr2) compares kernel regression absolute values of residuals. Criterion 3 (Cr3) compares kernel regression based r*(x|y) with r*(y|x).
rnam

Logical variable, default rnam=FALSE means the user does not want the row names to be (somewhat too cleverly) assigned by the function.

Value

A 7-column matrix called 'outcause' with names of variables X and Y in the first two columns and the name of the 'causal' variable in 3rd col. Remaining four columns report numerical computations of SD1 to SD4, r*(x|y), r*(y|x). Pearson r and p-values for its traditional significance testing.

Note

The cause reported in the third column is identified from the sign of the first SD1 only, ignoring SD2, SD3 and SD4 under both Cr1 and Cr2. It is a good idea to loop a call to this function with typ=1:3. One can print the resulting 'outcause' matrix with the xtable(outcause) for the Latex output. A similar deprecated function included in this package, called some0Pairs, incorporates all SD1 to SD4 and all three criteria Cr1 rto Cr3 to report a ‘sum’ of indexes representing the signed number whose sign can more comprehensively help determine the causal direction(s). Since the Cr1 here is revised in later work, this is deprecated.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D.'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

See Also

See Also somePairs, some0Pairs causeSummary

Examples

data(mtcars)
options(np.messages=FALSE)
for(j in 1:3){
a1=allPairs(mtcars[,1:3], typ=j)
print(a1)}

internal badCol

Description

intended for internal use

Usage

data(badCol)

Format

The format is: int 4

Compute the numerical integration by the trapezoidal rule.

Description

See page 220 of Vinod (2008) “Hands-on Intermediate Econometrics Using R,” for the trapezoidal integration formula needed for stochastic dominance. The book explains pre-multiplication by two large sparse matrices denoted by IF,IfI_F, I_f. Here we accomplish the same computation without actually creating the large sparse matrices. For example, the IfI_f is replaced by cumsum in this code (unlike the R code in my textbook).

Usage

bigfp(d, p)

Arguments

d

A vector of consecutive interval lengths, upon combining both data vectors
p

Vector of probabilities of the type 1/2T, 2/2T, 3/2T, etc. to 1.

Value

Returns a result after pre-multiplication by IF,IfI_F, I_f matrices, without actually creating the large sparse matrices. This is an internal function.

Note

This is an internal function, called by the function stochdom2, for comparison of two portfolios in terms of stochastic dominance (SD) of orders 1 to 4. Typical usage is: sd1b=bigfp(d=dj, p=rhs) sd2b=bigfp(d=dj, p=sd1b) sd3b=bigfp(d=dj, p=sd2b) sd4b=bigfp(d=dj, p=sd3b). This produces numerical evaluation vectors for the four orders, SD1 to SD4.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D.', 'Hands-On Intermediate Econometrics Using R' (2008) World Scientific Publishers: Hackensack, NJ. https://www.worldscientific.com/worldscibooks/10.1142/12831

bootstrap confidence intervals for (x2-x1) exact SD1 to SD4 stochastic dominance .

Description

This calls the meboot package to create J=999 replications of portfolio return matrices and compute 95% confidence intervals on x1, x2 and their difference (x2-x1). If the interval on (x2-x1) conta.ins zero the choice between the two can reverse due to sampling variation

Usage

bootDom12(x1, x2, confLevel = 95, reps = 999)

Arguments

x1

a vector of n portfolio returns
x2

a vector of n portfolio returns
confLevel

confidene level confLevel=95 is default
reps

number of bootstrap resamples, default is reps=999

Value

A matrix with six columns. First two Low1 and Upp1 are confidence interval limits for x1. Next two columns have analogous limits for x2. The last but first columns entitled Lowx2mx1 means lower confidence limit for (x2-x1), where m=minus. The last column entitled Uppx2mx1 means upper confidence limit for (x2-x1).

For strong stochastic dominance of x2 over x1 dominance beyond sampling variability, zero should not be inside the confidence interval in the last two columns.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

See Also

see exactSdMtx

Compute vector of n999 nonlinear Granger causality paths

Description

Maximum entropy bootstrap (meboot) package is used for statistical inference The bootstrap output can be analyzed to estimate an approximate confidence interval on sample-based direction of the causal path. The LC in the function name stands for local constant. Kernel regression np package options regtype="lc" for local constant, and bwmethod="cv.ls" for least squares-based bandwidth selection are fixed.

Usage

bootGcLC(x1, x2, px2 = 4, px1 = 4, pwanted = 4, ctrl = 0, n999 = 9)

Arguments

x1

The data vector x1
x2

The data vector x2
px2

number of lags of x2 in the data, default px2=4
px1

number of lags of x1 in the data default px1=4
pwanted

number of lags of both x2 and x1 wanted for Granger causal analysis, default =4
ctrl

data matrix having control variable(s) if any
n999

Number of bootstrap replications (default=9)

Value

out is n999 X 3 matrix for 3 outputs of GcauseX12 resampled

Note

This computation is computer intensive and generally very slow. It may be better to use this function it at a later stage in the investigation, after a preliminary causal determination is already made. The 3 outputs of GauseX12 are two Rsquares and the difference between after subtracting the second from the first. Col. 1 has (RsqX1onX2) Col.2 has (RsqX2onX1), and Col.3 has dif=(RsqX1onX2 -RsqX2onX1) Note that R-squares are always positive. If dif>0, RsqX1onX2>RsqX2onX1, implying that x2 on RHS performs better that is, x2 –> x1 is the path, or x2 Granger-causes x1. If dif<0, x1 –> x2 holds. If dif is too close to zero, we may have bidirectional causality x1 <–> x2. The proportion of resamples (out of n999) having dif<0 suggests level of confidence in the conclusion x1 –> x2. The proportion of resamples (out of n999) having dif>0 suggests level of confidence in the conclusion x2 –> x1.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Zheng, S., Shi, N.-Z., and Zhang, Z. (2012). Generalized measures of correlation for asymmetry, nonlinearity, and beyond. Journal of the American Statistical Association, vol. 107, pp. 1239-1252.

Vinod, H. D. and Lopez-de-Lacalle, J. (2009). 'Maximum entropy bootstrap for time series: The meboot R package.' Journal of Statistical Software, Vol. 29(5), pp. 1-19.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See Also GcRsqX12c.

Examples

## Not run: 
library(Ecdat);options(np.messages=FALSE);attach(data.frame(MoneyUS))
bootGcLC(y,m,n999=9) 

## End(Not run)
## Not run: 
library(lmtest); data(ChickEgg);attach(data.frame(ChickEgg))
b2=bootGcLC(x1=chicken,x2=egg,pwanted=3,px1=3,px2=3,n999=99)

## End(Not run)

Compute vector of n999 nonlinear Granger causality paths

Description

Maximum entropy bootstrap (meboot) package is used for statistical inference The bootstrap output can be analyzed to estimate an approximate confidence interval on sample-based direction of the causal path. Kernel regression np package options regtype="ll" for local linear, and bwmethod="cv.aic" for AIC-based bandwidth selection are fixed.

Usage

bootGcRsq(x1, x2, px2 = 4, px1 = 4, pwanted = 4, ctrl = 0, n999 = 9)

Arguments

x1

The data vector x1
x2

The data vector x2
px2

number of lags of x2 in the data, default px2=4
px1

number of lags of x1 in the data default px1=4
pwanted

number of lags of both x2 and x1 wanted for Granger causal analysis, default =4
ctrl

data matrix having control variable(s) if any
n999

Number of bootstrap replications (default=9)

Value

out is n999 X 3 matrix for 3 outputs of GcauseX12 resampled

Note

This computation is computer intensive and generally very slow. It may be better to use this function it at a later stage in the investigation, after a preliminary causal determination is already made. The 3 outputs of GauseX12 are two Rsquares and the difference between them after subtracting the second from the first. Col. 1 has (RsqX1onX2), Col.2 has (RsqX2onX1), and Col.3 has dif=(RsqX1onX2 -RsqX2onX1) Note that R-squares are always positive. If dif>0, RsqX1onX2>RsqX2onX1, implying that x2 on RHS performs better that is, x2 –> x1 is the causal path. If dif<0, x1 –> x2 holds. If dif is too close to zero, we may have bidirectional causality x1 <–> x2. The proportion of resamples (out of n999) having dif<0 suggests level of confidence in the conclusion x1 –> x2. The proportion of resamples (out of n999) having dif>0 suggests level of confidence in the conclusion x2 –> x1.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Zheng, S., Shi, N.-Z., and Zhang, Z. (2012). Generalized measures of correlation for asymmetry, nonlinearity, and beyond. Journal of the American Statistical Association, vol. 107, pp. 1239-1252.

Vinod, H. D. and Lopez-de-Lacalle, J. (2009). 'Maximum entropy bootstrap for time series: The meboot R package.' Journal of Statistical Software, Vol. 29(5), pp. 1-19.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See Also GcRsqX12.

Examples

## Not run: 
library(Ecdat);options(np.messages=FALSE);attach(data.frame(MoneyUS))
bootGcRsq(y,m,n999=9) 

## End(Not run)
## Not run: 
library(lmtest); data(ChickEgg);attach(data.frame(ChickEgg))
options(np.messages=FALSE)
b2=bootGcLC(x1=chicken,x2=egg,pwanted=3,px1=3,px2=3,n999=99)
Fn=function(x)quantile(x,prob=c(0.025, 0.975))#confInt
apply(b1,2,Fn)#reports 95 percent confidence interval

## End(Not run)

Compute matrix of n999 rows and p-1 columns of bootstrap ‘sum’ (scores from Cr1 to Cr3).

Description

The ‘2’ in the name of the function suggests a second implementation of ‘bootPair,’ where exact stochastic dominance, decileVote, and momentVote are used. Maximum entropy bootstrap (meboot) package is used for statistical inference using the sum of three signs sg1 to sg3, from the three criteria Cr1 to Cr3, to assess preponderance of evidence in favor of a sign, (+1, 0, -1). The bootstrap output can be analyzed to assess the approximate preponderance of a particular sign which determines the causal direction.

Usage

bootPair2(mtx, ctrl = 0, n999 = 9)

Arguments

mtx

data matrix with two or more columns
ctrl

data matrix having control variable(s) if any
n999

Number of bootstrap replications (default=9)

Value

Function creates a matrix called ‘out’. If the input to the function called mtx has p columns, the output out of bootPair2(mtx) is a matrix of n999 rows and p-1 columns, each containing resampled ‘sum’ values summarizing the weighted sums associated with all three criteria from the function silentPair2(mtx) applied to each bootstrap sample separately.

Note

This computation is computer-intensive and generally very slow. It may be better to use it later in the investigation, after a preliminary causal determination is already made. A positive sign for j-th weighted sum reported in the column ‘sum’ means that the first variable listed in the argument matrix mtx is the ‘kernel cause’ of the variable in the (j+1)-th column of mtx.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Zheng, S., Shi, N.-Z., and Zhang, Z. (2012). Generalized measures of correlation for asymmetry, nonlinearity, and beyond. Journal of the American Statistical Association, vol. 107, pp. 1239-1252.

Vinod, H. D. and Lopez-de-Lacalle, J. (2009). 'Maximum entropy bootstrap for time series: The meboot R package.' Journal of Statistical Software, Vol. 29(5), pp. 1-19.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

Vinod, Hrishikesh D., R Package GeneralCorr Functions for Portfolio Choice (November 11, 2021). Available at SSRN: https://ssrn.com/abstract=3961683

Vinod, Hrishikesh D., Stochastic Dominance Without Tears (January 26, 2021). Available at SSRN: https://ssrn.com/abstract=3773309

See Also

See Also silentPair2.

Examples

## Not run: 
options(np.messages = FALSE)
set.seed(34);x=sample(1:10);y=sample(2:11)
bb=bootPair2(cbind(x,y),n999=29)
apply(bb,2,summary) #gives summary stats for n999 bootstrap sum computations

bb=bootPair2(airquality,n999=999);options(np.messages=FALSE)
apply(bb,2,summary) #gives summary stats for n999 bootstrap sum computations

data('EuroCrime')
attach(EuroCrime)
bootPair2(cbind(crim,off),n999=29)#First col. crim causes officer deployment,
#hence positives signs are most sensible for such call to bootPairs
#note that n999=29 is too small for real problems, chosen for quickness here.

## End(Not run)

Compute matrix of n999 rows and p-1 columns of bootstrap ‘sum’ (strength from Cr1 to Cr3).

Description

Maximum entropy bootstrap (meboot) package is used for statistical inference using the sum of three signs sg1 to sg3 from the three criteria Cr1 to Cr3 to assess preponderance of evidence in favor of a sign. (+1, 0, -1). The bootstrap output can be analyzed to assess approximate preponderance of a particular sign which determines the causal direction.

Usage

bootPairs(mtx, ctrl = 0, n999 = 9)

Arguments

mtx

data matrix with two or more columns
ctrl

data matrix having control variable(s) if any
n999

Number of bootstrap replications (default=9)

Value

out When mtx has p columns, out of bootPairs(mtx) is a matrix of n999 rows and p-1 columns each containing resampled ‘sum’ values summarizing the weighted sums associated with all three criteria from the function silentPairs(mtx) applied to each bootstrap sample separately.

Note

This computation is computer intensive and generally very slow. It may be better to use it at a later stage in the investigation when a preliminary causal determination is already made. A positive sign for j-th weighted sum reported in the column ‘sum’ means that the first variable listed in the argument matrix mtx is the ‘kernel cause’ of the variable in the (j+1)-th column of mtx.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Zheng, S., Shi, N.-Z., and Zhang, Z. (2012). Generalized measures of correlation for asymmetry, nonlinearity, and beyond. Journal of the American Statistical Association, vol. 107, pp. 1239-1252.

Vinod, H. D. and Lopez-de-Lacalle, J. (2009). 'Maximum entropy bootstrap for time series: The meboot R package.' Journal of Statistical Software, Vol. 29(5), pp. 1-19.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See Also silentPairs.

Examples

## Not run: 
options(np.messages = FALSE)
set.seed(34);x=sample(1:10);y=sample(2:11)
bb=bootPairs(cbind(x,y),n999=29)
apply(bb,2,summary) #gives summary stats for n999 bootstrap sum computations

bb=bootPairs(airquality,n999=999);options(np.messages=FALSE)
apply(bb,2,summary) #gives summary stats for n999 bootstrap sum computations

data('EuroCrime')
attach(EuroCrime)
bootPairs(cbind(crim,off),n999=29)#First col. crim causes officer deployment,
#hence positives signs are most sensible for such call to bootPairs
#note that n999=29 is too small for real problems, chosen for quickness here.

## End(Not run)

Compute matrix of n999 rows and p-1 columns of bootstrap ‘sum’ index (strength from older criterion Cr1, with newer Cr2 and Cr3).

Description

Maximum entropy bootstrap (meboot) package is used for statistical inference using the sum of three signs sg1 to sg3 from the three criteria Cr1 to Cr3 to assess preponderance of evidence in favor of a sign. (+1, 0, -1). The bootstrap output can be analyzed to assess approximate preponderance of a particular sign which determines the causal direction.

Usage

bootPairs0(mtx, ctrl = 0, n999 = 9)

Arguments

mtx

data matrix with two or more columns
ctrl

data matrix having control variable(s) if any
n999

Number of bootstrap replications (default=9)

Value

out When mtx has p columns, out of bootPairs(mtx) is a matrix of n999 rows and p-1 columns each containing resampled ‘sum’ values summarizing the weighted sums associated with all three criteria from the function silentPairs(mtx) applied to each bootstrap sample separately.

Note

This computation is computer intensive and generally very slow. It may be better to use it at a later stage in the investigation when a preliminary causal determination is already made. A positive sign for j-th weighted sum reported in the column ‘sum’ means that the first variable listed in the argument matrix mtx is the ‘kernel cause’ of the variable in the (j+1)-th column of mtx.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Zheng, S., Shi, N.-Z., and Zhang, Z. (2012). Generalized measures of correlation for asymmetry, nonlinearity, and beyond. Journal of the American Statistical Association, vol. 107, pp. 1239-1252.

Vinod, H. D. and Lopez-de-Lacalle, J. (2009). 'Maximum entropy bootstrap for time series: The meboot R package.' Journal of Statistical Software, Vol. 29(5), pp. 1-19.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See Also silentPairs0, bootPairs has the version with later version of Cr1.

Examples

## Not run: 
options(np.messages = FALSE)
set.seed(34);x=sample(1:10);y=sample(2:11)
bb=bootPairs0(cbind(x,y),n999=29)
apply(bb,2,summary) #gives summary stats for n999 bootstrap sum computations

bb=bootPairs0(airquality,n999=999);options(np.messages=FALSE)
apply(bb,2,summary) #gives summary stats for n999 bootstrap sum computations

data('EuroCrime')
attach(EuroCrime)
bootPairs0(cbind(crim,off),n999=29)#First col. crim causes officer deployment,
#hence positives signs are most sensible for such call to bootPairs
#note that n999=29 is too small for real problems, chosen for quickness here.

## End(Not run)

Compute confidence intervals [quantile(s)] of indexes from bootPairs output

Description

Begin with the output of bootPairs function, a (n999 by p-1) matrix when there are p columns of data, bootQuantile produces a (k by p-1) mtx of quantile(s) of bootstrap ouput assuming that there are k quantiles needed.

Usage

bootQuantile(out, probs = c(0.025, 0.975), per100 = TRUE)

Arguments

out

output from bootPairs with p-1 columns and n999 rows
probs

quantile evaluation probabilities. The default is k=2, probs=c(.025,0.975) for a 95 percent confidence interval. Note that there are k=2 quantiles desired for each column with this specification
per100

logical (default per100=TRUE) to change the range of 'sum' to [-100, 100] values which are easier to interpret

Value

CI k quantiles evaluated at probs as a matrix with k rows and quantile of pairwise p-1 indexes representing p-1 column pairs (fixing the first column in each pair) This function summarizes the output of of bootPairs(mtx) (a n999 by p-1 matrix) each containing resampled ‘sum’ values summarizing the weighted sums associated with all three criteria from the function silentPairs(mtx) applied to each bootstrap sample separately. #'

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. and Lopez-de-Lacalle, J. (2009). 'Maximum entropy bootstrap for time series: The meboot R package.' Journal of Statistical Software, Vol. 29(5), pp. 1-19.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See Also silentPairs.

Examples

## Not run: 
options(np.messages = FALSE)
set.seed(34);x=sample(1:10);y=sample(2:11)
bb=bootPairs(cbind(x,y),n999=29)
bootQuantile(bb) #gives summary stats for n999 bootstrap sum computations

bb=bootPairs(airquality,n999=999);options(np.messages=FALSE)
bootQuantile(bb,tau=0.476)#signs for n999 bootstrap sum computations

data('EuroCrime')
attach(EuroCrime)
bb=bootPairs(cbind(crim,off),n999=29) #col.1= crim causes off 
#hence positive signs are more intuitively meaningful.
#note that n999=29 is too small for real problems, chosen for quickness here.
bootQuantile(bb)# quantile matrix for n999 bootstrap sum computations

## End(Not run)

Probability of unambiguously correct (+ or -) sign from bootPairs output

Description

If there are p columns of data, bootSign produces a p-1 by 1 vector of probabilities of correct signs assuming that the mean of n999 values has the correct sign and assuming that m of the 'sum' index values inside the range [-tau, tau] are neither positive nor negative but indeterminate or ambiguous (being too close to zero). That is, the denominator of P(+1) or P(-1) is (n999-m) if m signs are too close to zero. Thus it measures the bootstrap success rate in identifying the correct sign, when the sign of the average of n999 bootstraps is assumed to be correct.

Usage

bootSign(out, tau = 0.476)

Arguments

out

output from bootPairs with p-1 columns and n999 rows
tau

threshold to determine what value is too close to zero, default tau=0.476 is equivalent to 15 percent threshold for the unanimity index ui

Value

sgn When mtx has p columns, sgn reports pairwise p-1 signs representing (fixing the first column in each pair) the average sign after averaging the output of of bootPairs(mtx) (a n999 by p-1 matrix) each containing resampled ‘sum’ values summarizing the weighted sums associated with all three criteria from the function silentPairs(mtx) applied to each bootstrap sample separately. #'

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. and Lopez-de-Lacalle, J. (2009). 'Maximum entropy bootstrap for time series: The meboot R package.' Journal of Statistical Software, Vol. 29(5), pp. 1-19.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See Also silentPairs, bootQuantile, bootSignPcent.

Examples

## Not run: 
options(np.messages = FALSE)
set.seed(34);x=sample(1:10);y=sample(2:11)
bb=bootPairs(cbind(x,y),n999=29)
bootSign(bb,tau=0.476) #gives success rate in n999 bootstrap sum computations

bb=bootPairs(airquality,n999=999);options(np.messages=FALSE)
bootSign(bb,tau=0.476)#signs for n999 bootstrap sum computations

data('EuroCrime');options(np.messages=FALSE)
attach(EuroCrime)
bb=bootPairs(cbind(crim,off),n999=29) #col.1= crim causes off 
#hence positive signs are more intuitively meaningful.
#note that n999=29 is too small for real problems, chosen for quickness here.
bootSign(bb,tau=0.476)#gives success rate in n999 bootstrap sum computations

## End(Not run)

Probability of unambiguously correct (+ or -) sign from bootPairs output transformed to percentages.

Description

If there are p columns of data, bootSignPcent produces a p-1 by 1 vector of probabilities of correct signs assuming that the mean of n999 values has the correct sign and assuming that m of the 'ui' index values inside the range [-tau, tau] are neither positive nor negative but indeterminate or ambiguous (being too close to zero). That is, the denominator of P(+1) or P(-1) is (n999-m) if m signs are too close to zero. Thus it measures the bootstrap success rate in identifying the correct sign, when the sign of the average of n999 bootstraps is assumed to be correct.

Usage

bootSignPcent(out, tau = 5)

Arguments

out

output from bootPairs with p-1 columns and n999 rows
tau

threshold to determine what value is too close to zero, default tau=5 is 5 percent threshold for the unanimity index ui

Value

sgn When mtx has p columns, sgn reports pairwise p-1 signs representing (fixing the first column in each pair) the average sign after averaging the output of of bootPairs(mtx) (a n999 by p-1 matrix) each containing resampled ‘sum’ values summarizing the weighted sums associated with all three criteria from the function silentPairs(mtx) applied to each bootstrap sample separately. #'

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. and Lopez-de-Lacalle, J. (2009). 'Maximum entropy bootstrap for time series: The meboot R package.' Journal of Statistical Software, Vol. 29(5), pp. 1-19.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See Also silentPairs, bootQuantile, bootSign.

Examples

## Not run: 
options(np.messages = FALSE)
set.seed(34);x=sample(1:10);y=sample(2:11)
bb=bootPairs(cbind(x,y),n999=29)
bootSignPcent(bb,tau=5) #gives success rate in n999 bootstrap sum computations

bb=bootPairs(airquality,n999=999);options(np.messages=FALSE)
bootSignPcent(bb,tau=5)#success rate for signs from n999 bootstraps

data('EuroCrime');options(np.messages=FALSE)
attach(EuroCrime)
bb=bootPairs(cbind(crim,off),n999=29) #col.1= crim causes off 
#hence positive signs are more intuitively meaningful.
#note that n999=29 is too small for real problems, chosen for quickness here.
bootSignPcent(bb,tau=5)#successful signs from n999 bootstraps

## End(Not run)

Compute usual summary stats of 'sum' indexes from bootPairs output

Description

Begin with the output of bootPairs function, a (n999 by p-1) matrix when there are p columns of data, bootSummary produces a (6 by p-1) mtx of summary of bootstrap ouput (Min, 1st Qu,Median, Mean, 3rd Qi.,Max)

Usage

bootSummary(out, per100 = TRUE)

Arguments

out

output from bootPairs with p-1 columns and n999 rows in input here
per100

logical (default per100=TRUE) to change the range of 'sum' to [-100, 100] values which are easier to interpret

Value

summ summary output from the (n999 by p-1) matrix output of bootPairs(mtx) each containing resampled ‘sum’ values summarizing the weighted sums associated with all three criteria from the function silentPairs(mtx) applied to each bootstrap sample separately.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. and Lopez-de-Lacalle, J. (2009). 'Maximum entropy bootstrap for time series: The meboot R package.' Journal of Statistical Software, Vol. 29(5), pp. 1-19.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See Also silentPairs.

Examples

## Not run: 
options(np.messages = FALSE)
set.seed(34);x=sample(1:10);y=sample(2:11)
bb=bootPairs(cbind(x,y),n999=29)
bootSummary(bb) #gives summary stats for n999 bootstrap sum computations

bb=bootPairs(airquality,n999=999);options(np.messages=FALSE)
bootSummary(bb)#signs for n999 bootstrap sum computations

data('EuroCrime')
attach(EuroCrime)
bb=bootPairs(cbind(crim,off),n999=29) #col.1= crim causes off 
#hence positive signs are more intuitively meaningful.
#note that n999=29 is too small for real problems, chosen for quickness here.
bootSummary(bb)#signs for n999 bootstrap sum computations

## End(Not run)

Compute usual summary stats of 'sum' index in (-100, 100) from bootPair2

Description

The ‘2’ in the name of the function suggests a second implementation where exact stochastic dominance, decileVote and momentVote are used. Begin with the output of bootPairs function, a (n999 by p-1) matrix when there are p columns of data, bootSummary produces a (6 by p-1) mtx of summary of bootstrap ouput (Min, 1st Qu,Median, Mean, 3rd Qi.,Max)

Usage

bootSummary2(out, per100 = TRUE)

Arguments

out

output from bootPair2 with p-1 columns and n999 rows in input here
per100

logical (default per100=TRUE) to change the range of 'sum' to [-100, 100] values which are easier to interpret

Value

summ a summary matrix (n999 by p-1) having usual parameters using the output of bootPair2(mtx) Each containing resampled ‘sum’ values summarizing the weighted sums associated with all three criteria from the function silentPair2(mtx) applied to each bootstrap sample separately.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. and Lopez-de-Lacalle, J. (2009). 'Maximum entropy bootstrap for time series: The meboot R package.' Journal of Statistical Software, Vol. 29(5), pp. 1-19.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See Also silentPairs.

Examples

## Not run: 
options(np.messages = FALSE)
set.seed(34);x=sample(1:10);y=sample(2:11)
bb=bootPair2(cbind(x,y),n999=29)
bootSummary2(bb) #gives summary stats for n999 bootstrap sum computations

bb=bootPair2(airquality,n999=999);options(np.messages=FALSE)
bootSummary2(bb)#signs for n999 bootstrap sum computations

data('EuroCrime')
attach(EuroCrime)
bb=bootPair2(cbind(crim,off),n999=29) #col.1= crim causes off 
#hence positive signs are more intuitively meaningful.
#note that n999=29 is too small for real problems, chosen for quickness here.
bootSummary2(bb)#signs for n999 bootstrap sum computations

## End(Not run)

Generalized canonical correlation, estimating alpha, beta, rho.

Description

What exactly is generalized? Canonical correlations start with Rij, a symmetric matrix of Pearson correlation coefficients based on linear relations. This function starts with a more general non-symmetric R*ij produced by gmcmtx0() as an input. This is a superior measure of dependence, allowing for nonlinear dependencies. It generalizes Hotelling's derivation for the nonlinear case. This function uses data on two sets of column vectors. LHS set [x1, x2 .. xr] has r=nLHS number of columns with coefficients alpha, and the larger RHS set [xr+1, xr+2, .. xp] has nRHS=(p-r) columns and RHS coefficients beta. Must arrange the sets so that the larger set in on RHS with coefficients beta estimated first from an eigenvector of the problem [A* beta = rho^2 beta], where A* is a partitioning of our generalized matrix of (non-symmetric) correlation coefficients.

Usage

canonRho(mtx, nLHS = 2, sgn = 1, verbo = FALSE, ridg = c(0, 0))

Arguments

mtx

Input matrix of generalized correlation coefficients R*
nLHS

number of columns in the LHS set, default=2
sgn

preferred sign of coefficients default=1 for positive, use sgn= -1 if prior knowledge suggests that negative signs of coefficients are more realistic
verbo

logical, verbo=FALSE default means do not print results
ridg

two regularization constants added before computing matrix inverses of S11 and S22, respectively, with default=c(0,0). Some suggest ridg=c(0.01,0.01) for stable results

Value

A

eigenvalue computing matrix for Generalized canonical correlations
rho

Generalized canonical correlation coefficient
bet

RHS coefficient vector
alp

LHS coefficient vector

Note

This function calls kern,

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in 'Handbook of Statistics: Computational Statistics with R', Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Vinod, H. D. 'Canonical ridge and econometrics of joint production,' Journal of Econometrics, vol. 4, 147–166.

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

See Also

See gmcmtx0.

Examples

## Not run: 
set.seed(99)
mtx2=matrix(sample(1:25),nrow=5)
g1=gmcmtx0(mtx2)
canonRho(g1,verbo=TRUE)

## End(Not run)#'

All Pair Version Kernel (block) causality summary paths from three criteria

Description

Allowing input matrix of control variables, this function produces a 5 column matrix summarizing the results where the estimated signs of stochastic dominance order values, (+1, 0, -1), are weighted by wt=c(1.2,1.1, 1.05, 1) to compute an overall result for all orders of stochastic dominance by a weighted sum for the criteria Cr1 and Cr2 and added to the Cr3 estimate as: (+1, 0, -1). The final range for the unanimity of sign index is [–100, 100].

Usage

causeAllPair(
  mtx,
  nam = colnames(mtx),
  blksiz = 10,
  ctrl = 0,
  dig = 6,
  wt = c(1.2, 1.1, 1.05, 1),
  sumwt = 4
)

Arguments

mtx

The data matrix with many columns, We consider causal paths among all possible pairs of mtx columns.
nam

vector of column names for mtx. Default: colnames(mtx)
blksiz

block size, default=10, if chosen blksiz >n, where n=rows in matrix then blksiz=n. That is, no blocking is done
ctrl

data matrix for designated control variable(s) outside causal paths
dig

Number of digits for reporting (default dig=6).
wt

Allows user to choose a vector of four alternative weights for SD1 to SD4.
sumwt

Sum of weights can be changed here =4(default).

Details

The reason for slightly declining weights on the signs from SD1 to SD4 stochastic dominance orders is simply their slightly increasing sampling unreliability due to higher order trapezoidal approximations of integrals of densities involved in definitions of SD1 to SD4. The summary results for all three criteria are reported in one matrix called out:

Value

If there are p columns in the input matrix, x1, x2, .., xp, say, there are choose(p,2) or [p*(p-1)/2] possible pairs and as many causal paths. This function returns a matrix of p*(p-1)/2 rows and 5 columns entitled: “cause", “response", “strength", “corr." and “p-value", respectively with self-explanatory titles. The first two columns have names of variables x1 or x(1+j), depending on which is the cause. The ‘strength’ column has absolute value of summary index in range [0,100] providing summary of causal results based on preponderance of evidence from criteria Cr1 to Cr3 from four orders of stochastic dominance, etc. The fourth column ‘corr.’ reports the Pearson correlation coefficient while the fifth column has the p-value for testing the null of zero Pearson coeff. This function merely calls causeSumNoP repeatedly to include all pairs. The background function siPairsBlk allows for control variables. The output of this function can be sent to ‘xtable’ for a nice Latex table.

Note

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers. Since Cr1 to Cr3 near unanimously suggest ‘crim’ as the cause of ‘off’, strength index 100 suggests unanimity. attach(EuroCrime); causeSummary(cbind(crim,off))

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See bootPairs, causeSummBlk

See someCPairs

siPairsBlk, causeSummary

Examples

## Not run: 
mtx=data.frame(mtcars[,1:3]) #make sure columns of mtx have names
ctrl=data.frame(mtcars[,4:5])
 causeAllPair(mtx=mtx,ctrl=ctrl)

## End(Not run)

options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
causeAllPair(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

Block Version 2: Kernel causality summary of causal paths from three criteria

Description

The ‘2’ in the name of the function suggests a second implementation where exact stochastic dominance, ‘decileVote’ and ‘momentVote’ functions are used, Block version allows a new bandwidth (chosen by the np package) while fitting kernel regressions for each block of data. This may not be appropriate in all situations. Block size is flexible. The function develops a unanimity index regarding which regression flip, (y on xi) or (xi on y) is the best. The “cause” is always on the right-hand side of a regression equation, and the superior flip gives the correct sign. The summary of all signs determines the causal direction and unanimity index among three criteria. This is a block version of causeSummary2(). While allowing the researcher to keep some variables as controls, or outside the scope of causal path determination (e.g., age or latitude) this function produces detailed causal path information in a 5 column matrix identifying the names of variables, causal path directions, path strengths re-scaled to be in the range [–100, 100], (table reports absolute values of the strength) plus Pearson correlation and its p-value.

The algorithm determines causal path directions from the sign of the strength index and strength index values by comparing three aspects of flipped kernel regressions: [x1 on (x2, x3, .. xp)] and its flipped version [x2 on (x1, x3, .. xp)] We compare (i) formal exogeneity test criterion, (ii) absolute residuals, and (iii) R-squares of the flipped regressions implying three criteria Cr1, to Cr3. The criteria are quantified by new methods using four orders of stochastic dominance, SD1 to SD4. See Vinod (2021) two SSRN papers.

Usage

causeSum2Blk(mtx, nam = colnames(mtx), blksiz = 10, ctrl = 0, dig = 6)

Arguments

mtx

The data matrix with many columns, y the first column is a fixed target, and then it is paired with all other columns, one by one, and still called x for flipping.
nam

vector of column names for mtx. Default: colnames(mtx)
blksiz

block size, default=10, if chosen blksiz >n, where n=rows in the matrix then blksiz=n. That is, no blocking is done
ctrl

data matrix for designated control variable(s) outside causal paths
dig

The number of digits for reporting (default dig=6).

Value

If there are p columns in the input matrix, x1, x2, .., xp, say, and if we keep x1 as a common member of all causal-direction-pairs (x1, x(1+j)) for (j=1, 2, .., p-1) which can be flipped. That is, either x1 is the cause or x(1+j) is the cause in a chosen pair. The control variables are not flipped. The printed output of this function reports the results for p-1 pairs indicating which variable (by name) causes which other variable (also by name). It also prints the strength or signed summary strength index in the range [-100,100]. A positive sign of the strength index means x1 kernel causes x(1+j), whereas negative strength index means x(1+j) kernel causes x1. The function also prints Pearson correlation and its p-value. This function also returns a matrix of p-1 rows and 5 columns entitled: “cause", “response", “strength", “corr." and “p-value", respectively with self-explanatory titles. The first two columns have names of variables x1 or x(1+j), depending on which is the cause. The ‘strength’ column has an absolute value of the summary index in the range [0,100], providing a summary of causal results based on the preponderance of evidence from Cr1 to Cr3 from deciles, moments, from four orders of stochastic dominance. The order of input columns in "mtx" matters. The fourth column, ‘corr.’, reports the Pearson correlation coefficient, while the fifth column has the p-value for testing the null of zero Pearson coefficient. This function calls siPairsBlk, allowing for control variables. The output of this function can be sent to ‘xtable’ for a nice Latex table.

Note

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers. If Cr1 to Cr3 near-unanimously suggest ‘crim’ as the cause of ‘off’, strength index would be near 100 suggesting unanimity. attach(EuroCrime); causeSum2Blk(cbind(crim,off))

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

Vinod, Hrishikesh D., R Package GeneralCorr Functions for Portfolio Choice (November 11, 2021). Available at SSRN: https://ssrn.com/abstract=3961683

Vinod, Hrishikesh D., Stochastic Dominance Without Tears (January 26, 2021). Available at SSRN: https://ssrn.com/abstract=3773309

See Also

See bootPairs, causeSummary has an older version of this function.

See someCPairs

siPair2Blk, causeSummary2

Examples

## Not run: 
mtx=as.matrix(mtcars[,1:3])
ctrl=as.matrix(mtcars[,4:5])
causeSum2Blk(mtx,ctrl,nam=colnames(mtx))

## End(Not run)

Kernel regressions based causal paths in Panel Data.

Description

The algorithm of this function uses an internal function fminmax=function(x)min(x)==max(x). The subsets mtx2 of the original data da for a specific time or space can become degenerate if the columns of mtx2 have no variability. The apply function of R is applied to the columns of mtx2 as follows. "ap1=apply(mtx2,2,fminmax)." Now, "sumap1=sum(ap1)" counts how many columns of the data matrix are degenerate. We have a degeneracy problem only if sumap1 is >1 or =1. For example, the panel consists of data on 50 United States and 20 years. Now, consumer price index (cpi) data may be common for all states. That is, the min(cpi) equals max(cpi) for all states. Then, the variance of cpi is zero, and we have degeneracy. When this happens, the regressor cpi should not be involved in determining causal paths. We identify degeneracy using "fminmax=function(x)min(x)==max(x)"

Usage

causeSum2Panel(
  da,
  fn = causeSummary2NoP,
  rowfnout,
  colfnout,
  fnoutNames,
  namXs,
  namXt,
  namXy,
  namXc = 0,
  namXjmtx,
  chosenTimes = NULL,
  chosenSpaces = NULL,
  ylag = 0,
  verbo = FALSE
)

Arguments

da

panel dat having a named column for space and time
fn

an R function causeSummary2NoP(mtx)
rowfnout

the number of rows output by fn
colfnout

the number of columns output by fn
fnoutNames

the column names of output by fn, for example, fnoutNames=c("cause","effect","strength","r","p-val")
namXs

title of the column in da having the space variable
namXt

title of the column in da having the time variable
namXy

title of the column in da having the dependent y variable
namXc

title(s) of the column(s) in da having control variable(s), default=0 means none specified
namXjmtx

title(s) of the column(s) in da having regressor(s)
chosenTimes

subset of values of time variable chosen for quick results, There are NchosenTimes values chosen in the subset. default=NULL means all time identifiers in the data are included.
chosenSpaces

subset of values of space variable chosen for quick results, There are NchosenSpaces values chosen in the subset. default=NULL means all space identifiers are included. The degrees of freedom for Studentized statistic for Granger causality tests are df=(NchosenSpaces -1).
ylag

time lag in Granger causality study of time dimension the default ylag=0 is not really zero. It means ylag= min(4, round(NchosenTimes/5,0)), where NchosenTimes is the length of chosenTimes vector
verbo

print detail results along the way, default=FALSE

Details

We assume that panel data have space (space=individual region) and time (e.g., year) dimensions. We use upper case X to denote a common prefix in the panel data. Xs =name of the space variable, e.g., state or individual. The range of values for s is 1 to nspace. Xt =name of the time variable, e.g., year. The range of values for t is 1 to ntime. Xy =the dependent variable(s) value at time t in state s. Since panel data causal analysis can take a long computer time, we allow the user to choose subsets of time and space values called chosenTimes and chosenSpaces, respectively. Various input parameters starting with "nam" specify the names of variables in the panel study.

The algorithm calls some function fn(mtx) where mtx is the data matrix, and fn is causeSummary2NoP(mtx). The causal paths between (y, xj) pairs of variables in mtx are computed following 3 sophisticated criteria involving exact stochastic dominance. Type "?causeSummary2" on the R console to get details (omitted here for brevity). Panel data consist of a time series of cross-sections and are also called longitudinal data. We provide estimates of causal path directions and strengths for both the time-series and cross-sectional views of panel data. Since our regressions are kernel type with no functional forms, fixed effects for time and space are being suppressed when computing the causality.

Value

The causeSum2Panel(.) produces many output matrices and vectors. The first "outt" gives a 3-dimensional array of panel causal path output focused on time series for each space value using fixed space value. It reports causal path directions, and strengths for (y, xj) pairs. The second output array, called "outs", gives similar 3D panel causal path output focused on space cross sections using fixed time value. The third output matrix called "outdif" gives causal paths using Granger causality for each pair (y, xj). They are not causal strengths but differences between Rsquare values of two flipped kernel regressions. The summary of Granger causality answer is an output matrix called grangerAns (first row average of differences in R-squares and second row has its test statistic with degrees of freedom n-1), and grangerStat for related t-statistic for formal inference. based on column means and variances of "outdif". This function also produces a matrix summarizing "outt" and "outs" into two-dimensional matrices reporting averages of signed strengths as "strentime" and "strenspace", Also, "pearsontime" reports the Pearson correlation coefficients for various time values and their average in the last column. It determines the overall direction of the causal relation between y and xj. For example, a negative average correlation means y and xj are negatively correlated (xj goes up, y goes down). Similarly, "pearsonspace" summarizes "outs" correlations.

Note

The function prints to the screen some summaries of the three output matrices. It reports how often a variable is a cause in various pairs as time series or as cross sections. It also reports the average strengths of causal paths for "outt" and "outs" matrices. We compute the difference between two R-square values to find which causal direction is more plausible. This involves kernel regressions of y on its own lags and lags of a regressor. Unlike the usual Granger causality we estimate better-fitting nonlinear kernel regressions. If the averages in "outdif" matrix are negative, the Granger causal paths go from y to xj. This may be unexpected when the model assumes that y depends on x1 to xp, that is, the causal paths go from xj to y. In studying the causal pairs, the function creates mixtures of names y and xj. Character vectors containing the mixed names are are column names or row names depending on the context. For example, the output matrix grangerAns column names help identify the relevant regressor name. The first row of the grangerAns matrix has column averages of outdiff matrix to help get an overall estimate of the Granger-causal paths. The second row of the grangerAns has the Studentized test statistic for formal testing of the significance of Granger causal paths. Collecting the results for the time dimension strengths with suitable sign (negative strength means cause reversal xj->y) is output named strentime. The corresponding Pearson correlations as an output is named pearsontime. Collecting the results for the space dimension strengths with suitable sign (negative strength means cause reversal xj->y) is output named strenspace. The corresponding Pearson correlations are named pearsonspace. A grand summary of average strengths and correlations is output matrix named grandsum. It is intended to provide an overall picture of causal paths in Panel data. These paths should not be confused with Granger causal paths which always involve time lags and causes are presumed to precede effects in time.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

Vinod, Hrishikesh D., R Package GeneralCorr Functions for Portfolio Choice (November 11, 2021). Available at SSRN: https://ssrn.com/abstract=3961683

Vinod, Hrishikesh D., Stochastic Dominance Without Tears (January 26, 2021). Available at SSRN: https://ssrn.com/abstract=3773309

See Also

See causeSummary2

See causeSummary is subject to trapezoidal approximation.

Examples

## Not run: 
library(plm);data(Grunfeld)
options(np.messages=FALSE)
namXs="firm"
print("initial values identifying the space variable")
head(da[,namXs],3)
print(str(da[,namXs]))
chosenSpaces=(3:10)                        
if(is.numeric(da[,namXs])){
  chosenSpaces=as.numeric(chosenSpaces)}
if(!is.numeric(da[,namXs])){
  chosenSpaces=as.character(chosenSpaces)}

namXt="year"
print("initial values identifying the time variable")
head(da[,namXt],3)
print(str(da[,namXt]))
chosenTimes=1940:1949
if(is.numeric(da[,namXt])){
  chosenTimes=as.numeric(chosenTimes)}
if(!is.numeric(da[,namXt])){
  chosenTimes=as.character(chosenTimes)}

namXy="inv"
namXc=0
namXjmtx=c("value","capital")
p=length(namXjmtx)
fn=causeSummary2NoP
fnout=matrix(NA,nrow=p,ncol=5)
fnoutNames=c("cause","effect","strength","r","p-val")
causeSum2Panel(da, fn=causeSummary2NoP,
               rowfnout=p, colfnout=5, 
               fnoutNames=c("cause","effect","strength","r","p-val"),
               namXs=namXs,
               namXt=namXt,
               namXy=namXy,
               namXc=namXc,
               namXjmtx=namXjmtx,
               chosenTimes=chosenTimes,
               chosenSpaces=chosenSpaces,
               verbo=FALSE)

## End(Not run)

Kernel causality summary of evidence for causal paths from three criteria

Description

While allowing the researcher to keep some variables as controls, or outside the scope of causal path determination (e.g., age or latitude) this function produces detailed causal path information in a 5 column matrix identifying the names of variables, causal path directions, path strengths re-scaled to be in the range [–100, 100], (table reports absolute values of the strength) plus Pearson correlation and its p-value.

Usage

causeSummary(
  mtx,
  nam = colnames(mtx),
  ctrl = 0,
  dig = 6,
  wt = c(1.2, 1.1, 1.05, 1),
  sumwt = 4
)

Arguments

mtx

The data matrix with many columns, y the first column is fixed and then paired with all columns, one by one, and still called x for the purpose of flipping.
nam

vector of column names for mtx. Default: colnames(mtx)
ctrl

data matrix for designated control variable(s) outside causal paths
dig

Number of digits for reporting (default dig=6).
wt

Allows user to choose a vector of four alternative weights for SD1 to SD4.
sumwt

Sum of weights can be changed here =4(default).

Details

The algorithm determines causal path directions from the sign of the strength index and strength index values by comparing three aspects of flipped kernel regressions: [x1 on (x2, x3, .. xp)] and its flipped version [x2 on (x1, x3, .. xp)] We compare (i) formal exogeneity test criterion, (ii) absolute residuals, and (iii) R-squares of the flipped regressions implying three criteria Cr1, to Cr3. The criteria are quantified by sophisticated methods using four orders of stochastic dominance, SD1 to SD4. We assume slightly declining weights on causal path signs because known reliability ranking. SD1 is better than SD2, better than SD3, better than SD4. The user can optionally change our weights.

Value

If there are p columns in the input matrix, x1, x2, .., xp, say, and if we keep x1 as a common member of all causal direction pairs (x1, x(1+j)) for (j=1, 2, .., p-1) which can be flipped. That is, either x1 is the cause or x(1+j) is the cause in a chosen pair. The control variables are not flipped. The printed output of this function reports the results for p-1 pairs indicating which variable (by name) causes which another variable (also by name). It also prints a signed summary strength index in the range [-100,100]. A positive sign of the strength index means x1 kernel causes x(1+j), whereas negative strength index means x(1+j) kernel causes x1. The function also prints Pearson correlation and its p-value. In short, function returns a matrix of p-1 rows and 5 columns entitled: “cause", “response", “strength", “corr." and “p-value", respectively with self-explanatory titles. The first two columns have names of variables x1 or x(1+j), depending on which is the cause. The ‘strength’ column reports the absolute value of summary index, now in the range [0,100] providing summary of causal results based on preponderance of evidence from Cr1 to Cr3 from four orders of stochastic dominance, etc. The order of input columns matters. The fourth column ‘corr.’ reports the Pearson correlation coefficient while the fifth column has the p-value for testing the null of zero Pearson coeff. This function calls silentPairs allowing for control variables. The output of this function can be sent to ‘xtable’ for a nice Latex table.

Note

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers. Since Cr1 to Cr3 near unanimously suggest ‘crim’ as the cause of ‘off’, strength index 100 suggests unanimity. In portfolio applications of stochastic dominance one wants higher returns. Here we are comparing two probability distributions of absolute residuals for two flipped models. We choose that flip which has smaller absolute residuals or better fit. attach(EuroCrime); causeSummary(cbind(crim,off))

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See bootPairs, causeSummary0 has an older version of this function.

See someCPairs

silentPairs

Examples

## Not run: 
mtx=as.matrix(mtcars[,1:3])
ctrl=as.matrix(mtcars[,4:5])
 causeSummary(mtx,ctrl,nam=colnames(mtx))

## End(Not run)

options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
causeSummary(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

Older Kernel causality summary of evidence for causal paths from three criteria

Description

Allowing input matrix of control variables, this function produces a 5 column matrix summarizing the results where the estimated signs of stochastic dominance order values, (+1, 0, -1), are weighted by wt=c(1.2,1.1, 1.05, 1) to compute an overall result for all orders of stochastic dominance by a weighted sum for the criteria Cr1 and Cr2 and added to the Cr3 estimate as: (+1, 0, -1). The final range for the unanimity of sign index is [–100, 100].

Usage

causeSummary0(
  mtx,
  nam = colnames(mtx),
  ctrl = 0,
  dig = 6,
  wt = c(1.2, 1.1, 1.05, 1),
  sumwt = 4
)

Arguments

mtx

The data matrix with many columns, y the first column is fixed and then paired with all columns, one by one, and still called x for the purpose of flipping.
nam

vector of column names for mtx. Default: colnames(mtx)
ctrl

data matrix for designated control variable(s) outside causal paths
dig

Number of digits for reporting (default dig=6).
wt

Allows user to choose a vector of four alternative weights for SD1 to SD4.
sumwt

Sum of weights can be changed here =4(default).

Details

The reason for slightly declining weights on the signs from SD1 to SD4 is simply that the local mean comparisons implicit in SD1 are known to be more reliable than local variance implicit in SD2, local skewness implicit in SD3 and local kurtosis implicit in SD4. The reason for slightly declining sampling unreliability of higher moments is simply that SD4 involves fourth power of the deviations from the mean and SD3 involves 3rd power, etc. The summary results for all three criteria are reported in one matrix called out:

Value

If there are p columns in the input matrix, x1, x2, .., xp, say, and if we keep x1 as a common member of all causal direction pairs (x1, x(1+j)) for (j=1, 2, .., p-1) which can be flipped. That is, either x1 is the cause or x(1+j) is the cause in a chosen pair. The control variables are not flipped. The printed output of this function reports the results for p-1 pairs indicating which variable (by name) causes which other variable (also by name). It also prints strength or signed summary strength index in range [-100,100]. A positive sign of the strength index means x1 kernel causes x(1+j), whereas negative strength index means x(1+j) kernel causes x1. The function also prints Pearson correlation and its p-value. This function also returns a matrix of p-1 rows and 5 columns entitled: “cause", “response", “strength", “corr." and “p-value", respectively with self-explanatory titles. The first two columns have names of variables x1 or x(1+j), depending on which is the cause. The ‘strength’ column has absolute value of summary index in range [0,100] providing summary of causal results based on preponderance of evidence from Cr1 to Cr3 from four orders of stochastic dominance, etc. The order of input columns matters. The fourth column ‘corr.’ reports the Pearson correlation coefficient while the fifth column has the p-value for testing the null of zero Pearson coeff. This function calls silentPairs0 (the older version) allowing for control variables. The output of this function can be sent to ‘xtable’ for a nice Latex table.

Note

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers. Since Cr1 to Cr3 near unanimously suggest ‘crim’ as the cause of ‘off’, strength index 100 suggests unanimity. attach(EuroCrime); causeSummary0(cbind(crim,off)). Both versions give identical result for this example. Old version of Cr1 using gradients was also motivated by the same Hausman-Wu test statistic.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See bootPairs

See someCPairs

silentPairs

Examples

## Not run: 
mtx=as.matrix(mtcars[,1:3])
ctrl=as.matrix(mtcars[,4:5])
 causeSummary0(mtx,ctrl,nam=colnames(mtx))

## End(Not run)

options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
causeSummary0(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

Kernel causality summary of evidence for causal paths from three criteria using new exact stochastic dominance. The function develops a unanimity index for deciding which flip (y on xi) or (xi on y) is best. Relevant signs determine the causal direction and unanimity index among three criteria. While allowing the researcher to keep some variables as controls, or outside the scope of causal path determination (e.g., age or latitude) this function produces detailed causal path information in a 5 column matrix identifying the names of variables, causal path directions, path strengths re-scaled to be in the range [–100, 100], (table reports absolute values of the strength) plus Pearson correlation and its p-value. The ‘2’ in the name of the function suggests a second implementation where exact stochastic dominance, decileVote, and momentVote are used and where we avoid Anderson's trapezoidal approximation.

Description

The algorithm determines causal path directions from the sign of the strength index and strength index values by comparing three aspects of flipped kernel regressions: [x1 on f(x2, x3, .. xp)] and its flipped version [x2 on f(x1, x3, .. xp)] We compare (i) formal exogeneity test criterion, (ii) absolute residuals, and (iii) R-squares of the flipped regressions implying three criteria Cr1, to Cr3. The criteria are quantified by newer exact methods using four orders of stochastic dominance, SD1 to SD4. See Vinod's (2021) SSRN papers. In portfolio applications of stochastic dominance, one wants higher values. Here, we are comparing two probability distributions of absolute residuals for two flipped models. We choose that flip, which has smaller absolute residuals that will have a better fit.

Usage

causeSummary2(mtx, nam = colnames(mtx), ctrl = 0, dig = 6)

Arguments

mtx

The data matrix with many columns, y the first column is fixed and then paired with all columns, one by one, and still called x for the purpose of flipping.
nam

vector of column names for mtx. Default: colnames(mtx)
ctrl

data matrix for designated control variable(s) outside causal paths
dig

Number of digits for reporting (default dig=6).

Value

If there are p columns in the input matrix, x1, x2, .., xp, say, and if we keep x1 as a common member of all causal direction pairs (x1, x(1+j)) for (j=1, 2, .., p-1) which can be flipped. That is, either x1 is the cause or x(1+j) is the cause in a chosen pair. The control variables are not flipped. The printed output of this function reports the results for p-1 pairs indicating which variable (by name) causes which other variable (also by name). It also prints a signed summary strength index in the range [-100,100]. A positive sign of the strength index means x1 kernel causes x(1+j), whereas a negative strength index means x(1+j) kernel causes x1. The function also prints the Pearson correlation and its p-value. In short, function returns a matrix of p-1 rows and 5 columns entitled: “cause", “response", “strength", “corr." and “p-value", respectively with self-explanatory titles. The first two columns have names of variables x1 or x(1+j), depending on which is the cause. The ‘strength’ column reports the absolute value of the summary index, in the range [0,100], providing a summary of causal results based on the preponderance of evidence from Cr1 to Cr3 from four orders of stochastic dominance, moments, deciles, etc. The order of input columns in mtx matters. The fourth column, ‘corr.’ of ‘out’, reports the Pearson correlation coefficient. The fifth column has the p-value for testing the null of zero Pearson coeff. This function calls silentPair2, allowing for control variables. The output of this function can be sent to ‘xtable’ for a nice Latex table.

Note

The European Crime data has all three criteria correctly suggesting that a high crime rate kernel causes the deployment of a large number of police officers. Since Cr1 to Cr3 nearly unanimously suggest ‘crim’ as the cause of ‘off’, strength index 100 suggests unanimity among the criteria. attach(EuroCrime); causeSummary(cbind(crim,off))

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

Vinod, Hrishikesh D., R Package GeneralCorr Functions for Portfolio Choice (November 11, 2021). Available at SSRN: https://ssrn.com/abstract=3961683

Vinod, Hrishikesh D., Stochastic Dominance Without Tears (January 26, 2021). Available at SSRN: https://ssrn.com/abstract=3773309

See Also

See siPair2Blk for a block version

See causeSummary is subject to trapezoidal approximation.

see silentPair2 called by this function.

Examples

## Not run: 
mtx=as.matrix(mtcars[,1:3])
ctrl=as.matrix(mtcars[,4:5])
 causeSummary2(mtx,ctrl,nam=colnames(mtx))

## End(Not run)

options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
causeSummary2(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

No Print version Kernel causality summary of evidence for causal paths from three criteria using new exact stochastic dominance.

Description

The function develops a unanimity index for deciding which flip (y on xi) or (xi on y) is best. Relevant signs determine the causal direction and unanimity index among three criteria. While allowing the researcher to keep some variables as controls, or outside the scope of causal path determination (e.g., age or latitude) this function produces detailed causal path information in a 5 column matrix identifying the names of variables, causal path directions, path strengths re-scaled to be in the range [–100, 100], (table reports absolute values of the strength) plus Pearson correlation and its p-value. The ‘2’ in the name of the function suggests a second implementation where exact stochastic dominance, decileVote, and momentVote are used and where we avoid Anderson's trapezoidal approximation.

Usage

causeSummary2NoP(mtx, nam = colnames(mtx), ctrl = 0, dig = 6)

Arguments

mtx

The data matrix with many columns, y the first column is fixed and then paired with all columns, one by one, and still called x for the purpose of flipping.
nam

vector of column names for mtx. Default: colnames(mtx)
ctrl

data matrix for designated control variable(s) outside causal paths
dig

Number of digits for reporting (default dig=6).

Details

The algorithm determines causal path directions from the sign of the strength index and strength index values by comparing three aspects of flipped kernel regressions: [x1 on f(x2, x3, .. xp)] and its flipped version [x2 on f(x1, x3, .. xp)] We compare (i) formal exogeneity test criterion, (ii) absolute residuals, and (iii) R-squares of the flipped regressions implying three criteria Cr1, to Cr3. The criteria are quantified by newer exact methods using four orders of stochastic dominance, SD1 to SD4. See Vinod's (2021) SSRN papers. In portfolio applications of stochastic dominance, one wants higher values. Here, we are comparing two probability distributions of absolute residuals for two flipped models. We choose that flip, which has smaller absolute residuals that will have a better fit.

Value

If there are p columns in the input matrix, x1, x2, .., xp, say, and if we keep x1 as a common member of all causal direction pairs (x1, x(1+j)) for (j=1, 2, .., p-1) which can be flipped. That is, either x1 is the cause or x(1+j) is the cause in a chosen pair. The control variables are not flipped. The printed output of this function reports the results for p-1 pairs indicating which variable (by name) causes which other variable (also by name). It also prints a signed summary strength index in the range [-100,100]. A positive sign of the strength index means x1 kernel causes x(1+j), whereas a negative strength index means x(1+j) kernel causes x1. The function also prints the Pearson correlation and its p-value. In short, function returns a matrix of p-1 rows and 5 columns entitled: “cause", “response", “strength", “corr." and “p-value", respectively with self-explanatory titles. The first two columns have names of variables x1 or x(1+j), depending on which is the cause. The ‘strength’ column reports the absolute value of the summary index, in the range [0,100], providing a summary of causal results based on the preponderance of evidence from Cr1 to Cr3 from four orders of stochastic dominance, moments, deciles, etc. The order of input columns in mtx matters. The fourth column, ‘corr.’ of ‘out’, reports the Pearson correlation coefficient. The fifth column has the p-value for testing the null of zero Pearson coeff. This function calls silentPair2, allowing for control variables. The output of this function can be sent to ‘xtable’ for a nice Latex table.

Note

The European Crime data has all three criteria correctly suggesting that a high crime rate kernel causes the deployment of a large number of police officers. Since Cr1 to Cr3 nearly unanimously suggest ‘crim’ as the cause of ‘off’, strength index 100 suggests unanimity among the criteria. attach(EuroCrime); causeSummary(cbind(crim,off))

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

Vinod, Hrishikesh D., R Package GeneralCorr Functions for Portfolio Choice (November 11, 2021). Available at SSRN: https://ssrn.com/abstract=3961683

Vinod, Hrishikesh D., Stochastic Dominance Without Tears (January 26, 2021). Available at SSRN: https://ssrn.com/abstract=3773309

See Also

See siPair2Blk for a block version

See causeSummary is subject to trapezoidal approximation.

see silentPair2 called by this function.

Examples

## Not run: 
mtx=as.matrix(mtcars[,1:3])
ctrl=as.matrix(mtcars[,4:5])
 causeSummary2(mtx,ctrl,nam=colnames(mtx))

## End(Not run)

options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
causeSummary2(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

Block Version 2: Kernel causality summary of causal paths from three criteria

Description

A block version of causeSummary() chooses new bandwidth for every ten (blksiz=10) observations chosen by the ‘np’ package injecting flexibility. While allowing the researcher to keep some variables as controls, or outside the scope of causal path determination (e.g., age or latitude), this function produces detailed causal path information. The output table is a 5-column matrix identifying the names of variables, causal path directions, and path strengths re-scaled to be in the range [–100, 100], (table reports absolute values of the strength) plus Pearson correlation coefficient and its p-value.

Usage

causeSummBlk(
  mtx,
  nam = colnames(mtx),
  blksiz = 10,
  ctrl = 0,
  dig = 6,
  wt = c(1.2, 1.1, 1.05, 1),
  sumwt = 4
)

Arguments

mtx

The data matrix with many columns, y the first column is a fixed target, and then it is paired with all other columns, one by one, and still called x for the purpose of flipping.
nam

vector of column names for mtx. Default: colnames(mtx)
blksiz

block size, default=10, if chosen blksiz >n, where n=rows in the matrix then blksiz=n. That is, no blocking is done
ctrl

data matrix for designated control variable(s) outside causal paths
dig

The number of digits for reporting (default dig=6).
wt

Allows user to choose a vector of four alternative weights for SD1 to SD4.
sumwt

Sum of weights can be changed here =4(default).

Details

The algorithm determines causal path directions from the sign of the strength index. The strength index magnitudes are computed by comparing three aspects of flipped kernel regressions: [x1 on (x2, x3, .. xp)] and its flipped version [x2 on (x1, x3, .. xp)]. The cause should be on the right-hand side of the regression equation. The properties of regression fit determine which flip is superior. We compare (Cr1) formal exogeneity test criterion, (residuals times RHS regressor, where smaller in absolute value is better) (Cr2) absolute values of residuals, where smaller in absolute value is better, and (Cr3) R-squares of the flipped regressions implying three criteria Cr1, to Cr3. The criteria are quantified by sophisticated methods using four orders of stochastic dominance, SD1 to SD4. We assume slightly declining weights on the sign observed by Cr1 to Cr3. The user can change default weights.

Value

If there are p columns in the input matrix, x1, x2, .., xp, say, and if we keep x1 as a common member of all causal-direction-pairs (x1, x(1+j)) for (j=1, 2, .., p-1) which can be flipped. That is, either x1 is the cause or x(1+j) is the cause in a chosen pair. The control variables are not flipped. The printed output of this function reports the results for p-1 pairs indicating which variable (by name) causes which other variable (also by name). It also prints a strength, or signed summary strength index forced to be in the range [-100,100] for easy interpretation. A positive sign of the strength index means x1 kernel causes x(1+j), whereas negative strength index means x(1+j) kernel causes x1. The function also prints Pearson correlation and its p-value. This function also returns a matrix of p-1 rows and 5 columns entitled: “cause", “response", “strength", “corr." and “p-value", respectively with self-explanatory titles. The first two columns have names of variables x1 or x(1+j), depending on which is the cause. The ‘strength’ column has the absolute value of a summary index in the range [0,100], providing a summary of causal results based on the preponderance of evidence from Cr1 to Cr3 from four orders of stochastic dominance, etc. The order of input columns matters. The fourth column of the output matrix entitled ‘corr.’ reports the Pearson correlation coefficient, while the fifth column of the output matrix has the p-value for testing the null hypothesis of a zero Pearson coefficient. This function calls siPairsBlk, allowing for control variables. The output of this function can be sent to ‘xtable’ for a nice Latex table.

Note

The European Crime data has all three criteria correctly suggesting that a high crime rate kernel causes the deployment of a large number of police officers. Since Cr1 to Cr3 near-unanimously suggest ‘crim’ as the cause of ‘off’, a strength index of 100 suggests unanimity. attach(EuroCrime); causeSummBlk(cbind(crim,off))

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See bootPairs, causeSummary has an older version of this function.

See someCPairs

siPairsBlk, causeSummary

Examples

## Not run: 
mtx=as.matrix(mtcars[,1:3])
ctrl=as.matrix(mtcars[,4:5])
 causeSummBlk(mtx,ctrl,nam=colnames(mtx))

## End(Not run)

options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
causeSummBlk(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

No print (NoP) version of causeSummBlk summary causal paths from three criteria

Description

Allowing input matrix of control variables, this function produces a 5 column matrix summarizing the results where the estimated signs of stochastic dominance order values, (+1, 0, -1), are weighted by wt=c(1.2,1.1, 1.05, 1) to compute an overall result for all orders of stochastic dominance by a weighted sum for the criteria Cr1 and Cr2 and added to the Cr3 estimate as: (+1, 0, -1). The final range for the unanimity of sign index is [–100, 100].

Usage

causeSumNoP(
  mtx,
  nam = colnames(mtx),
  blksiz = 10,
  ctrl = 0,
  dig = 6,
  wt = c(1.2, 1.1, 1.05, 1),
  sumwt = 4
)

Arguments

mtx

The data matrix with many columns, y the first column is a fixed target and then it is paired with all other columns, one by one, and still called x for the purpose of flipping.
nam

vector of column names for mtx. Default: colnames(mtx)
blksiz

block size, default=10, if chosen blksiz >n, where n=rows in matrix then blksiz=n. That is, no blocking is done
ctrl

data matrix for designated control variable(s) outside causal paths
dig

Number of digits for reporting (default dig=6).
wt

Allows user to choose a vector of four alternative weights for SD1 to SD4.
sumwt

Sum of weights can be changed here =4(default).

Details

The reason for slightly declining weights on the signs from SD1 to SD4 is simply that the higher order stochastic dominance numbers are less reliable. The summary results for all three criteria are reported in one matrix called out but not printed:

Value

If there are p columns in the input matrix, x1, x2, .., xp, say, and if we keep x1 as a common member of all causal-direction-pairs (x1, x(1+j)) for (j=1, 2, .., p-1) which can be flipped. That is, either x1 is the cause or x(1+j) is the cause in a chosen pair. The control variables are not flipped. The printed output of this function reports the results for p-1 pairs indicating which variable (by name) causes which other variable (also by name). It also prints strength or signed summary strength index in range [-100,100]. A positive sign of the strength index means x1 kernel causes x(1+j), whereas negative strength index means x(1+j) kernel causes x1. The function also prints Pearson correlation and its p-value. This function also returns a matrix of p-1 rows and 5 columns entitled: “cause", “response", “strength", “corr." and “p-value", respectively with self-explanatory titles. The first two columns have names of variables x1 or x(1+j), depending on which is the cause. The ‘strength’ column has absolute value of summary index in range [0,100] providing summary of causal results based on preponderance of evidence from Cr1 to Cr3 from four orders of stochastic dominance, etc. The order of input columns matters. The fourth column ‘corr.’ reports the Pearson correlation coefficient while the fifth column has the p-value for testing the null of zero Pearson coeff. This function calls siPairsBlk allowing for control variables. The output of this function can be sent to ‘xtable’ for a nice Latex table.

Note

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers. Since Cr1 to Cr3 near unanimously suggest ‘crim’ as the cause of ‘off’, strength index 100 suggests unanimity. attach(EuroCrime); causeSummary(cbind(crim,off))

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See bootPairs, causeSummary0 has an older version of this function.

See causeAllPair

siPairsBlk, causeSummary

Examples

## Not run: 
mtx=data.frame(mtcars[,1:3])
ctrl=data.frame(mtcars[,4:5])
 causeSumNoP(mtx=mtx,ctrl=ctrl)

## End(Not run)

options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
causeSumNoP(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

Compute cofactor of a matrix based on row r and column c.

Description

Compute cofactor of a matrix based on row r and column c.

Usage

cofactor(x, r, c)

Arguments

x

matrix whose cofactor is desired to be computed
r

row number
c

column number

Value

cofactor of x, w.r.t. row r and column c.

Note

needs the function 'minor” in memory. attaches sign (-1)^(r+c) to the minor.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

See Also

minor(x,r,c)

Examples

## The function is currently defined as
function (x, r, c) 
{
    out = minor(x, r, c) * ((-1)^(r + c))
    return(out)
  }

Compares two vectors (portfolios) using stochastic dominance of orders 1 to 4.

Description

Given two vectors of portfolio returns this function calls the internal function wtdpapb to report the simple means of four sophisticated measures of stochastic dominance. as explained in Vinod (2008).

Usage

comp_portfo2(xa, xb)

Arguments

xa

Data on returns for portfolio A in the form of a T by 1 vector
xb

Data on returns for portfolio B in the form of a T by 1 vector

Value

Returns four numbers which are averages of four sophisticated measures of stochastic dominance measurements called SD1 to SD4.

Note

It is possible to modify this function to report the median or standard deviation or any other descriptive statistic by changing the line in the code 'oumean = apply(outb, 2, mean)' toward the end of this function. A trimmed mean may be of interest when outliers are suspected.

require(np)

Make sure that functions wtdpapb, bigfp, stochdom2 are in the memory. and options(np.messages=FALSE)

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D.", "Hands-On Intermediate Econometrics Using R" (2008) World Scientific Publishers: Hackensack, NJ. (Chapter 4) https://www.worldscientific.com/worldscibooks/10.1142/12831

See Also

stochdom2

Examples

set.seed(30)
xa=sample(20:30)#generally lower returns
xb=sample(32:40)# higher returns in xb
gp = comp_portfo2(xa, xb)#all Av(sdi) positive means xb dominates
##positive SD1 to SD4 means xb dominates xa as it should

Compares two vectors (portfolios) using momentVote, DecileVote and exactSdMtx functions.

Description

Given two vectors of portfolio returns this function summarizes their ranks based on moments, deciles and exact measures of stochastic dominance. as explained in Vinod (2021). This algorithm has model selection applications.

Usage

compPortfo(xa, xb)

Arguments

xa

Data on returns for portfolio A in the form of a T by 1 vector
xb

Data on returns for portfolio B in the form of a T by 1 vector

Value

Returns three numbers which represent signs based differences in ranks (rank=1 for most desirable) measured by [rank(xa)-rank(xb)] using momentVote, decileVote, and exactSdMtx which are weighted averages of four moments, nine deciles and exact measures of stochastic dominance (from ECDFs of four orders, SD1 to SD4) respectively.

Note

There are model-selection applications where two models A and B are compared and one wants to choose the model smaller absolute value of residuals. This function when applied for model-selection will have he inputs xa and xb as absolute residuals. We can compare the entire probability distributions of absolute residuals by moments, deciles or SD1 to SD4. Of course, care must be taken to choose xa or xb depending on which model has smaller absolute residuals. This choice is the exact opposite of portfolio choice application where larger return is more desirable. silentPair2() and siPair2Blk call this function for model selection application.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D.", "Hands-On Intermediate Econometrics Using R" (2008) World Scientific Publishers: Hackensack, NJ. (Chapter 4) https://www.worldscientific.com/worldscibooks/10.1142/12831

Vinod, Hrishikesh D., R Package GeneralCorr Functions for Portfolio Choice (November 11, 2021). Available at SSRN: https://ssrn.com/abstract=3961683

See Also

exactSdMtx

momentVote

decileVote

Examples

set.seed(30)
xa=sample(20:30)#generally lower returns
xb=sample(32:40)# higher returns in xb
gp = compPortfo(xa, xb)#all Av(sdi) positive means xb dominates
##output (1,1,1) means xb dominates xa. xb are larger by consruction

internal da

Description

intended for internal use only

Usage

da

internal da2Lag

Description

intended for internal use

Usage

data(da2Lag)

Format

The format is: int 4

Function compares nine deciles of stock return distributions.

Description

The first step computes a minimum reference return and nine deciles. The input x must be a matrix having p columns (with a name for each column) and n rows as in the data. If data are missing for some columns, insert NA's. Thus x has p column of the data matrix ready for comparison and ranking. For example, x has a matrix of stock returns. The output matrix produced by this function also has p columns for each column (i.e., for each stock being compared). The output matrix has nineteen rows. The top nine rows have the magnitudes of deciles. Rows 10 to 18 have respective ranks of the decile magnitudes. The next (19-th) row of the output reports a weighted sum of ranks. Ranking always gives the smallest number 1 to the most desirable outcome. We suggest that a higher portfolio weight be given to the column having smallest rank value (along the 19th line). The 20-th row further ranks the weighted sums of ranks in row 19. Investor should choose the stock (column) representing the smallest rank value along the last (20th) row of the ‘out’ matrix.

Usage

decileVote(mtx, howManySd = 0.1)

Arguments

mtx

(n X p) matrix of data. For example, returns on p stocks n months
howManySd

used to define ‘fixmin’= imaginary lowest return defined by going howManySd=default=0.1 maximum of standard deviations of all stocks below the minimum return for all stocks in the data

Value

out is a matrix with p columns (same as in the input matrix) and twenty rows. Top nine rows have 9 deciles, next nine rows have their ranks. The 19-th row of ‘out’ has a weighted sum of 9 ranks. All columns refer to one stock. The weighted sum for each stock is then ranked. A portfolio manager is assumed to prefer higher return represented by high decile values represented by the column with the largest weighted sum. can give largest weight to the column with the smallest bottom line. The bottom line (20-th) labeled “choice" of the ‘out’ matrix is defined so that choice =1 suggests the stock deserving the highest weight in the portfolio. The portfolio manager will generally give the lowest weight (=0?) to the stock representing column having number p as the choice number. The manager may want to sell this stock. Another output of the ‘decileVote’ function is ‘fixmin’ representing the smallest possible return of all the stocks in the input ‘mtx’ of returns. It is useful as a reference stock. We compute stochastic dominance numbers for each stock with this imaginary stock yielding fixmin return for all time periods.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

x1=c(1,4,7,2,6)
x2=c(3,4,8,4,7)
decileVote(cbind(x1,x2))

depMeas Signed measure of nonlinear nonparametric dependence between two vectors.

Description

An infant may depend on the mother for survival, but not vice versa. Dependence relations need not be symmetric, yet correlation coefficients are symmetric. One way to measure the extent of dependence is to find the max of the absolute values of the two asymmetric correlations using Vinod's (2015) definition of generalized (asymmetric) correlation coefficients. It requires a kernel regression of x on y obtained by using the ‘np’ package and its flipped version (regress y on x). We use a block version of ‘gmcmtx0’ called 'gmcmtxBlk' to admit several bandwidths for every ten observations if the user sets blksiz=10, a recommended choice here.

Usage

depMeas(x, y, blksiz = length(x))

Arguments

x

Vector of data on the first variable
y

Vector of data on the second variable
blksiz

block size, default blksiz =n, where n=rows in the matrix or no blocking is done

Value

A measure of dependence having the same sign as Pearson correlation. Its magnitude equals the larger of the two generalized correlation coefficients.

Note

This function needs the gmcmtxBlk function, which in turn needs the np package.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Vinod, H. D. (2021) 'Generalized, Partial and Canonical Correlation Coefficients' Computational Economics, 59(1), 1–28.

See Also

See Also gmcmtx0 and gmcmtxBlk

Examples

library(generalCorr)
options(np.messages = FALSE)
x=1:20;y=sin(x)
depMeas(x,y,blksiz=20)

order 4 differencing of a time series vector

Description

This is for momentum traders who focus on growth, acceleration, its gorwth and further acceleration. The diff function of R seems to do recycling of available numbers, not wanted for our purposes.

Usage

dif4(x)

Arguments

x

(n X 1) vector of time series (market returns) with n items each

Value

ou2 matrix having five columns, first for x, the next four columns have diff(x), diff-squared(x), diff-cubed(x) and diff-fourth(x)

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

x=c(2,8,3,5,1,8,19,22,23)
dif4(x)

order four differencing of a matrix of time series

Description

This is for momentum traders who focus on growth, acceleration, its growth and further acceleration. The diff function of R seems to do recycling of available numbers, not wanted for our purposes. Hence, this function is needed in portfolio studies based on time series.

Usage

dif4mtx(mtx)

Arguments

mtx

(n X p) matrix of p time series (market returns) with n items each

Value

out matrix having 12 rows, (data, D1 to D4 and ranks of D1 to D4 The column names of out are those of input matrix mtx.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

x=c(2,8,3,5,1,8,19,22,23)
y=c(3,11,2,6,7,9,20,25,21)
dif4mtx(cbind(x,y))

Internal diff.e0

Description

Internal diff.e0

Usage

data(diff.e0)

Internal dig

Description

Intended for internal use

Usage

data(dig)

Format

The format digs: int 78

internal e0

Description

intended for internal use only

Usage

e0

European Crime Data

Description

This data set refers to crime in European countries during 2008. The sources are World Bank and Eurostat. The crime statistics refers to homicides. It avoids possible reporting bias from the presence of police officers, because homicide reporting in most countries is standardized. Typical usage is: data(EuroCrime);attach(EuroCrime). The secondary source ‘quandl.com’ was used for collecting these data.

Details

The variables included in the dataset are:

  • Country Name of the European country

  • crim Per capita crime rate

  • off Per capita deployment of police officers

Exact stochastic dominance computation from areas above ECDF pillars.

Description

ECDF=empirical cumulative distribution functions. These are sufficient statistics representing probability density functions defined by observable finite data (e.g., stock returns). The exact computation of stochastic dominance orders SD1 to SD4 needs areas between two ECDFs, since such areas represent integrals. Higher-order SDs with continuous variables involve repeated integrals. Our quantification needs areas of ECDFs defined from areas of lower-order ECDFs. We argue that these computations are convenient if there is an ECDF of an imaginary reference minimum (x.ref) return, whose ECDF is a rectangle common for all stock comparisons. A common (x.ref) avoids having to compute all possible pairs of p stocks. Choosing a common reference as SP500 index stock cannot avoid a slower trapezoidal approximation for integrals, since its returns vary over time. We want exact areas of rectangles and fast.

Usage

exactSdMtx(mtx, howManySd = 0.1)

Arguments

mtx

(n X p) matrix of data. For example, returns on p stocks over n months
howManySd

used to define (x.ref)= lowest return number. If the grand minimum of all returns in ‘mtx’ is denoted GrMin, then howManySd equals the number of max(sd) (maximum standard deviation for data columns) below the GrMin used to define (x.ref). Thus, (x.ref)=GrMin-howManySd*max(sd). default howManySd=0.1

Details

The exactSdMtx function inputs ‘mtx’ (n X p) matrix data (e.g., n monthly returns on p stocks). Its output has four matrices SD1 to SD4, each with dimension (n X p). They measure exact dominance areas between empirical CDF for each column to the ECDF of (x.ref), an artificial stock with minimal return in all time periods. A fifth output matrix called ‘out’ produced by exactSdMtx has 4 rows and p columns containing column sums of SD1 to SD4. We intend that this ‘out’ matrix produced by exactSdMtx is then input to another function summaryRank() in the package designed for practitioners. For example, it indicates the best and the worst columns representing (the best stock to buy and best stock to sell) from the input data ‘mtx’ for investment based on a sophisticated computation of their ranks.

Value

five matrices. SD1 to SD4 contain four orders of stochastic dominance areas using the ECDF pillars and a common (x.ref). The fifth "out" matrix is another output with 4 rows for SD1 to SD4, and p columns (p=No. of columns in data matrix mtx) having a summary of ranks using all four, SD1 to SD4.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

x1=c(2,5,6,9,13,18,21)
x2=c(3,6,9,12,14,19,27) 
st1=exactSdMtx(cbind(x1,x2))

Generalized Granger-Causality. If dif>0, x2 Granger-causes x1.

Description

The usual Granger-causality assumes linear regressions. This function allows nonlinear nonparametric kernel regressions using a local linear (ll) option. Granger-causality (Gc) is generalized using nonlinear kernel regressions using local linear (ll) option. This functionn computes two R^2 values. (i) R12 or kernel regression R^2 of x1t on its own lags and x2t and its lags. (ii) R21 or kernel regression R^2 of x2t on its own lags and x1t and its lags. (iii) dif=R12-R21, the difference between the two R^2 values. If dif>0 then x2 Granger-causes x1.

Usage

GcRsqX12(x1, x2, px1 = 4, px2 = 4, pwanted = 4, ctrl = 0)

Arguments

x1

The data vector x1
x2

The data vector x2
px1

The number of lags of x1 in the data default px1=4
px2

The number of lags of x2 in the data, default px2=4
pwanted

number of lags of both x2 and x1 wanted for Granger causal analysis, default =4
ctrl

data matrix for designated control variable(s) outside causal paths default=0 means no control variables are present

Details

Calls GcRsqYX for R-square from kernel regression (local linear version) R^2[x1=f(x1,x2)] choosing GcRsqYX(y=x1, x=x2). It predicts x1 from both x1 and x2 using all information till time (t-1). It also calls GcRsqYX again after flipping x1 and x2. It returns RsqX1onX2, RsqX2onX1 and the difference dif=(RsqX1onX2-RsqX2onX1) If (dif>0) the regression y=f(x1,x2) is better than the flipped version implying that x1 is more predictable or x2 Granger-causes x1, x2 –> x1, rather than vice versa. The kernel regressions use regtype="ll" for local linear, bwmethod="cv.aic" for AIC-based bandwidth selection.

Value

This function returns 3 numbers: RsqX1onX2, RsqX2onX1 and dif

returns a list of 3 numbers. RsqX1onX2=(Rsquare of kernel regression of X1 on lags of X1 and X2 and its lags), RsqX2onX1= (Rsquare of kernel regression of x2 on own lags of X2 and X1), and the difference between the two Rquares (first minus second) called ‘dif.’ If dif>0 then x2 Granger-causes x1

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North-Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

Zheng, S., Shi, N.-Z., Zhang, Z., 2012. Generalized measures of correlation for asymmetry, nonlinearity, and beyond. Journal of the American Statistical Association 107, 1239-1252. -at-note internal routine

See Also

bootGcRsq, causeSummary, GcRsqYX.

Examples

## Not run: 
library(Ecdat);options(np.messages=FALSE);attach(data.frame(MoneyUS))
GcRsqX12(y,m)   

## End(Not run)

Generalized Granger-Causality. If dif>0, x2 Granger-causes x1.

Description

The usual Granger-causality assumes linear regressions. This allows nonlinear nonparametric kernel regressions using a local constat (lc) option. Calls GcRsqYXc for R square from kernel regression. R^2[x1=f(x1,x2)] choosing GcRsqYXc(y=x1, x=x2). The name ‘c’ in the function refers to local constant option of kernel regressions.' It predicts x1 from both x1 and x2 using all information till time (t-1). It also calls GcRsqYXc again after flipping x1 and x2. It returns RsqX1onX2, RsqX2onX1 and the difference dif=(RsqX1onX2-RsqX2onX1) If (dif>0) the regression x1=f(x1,x2) is better than the flipped version implying that x1 is more predictable or x2 Granger-causes x1 x2 –> x1, rather than vice versa. The kernel regressions use regtype="lc" for local constant, bwmethod="cv.ls" for least squares-based bandwidth selection.

Usage

GcRsqX12c(x1, x2, px1 = 4, px2 = 4, pwanted = 4, ctrl = 0)

Arguments

x1

The data vector x1
x2

The data vector x2
px1

number of lags of x1 in the data default px1=4
px2

number of lags of x2 in the data, default px2=4
pwanted

number of lags of both x2 and x1 wanted for Granger causal analysis, default =4
ctrl

data matrix for designated control variable(s) outside causal paths default=0 means no control variables are present

Value

This function returns 3 numbers: RsqX1onX2, RsqX2onX1 and dif

returns a list of 3 numbers. RsqX1onX2=(Rsquare of kernel regression of X1 on X1 and X2), RsqX2onX1= (Rsquare of kernel regression of x2 on X2 and X1), and the difference between the two Rquares called dif

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

Zheng, S., Shi, N.-Z., Zhang, Z., 2012. Generalized measures of correlation for asymmetry, nonlinearity, and beyond. Journal of the American Statistical Association 107, 1239-1252. -at-note internal routine

See Also

causeSummary

GcRsqYXc

Examples

## Not run: 
library(Ecdat);options(np.messages=FALSE);attach(data.frame(MoneyUS))
GcRsqX12c(y,m)   

## End(Not run)

Nonlinear Granger causality between two time series workhorse function.

Description

Function input is y=LHS=First time series and x=RHS=Second time series. Kernel regression np package options regtype="ll" for local linear, and bwmethod="cv.aic" for AIC-based bandwidth selection are fixed. Denote Rsq=Rsquare=R^2 in nonlinear kernel regression. GcRsqYX(.) computes the following two R^2 values. out[1]=Rsqyyx = R^2 when we regress y on own lags of y and x. out[2]=Rsqyy = R^2 when we regress y on lags of y alone.

Usage

GcRsqYX(y, x, px = 4, py = 4, pwanted = 4, ctrl = 0)

Arguments

y

The data vector y for the Left side or dependent or first variable
x

The data vector x for the right side or explanatory or second variable
px

number of lags of x in the data
py

number of lags of y in the data. px=4 for quarterly data
pwanted

number of lags of both x and y wanted for Granger causal analysis
ctrl

data matrix for designated control variable(s) outside causal paths default=0 means no control variables are present

Value

This function returns a set of 2 numbers measuring nonlinear Granger-causality for time series. out[1]=Rsqyyx, out[2]=Rsqyy.

Note

If data are annual or if no quarterly-type structure is present, use this function with pwanted=px=py. For example, the egg or chicken data from lmtest package.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

Zheng, S., Shi, N.-Z., Zhang, Z., 2012. Generalized measures of correlation for asymmetry, nonlinearity, and beyond. Journal of the American Statistical Association 107, 1239-1252.

See Also

GcRsqX12, kern2, kern2ctrl.

Examples

## Not run: 
library(Ecdat);options(np.messages=FALSE);attach(data.frame(MoneyUS))
GcRsqYX(y,m)  

## End(Not run)

Nonlinear Granger causality between two time series workhorse function.(local constant version)

Description

Function input is y=LHS=First time series and x=RHS=Second time series. Kernel regression np package options regtype="lc" for local constant, and bwmethod="cv.ls" for least squares-based bandwidth selection are fixed. Denote Rsq=Rsquare=R^2 in nonlinear kernel regression. GcRsqYXc(.) computes the following two R^2 values. out[1]=Rsqyyx = R^2 when we regress y on own lags of y and x. out[2]=Rsqyy = R^2 when we regress y on own lags of y alone.

Usage

GcRsqYXc(y, x, px = 4, py = 4, pwanted = 4, ctrl = 0)

Arguments

y

The data vector y for the Left side or dependent or first variable
x

The data vector x for the right side or explanatory or second variable
px

number of lags of x in the data
py

number of lags of y in the data. px=4 for quarterly data
pwanted

number of lags of both x and y wanted for Granger causal analysis
ctrl

data matrix for designated control variable(s) outside causal paths default=0 means no control variables are present

Value

This function returns a set of 2 numbers measuring nonlinear Granger-causality for time series. out[1]=Rsqyyx, out[2]=Rsqyy.

Note

If data are annual or if no quarterly-type structure is present, use this function with pwanted=px=py. For example, the egg or chicken data from lmtest package, Thurman W.N. and Fisher M.E. (1988)

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

Zheng, S., Shi, N.-Z., Zhang, Z., 2012. Generalized measures of correlation for asymmetry, nonlinearity, and beyond. Journal of the American Statistical Association 107, 1239-1252.

See Also

GcRsqX12c

kern_ctrl

Examples

## Not run: 
library(Ecdat);options(np.messages=FALSE);attach(data.frame(MoneyUS))
GcRsqYXc(y,m) 

## End(Not run)

generalCorr package description:

Description

This package provides convenient software tools for causal path determinations using Vinod (2014, 2015, 2018, 2021) and is explained in many package vignettes. causeSummary(mtx), causeSummary2(mtx),causeSum2Blk(mtx), causeSummBlk are various versions reporting pair-wise causal path directions and causal strengths. We fit a kernel regression of X1 on (X2, X3,..Xk) and another flipped regression of X2 on (X1, x3, ..Xk). We compare the two fits using three sophisticated criteria called Cr1 to Cr3. We rescale the weighted sum of the quantified three criteria to the [-100, 100] range. The sign of the weighted sum gives the direction of the causal path, and the magnitude of the weighted sum gives the strength of the causal path. A matrix of non-symmetric generalized correlations r*(x|y) is reported by the functions rstar() and gmcmtx0(). sudoCoefParcor() computes pseudo kernel regression coefficients based on generalized partial correlation coefficients (GPCC) depMeas() a measure of nonlinear nonparametric dependence between two vectors. parcorVec() has generalized partial correlation coefficients, Vinod (2021) parcorVecH() has a hybrid version of the above (using HGPCC). The usual partial correlations r(x,y|z) for regression of y on (x, z) measure the effect of y on x after removing the effect of z, where z can have several variables. Vinod (2021) suggests new generalized partial correlation coefficients (GPCC) using kernel regressions, r*(x,y|z).

Details

The criterion Cr1 uses observable values of standard exogeneity test criterion, namely, (kernel regression residual) times (regressor values) Cr2 computes absolute values kernel regression residuals. The quantification of Cr1 and Cr2 further uses four orders of stochastic dominance measures. Cr3 compares the R-square of the two fits. The package provides additional tools for matrix algebra, such as cofactor(), for outlier detection get0outlier(), for numerical integration by the trapezoidal rule, stochastic dominance stochdom2() and comp_portfo2(), etc. The package has a function pcause() for bootstrap-based statistical inference and another one for a heuristic t-test called heurist(). Pairwise deletion of missing data is done in napair(), while triplet-wise deletion is in naTriplet() intended for use when control variable(s) are also present. If one has panel data, functions PanelLag() and Panel2Lag() are relevant. pillar3D provides 3-dimensional plots of data that look more like surfaces, than usual plots with vertical pins.

Recent 2020 additions include canonRho() for generalized canonical correlations, and many functions for Granger causality between lagged time series including GcRsqX12(), bootGcRsq() and GcRsqYXc().

Recent additions include several functions for portfolio choice. causeSum2Panel() for panel data, sudoCoefParcor() for pseudo regression coefficients for kernel regressions. decileVote(), momentVote(), exactSdMtx() for exact computation of stochastic dominance from ECDF areas. The newer stochastic dominance tools are used in causeSummary2(mtx),causeSum2Blk(mtx) dif4mtx() computes growth, change in growth etc. up-to order 4 differencing of time series. outOFsamp() and outOFsell() pandemic-proof out-of-sample evaluation of portfolio returns using randomization. causeSum2Panel() exploits panel data features for causal paths.

Note

Eight vignettes provided with this package at CRAN describe the theory and usage of the package with examples. Read them using the command: vignette("generalCorr-vignette") to read the first vignette. vignettes 2 to 6 can be read by including the vignette number. For example, vignette("generalCorr-vignette6") to read the sixth vignette.

References

Vinod, H. D.'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in 'Handbook of Statistics: Computational Statistics with R', Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Zheng, S., Shi, N.-Z., and Zhang, Z. (2012). 'Generalized measures of correlation for asymmetry, nonlinearity, and beyond,' Journal of the American Statistical Association, vol. 107, pp. 1239-1252.

Vinod, H. D. (2021) 'Generalized, Partial and Canonical Correlation Coefficients' Computational Economics, 59(1), 1–28.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Function to compute outliers and their count using Tukey's method using 1.5 times interquartile range (IQR) to define boundaries.

Description

Function to compute outliers and their count using Tukey's method using 1.5 times interquartile range (IQR) to define boundaries.

Usage

get0outliers(x, verbo = TRUE, mult = 1.5)

Arguments

x

vector of data.
verbo

set to TRUE(default) assuming printed details are desired.
mult

=1.5(default), the number of times IQR is used in defining outlier boundaries.

Value

below

which items are lower than the lower limit
above

which items are larger than the upper limit
low.lim

the lower boundary for outlier detection
up.lim

the upper boundary for outlier detection
nUP

count of number of data points above upper boundary
nLO

count of number of data points below lower boundary

Note

The function removes the missing data before checking for outliers.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

set.seed(101);x=sample(1:100)[1:15];x[16]=150;x[17]=NA
get0outliers(x)#correctly identifies outlier=150

Two sequences: starting+ending values from n and blocksize (internal use)

Description

This is an auxiliary function for gmcmtxBlk. It gives sequences of starting and ending values

Usage

getSeq(n, blksiz)

Arguments

n

length of the range
blksiz

blocksize

Value

two vectors sqLO and sqUP

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

See Also

gmcmtxBlk

Examples

getSeq(n=99, blksiz=10)

internal gmc0

Description

intended for internal use only

Usage

gmc0

internal gmc1

Description

intended for internal use only

Usage

gmc1

Matrix R* of generalized correlation coefficients captures nonlinearities.

Description

This function checks for missing data for each pair individually. It then uses the kern function to kernel regress x on y, and conversely y on x. It needs the R package ‘np’, which reports the R-squares of each regression. gmcmtx0() function reports their square roots after assigning them the observed sign of the Pearson correlation coefficient. Its threefold advantages are: (i) It is asymmetric, yielding causal direction information by relaxing the assumption of linearity implicit in usual correlation coefficients. (ii) The r* correlation coefficients are generally larger upon admitting arbitrary nonlinearities. (iii) max(|R*ij|, |R*ji|) measures (nonlinear) dependence. For example, let x=1:20 and y=sin(x). This y has a perfect (100 percent) nonlinear dependence on x, and yet Pearson correlation coefficient r(xy) -0.0948372 is near zero, and the 95% confidence interval (-0.516, 0.363) includes zero, implying that r(xy) is not significantly different from zero. This shows a miserable failure of traditional r(x,y) to measure dependence when nonlinearities are present. gmcmtx0(cbind(x,y)) will correctly reveal perfect (nonlinear) dependence with generalized correlation coefficient =-1.

Usage

gmcmtx0(mym, nam = colnames(mym))

Arguments

mym

A matrix of data on variables in columns
nam

Column names of the variables in the data matrix

Value

A non-symmetric R* matrix of generalized correlation coefficients

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D.'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in 'Handbook of Statistics: Computational Statistics with R', Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Zheng, S., Shi, N.-Z., and Zhang, Z. (2012). 'Generalized measures of correlation for asymmetry, nonlinearity, and beyond,' Journal of the American Statistical Association, vol. 107, pp. 1239-1252.

See Also

See Also as gmcmtxBlk for a more general version using blocking allowing several bandwidths.

Examples

gmcmtx0(mtcars[,1:3])

## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
colnames(x)=c('V1', 'v2', 'V3')
gmcmtx0(x)
## End(Not run)

Matrix R* of generalized correlation coefficients captures nonlinearities using blocks.

Description

The algorithm uses two auxiliary functions, getSeq and NLhat. The latter uses the kern function to kernel regress x on y, and conversely y on x. It needs the package ‘np,’ which reports residuals and allows one to compute fitted values (xhat, yhat). Unlike gmcmtx0, this function considers blocks of blksiz=10 (default) pairs of data points separately with distinct bandwidths for each block, usually creating superior local fits.

Usage

gmcmtxBlk(mym, nam = colnames(mym), blksiz = 10)

Arguments

mym

A matrix of data on selected variables arranged in columns
nam

Column names of the variables in the data matrix
blksiz

block size, default=10, if chosen blksiz >n, where n=rows in matrix then blksiz=n. That is, no blocking is done

Details

This function does pairwise checks of missing data for all pairs. Assume that there are n rows in the input matrix ‘mym’ with some missing rows. If the columns of mym are denoted (X1, X2, ...Xp), we are considering all pairs (Xi, Xj), treated as (x, y), with ‘nv’ number of valid (non-missing) rows Note that each x and y is an (nv by 1) vector. This function further splits these (x, y) vectors into as many subgroups or blocks as are needed for the nv paired valid data points for the chosen block length (blksiz)

Next, the algorithm strings together various blocks of fitted value vectors (xhat, yhat) also of dimension nv by 1. Now for each pair of Xi Xj (column Xj= cause, row Xi=response, treated as x and y), the algorithm computes R*ij the simple Pearson correlation coefficient between (x, xhat) and as R*ji the correlation coeff. between (y, yhat). Next, it assigns |R*ij| and |R*ji| the observed sign of the Pearson correlation coefficient between x and y.

Its advantages discussed in Vinod (2015, 2019) are: (i) It is asymmetric yielding causal direction information, by relaxing the assumption of linearity implicit in usual correlation coefficients. (ii) The R* correlation coefficients are generally larger upon admitting arbitrary nonlinearities. (iii) max(|R*ij|, |R*ji|) measures (nonlinear) dependence. For example, let x=1:20 and y=sin(x). This y has a perfect (100 percent) nonlinear dependence on x and yet Pearson correlation coefficient r(x y)= -0.0948372 is near zero, and its 95% confidence interval (-0.516, 0.363) includes zero, implying that the population r(x,y) is not significantly different from zero. This example highlights a serious failure of the traditional r(x,y) in measuring dependence between x and y when nonlinearities are present. gmcmtx0 without blocking does work if x=1:n, and y=f(x)=sin(x) is used with n<20. But for larger n, the fixed bandwidth used by the kern function becomes a problem. The block version has additional bandwidths for each block, and hence it correctly quantifies the presence of high dependence even when x=1:n, and y=f(x) are defined for large n and complicated nonlinear functional forms for f(x).

Value

A non-symmetric R* matrix of generalized correlation coefficients

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D.'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in 'Handbook of Statistics: Computational Statistics with R', Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Vinod, H. D. 'New exogeneity tests and causal paths,' Chapter 2 in 'Handbook of Statistics: Conceptual Econometrics Using R', Vol.32, co-editors: H. D. Vinod and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2019, pp. 33-64.

Zheng, S., Shi, N.-Z., and Zhang, Z. (2012). 'Generalized measures of correlation for asymmetry, nonlinearity, and beyond,' Journal of the American Statistical Association, vol. 107, pp. 1239-1252.

Examples

## Not run: 
x=1:20; y=sin(x)
gmcmtxBlk(cbind(x,y),blksiz=10)
## End(Not run)

compute the matrix R* of generalized correlation coefficients.

Description

This function checks for missing data separately for each pair using kern function to kernel regress x on y, and conversely y on x. It needs the library ‘np’ which reports R-squares of each regression. This function reports their square roots with the sign of the Pearson correlation coefficients. Its appeal is that it is asymmetric yielding causal direction information. It avoids the assumption of linearity implicit in the usual correlation coefficients.

Usage

gmcmtxZ(mym, nam = colnames(mym))

Arguments

mym

A matrix of data on variables in columns
nam

Column names of the variables in the data matrix

Value

A non-symmetric R* matrix of generalized correlation coefficients

Note

This allows the user to change gmcmtx0 and further experiment with my code.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Examples

## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
colnames(x)=c('V1', 'v2', 'V3')
gmcmtxZ(x)

## End(Not run)

Function to compute generalized correlation coefficients r*(x|y) and r*(y|x) from two vectors (not matrices)

Description

This function uses the ‘np’ package and assumes that there are no missing data.

Usage

gmcxy_np(x, y)

Arguments

x

vector of x data
y

vector of y data

Value

corxy

r*(x|y) from regressing x on y, where y is the kernel cause.
coryx

r*(y|x) from regressing y on x, where x is the cause.

Note

This is provided if the user want to avoid calling kern.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R,' Chapter 4 in 'Handbook of Statistics: Computational Statistics with R,' Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Examples

## Not run: 
set.seed(34);x=sample(1:10);y=sample(2:11)
gmcxy_np(x,y)
## End(Not run)

internal goodCol

Description

intended for internal use only

Usage

goodCol

Heuristic t test of the difference between two generalized correlations.

Description

Function to run a heuristic t test of the difference between two generalized correlations.

Usage

heurist(rxy, ryx, n)

Arguments

rxy

generalized correlation r*(x|y) where y is the kernel cause.
ryx

generalized correlation r*(y|x) where x is the kernel cause.
n

Sample size needed to determine the degrees of freedom for the t test.

Value

Prints the t statistics and p-values.

Note

This function requires Revele's R package called ‘psych’ in memory. This test is known to be conservative (i.e., often fails to reject the null hypothesis of zero difference between the two generalized correlation coefficients.)

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

set.seed(34);x=sample(1:10);y=sample(2:11)
g1=gmcxy_np(x,y)
n=length(x)
h1=heurist(g1$corxy,g1$coryx,n)
print(h1)
print(h1$t) #t statistic
print(h1$p) #p-value

internal i

Description

intended for internal use

Usage

data(i)

Format

The format is: int 78

internal object

Description

intended for internal use

internal ii

Description

intended for internal use

internal j

Description

intended for internal use

Usage

data(j)

Format

The format is: int 4

Kernel regression with options for residuals and gradients.

Description

Function to run kernel regression with options for residuals and gradients asssuming no missing data.

Usage

kern(dep.y, reg.x, tol = 0.1, ftol = 0.1, gradients = FALSE, residuals = FALSE)

Arguments

dep.y

Data on the dependent (response) variable
reg.x

Data on the regressor (stimulus) variables
tol

Tolerance on the position of located minima of the cross-validation function (default =0.1)
ftol

Fractional tolerance on the value of cross validation function evaluated at local minima (default =0.1)
gradients

Make this TRUE if gradients computations are desired
residuals

Make this TRUE if residuals are desired

Value

Creates a model object ‘mod’ containing the entire kernel regression output. Type names(mod) to reveal the variety of outputs produced by ‘npreg’ of the ‘np’ package. The user can access all of them at will by using the dollar notation of R.

Note

This is a work horse for causal identification.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D.'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

See Also

See kern_ctrl.

Examples

## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:50],ncol=2)
require(np); options(np.messages=FALSE)
k1=kern(x[,1],x[,2])
print(k1$R2) #prints the R square of the kernel regression

## End(Not run)

Kernel regression with control variables and optional residuals and gradients.

Description

Allowing matrix input of control variables, this function runs kernel regression with options for residuals and gradients.

Usage

kern_ctrl(
  dep.y,
  reg.x,
  ctrl,
  tol = 0.1,
  ftol = 0.1,
  gradients = FALSE,
  residuals = FALSE
)

Arguments

dep.y

Data on the dependent (response) variable
reg.x

Data on the regressor (stimulus) variable
ctrl

Data matrix on the control variable(s) kept outside the causal paths. A constant vector is not allowed as a control variable.
tol

Tolerance on the position of located minima of the cross-validation function (default=0.1)
ftol

Fractional tolerance on the value of cross validation function evaluated at local minima (default=0.1)
gradients

Set to TRUE if gradients computations are desired
residuals

Set to TRUE if residuals are desired

Value

Creates a model object ‘mod’ containing the entire kernel regression output. If this function is called as mod=kern_ctrl(x,y,ctrl=z), the researcher can simply type names(mod) to reveal the large variety of outputs produced by ‘npreg’ of the ‘np’ package. The user can access all of them at will using the dollar notation of R.

Note

This is a work horse for causal identification.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

See Also

See kern.

Examples

## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:50],ncol=5)
require(np)
k1=kern_ctrl(x[,1],x[,2],ctrl=x[,4:5])
print(k1$R2) #prints the R square of the kernel regression

## End(Not run)

Kernel regression version 2 with optional residuals and gradients with regtype="ll" for local linear, bwmethod="cv.aic" for AIC-based bandwidth selection.

Description

Kernel regression version 2 with optional residuals and gradients with regtype="ll" for local linear, bwmethod="cv.aic" for AIC-based bandwidth selection.

Usage

kern2(
  dep.y,
  reg.x,
  tol = 0.1,
  ftol = 0.1,
  gradients = FALSE,
  residuals = FALSE
)

Arguments

dep.y

Data on the dependent (response) variable
reg.x

Data on the regressor (stimulus) variables
tol

Tolerance on the position of located minima of the cross-validation function (default =0.1)
ftol

Fractional tolerance on the value of cross validation function evaluated at local minima (default =0.1)
gradients

Make this TRUE if gradients computations are desired
residuals

Make this TRUE if residuals are desired

Value

Creates a model object ‘mod’ containing the entire kernel regression output. Type names(mod) to reveal the variety of outputs produced by ‘npreg’ of the ‘np’ package. The user can access all of them at will by using the dollar notation of R.

Note

This is version 2 ("ll","cv.aic") of a work horse for causal identification.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D.'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

See Also

See kern_ctrl.

Examples

## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:50],ncol=2)
require(np); options(np.messages=FALSE)
k1=kern(x[,1],x[,2])
print(k1$R2) #prints the R square of the kernel regression

## End(Not run)

Kernel regression with control variables and optional residuals and gradients. version 2 regtype="ll" for local linear, bwmethod="cv.aic" for AIC-based bandwidth selection. It admits control variables.

Description

Kernel regression with control variables and optional residuals and gradients. version 2 regtype="ll" for local linear, bwmethod="cv.aic" for AIC-based bandwidth selection. It admits control variables.

Usage

kern2ctrl(
  dep.y,
  reg.x,
  ctrl,
  tol = 0.1,
  ftol = 0.1,
  gradients = FALSE,
  residuals = FALSE
)

Arguments

dep.y

Data on the dependent (response) variable
reg.x

Data on the regressor (stimulus) variable
ctrl

Data matrix on the control variable(s) kept outside the causal paths. A constant vector is not allowed as a control variable.
tol

Tolerance on the position of located minima of the cross-validation function (default=0.1)
ftol

Fractional tolerance on the value of cross validation function evaluated at local minima (default=0.1)
gradients

Set to TRUE if gradients computations are desired
residuals

Set to TRUE if residuals are desired

Value

Creates a model object ‘mod’ containing the entire kernel regression output. If this function is called as mod=kern_ctrl(x,y,ctrl=z), the researcher can simply type names(mod) to reveal the large variety of outputs produced by ‘npreg’ of the ‘np’ package. The user can access all of them at will using the dollar notation of R.

Note

This is version 2 ("ll","cv.aic") of a work horse for causal identification.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

See Also

See kern.

Examples

## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:50],ncol=5)
require(np)
k1=kern_ctrl(x[,1],x[,2],ctrl=x[,4:5])
print(k1$R2) #prints the R square of the kernel regression

## End(Not run)

Approximate overall magnitudes of kernel regression partials dx/dy and dy/dx.

Description

Uses Vinod (2015) and runs kernel regression of x on y, and also of y on x by using the ‘np’ package. The function goes on to compute a summary magnitude of the overall approximate partial derivative dx/dy (and dy/dx), after adjusting for units by using an appropriate ratio of standard deviations. Of course, the real partial derivatives of nonlinear functions are generally distinct for each observation.

Usage

mag(x, y)

Arguments

x

Vector of data on the dependent variable
y

Vector of data on the regressor

Value

vector of two magnitudes of kernel regression partials dx/dy and dy/dx.

Note

This function is intended for use only after the direction of causal path is already determined by various functions in this package (e.g. somePairs). For example, if the researcher knows that x causes y, then only dy/dx denoted by dydx is relevant. The other output of the function dxdy is to be ignored. Similarly, only ‘dxdy’ is relevant if y is known to be the cause of x.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

See Also

See mag_ctrl.

Examples

set.seed(123);x=sample(1:10);y=1+2*x+rnorm(10)
mag(x,y)#dxdy approx=.5 and dydx approx=2 will be nice.

After removing control variables, magnitude of effect of x on y, and of y on x.

Description

Uses Vinod (2015) and runs kernel regressions: x~ y + ctrl and x~ ctrl to evaluate the ‘incremental change’ in R-squares. Let (rxy;ctrl) denote the square root of that ‘incremental change’ after its sign is made the same as that of the Pearson correlation coefficient from cor(x,y)). One can interpret (rxy;ctrl) as a generalized partial correlation coefficient when x is regressed on y after removing the effect of control variable(s) in ctrl. It is more general than the usual partial correlation coefficient, since this one allows for nonlinear relations among variables. Next, the function computes ‘dxdy’ obtained by multiplying (rxy;ctrl) by the ratio of standard deviations, sd(x)/sd(y). Now our ‘dxdy’ approximates the magnitude of the partial derivative (dx/dy) in a causal model where y is the cause and x is the effect. The function also reports entirely analogous ‘dydx’ obtained by interchanging x and y.

Usage

mag_ctrl(x, y, ctrl)

Arguments

x

Vector of data on the dependent variable.
y

Vector of data on the regressor.
ctrl

data matrix for designated control variable(s) outside causal paths. A constant vector is not allowed as a control variable.

Value

vector of two magnitudes ‘dxdy’ (effect when x is regressed on y) and ‘dydx’ for reverse regression. Both regressions remove the effect of control variable(s).

Note

This function is intended for use only after the causal path direction is already determined by various functions in this package (e.g. someCPairs). That is, after the researcher knows whether x causes y or vice versa. The output of this function is a vector of two numbers: (dxdy, dydx), in that order, representing the magnitude of effect of one variable on the other. We expect the researcher to use only ‘dxdy’ if y is the known cause, or ‘dydx’ if x is the cause. These approximate overall measures may not be well-defined in some applications, because the real partial derivatives of nonlinear functions are generally distinct for each evaluation point.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C. R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

See Also

See mag

Examples

set.seed(123);x=sample(1:10); z=runif(10); y=1+2*x+3*z+rnorm(10)
options(np.messages=FALSE)
mag_ctrl(x,y,z)#dx/dy=0.47 is approximately 0.5, but dy/dx=1.41 is not approx=2,

internal min.e0

Description

intended for internal use only

Usage

min.e0

Function to do compute the minor of a matrix defined by row r and column c.

Description

Function to do compute the minor of a matrix defined by row r and column c.

Usage

minor(x, r, c)

Arguments

x

The input matrix
r

The row number
c

The column number

Value

The appropriate ‘minor’ matrix defined from the input matrix.

Note

This function is needed by the cofactor function.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

## Not run: 
 x=matrix(1:20,ncol=4)
minor(x,1,2)
## End(Not run)

Function compares Pearson Stats and Sharpe Ratio for a matrix of stock returns

Description

The first step computes mean, std.dev, skewness, kurtosis (kurt),and the Sharpe Ratio (mean/sd) representing risk-adjusted return where sd measures the risk. The input x must be a matrix having p columns (col.names recommended). and n rows as in the data. If data are missing for some columns, insert NA's. Thus x has p column of data matrix ready for comparison and ranking. For example, x has a matrix of stock returns. The output matrix produced by this function has p columns for each data column (i.e. for each stock being compared). The output matrix has twelve rows. Top five rows have the magnitudes of mean, sd, skew, kurt, Sharpe ratios. Output matrix rows 6 to 10 have respective ranks of moment stats. The output 11-th row reports a weighted sum of ranks with following weights mean=1,sd=-1,skew=0.5,kurt=-0.5,Sharpe Ratio=1. User has the option to change the weights. They measure relative importance.

Usage

momentVote(mtx, weight = c(1, -1, 0.5, -0.5, 1))

Arguments

mtx

n by p matrix of data, For example, n stock returns for p stocks. The mtx columns should have some names (ticker symbols)
weight

vector of reliability weights. default: mean=1, sd=-1, skew=0.5,kurt=-0.5,sharpe=1

Details

Since skewness and kurtosis are measured relatively less reliably (have greater sampling variation due to higher powers) their weight is 0.5. Our ranking gives the smallest number 1 to the most desirable outcome. The 11-th line of the output matrix has weighted sum of ranks and we suggest higher portfolio weight be given to the column having smallest value (in the bottom line). The 12-th row of output matrix has ‘choice,’ where input weights give the number 1 is for the top choice column of data and all other choice numbers. The (p+1)-th column of the output matrix has the chosen weights. The argument weight to the ‘momentVote’ function allows one to change these weights.

Value

a matrix with same number of columns as in the input matrix x and eleven rows. Top five rows have moment quantities, next five are their ranks the eleventh row has weighted sum of ranks with the input weights (see default) and the 12-th row has choice numbers (choice=1 is best)

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

x1=c(1,4,7,2,6)
x2=c(3,4,8,4,7)
momentVote(cbind(x1,x2))

internal mtx

Description

intended for internal use only

Usage

mtx

internal mtx0

Description

intended for internal use only

Usage

mtx0

internal mtx2

Description

intended for internal use only

Usage

mtx2

internal n

Description

intended for internal use

Usage

n

Format

The format is: int 78

internal nall

Description

intended for internal use only

Usage

nall

internal nam.badCol

Description

intended for internal use only

Usage

nam.badCol

internal nam.goodCol

Description

intended for internal use only

Usage

nam.goodCol

internal nam.mtx0

Description

intended for internal use only

Usage

nam.mtx0

Function to do pairwise deletion of missing rows.

Description

The aim in pair-wise deletions is to retain the largest number of available data pairs with all non-missing data.

Usage

napair(x, y)

Arguments

x

Vector of x data
y

Vector of y data

Value

newx

A new vector x after removing pairwise missing data
newy

A new vector y after removing pairwise missing data

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

## Not run: 
x=sample(1:10);y=sample(1:10);x[2]=NA; y[3]=NA
napair(x,y)
## End(Not run)

Function to do matched deletion of missing rows from x, y and z variable(s).

Description

The aim in three-way deletions is to retain only the largest number of available data triplets with all non-missing data. This works where naTriplet fails (e.g.parcorVecH()). This is called by parcorHijk

Usage

naTriple(x, y, z)

Arguments

x

Vector of x data
y

Vector of y data
z

vector or a matrix of additional variable(s)

Value

newx

A new vector x after removing triplet-wise missing data
newy

A new vector or matrix y after removing triplet-wise missing data
newz

A new vector or matrix ctrl after removing triplet-wise missing data

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

See Also

See napair naTriplet.

Examples

## Not run: 
x=sample(1:10);y=sample(1:10);x[2]=NA; y[3]=NA
w=sample(2:11)
naTriple(x,y,w)
## End(Not run)

Function to do matched deletion of missing rows from x, y and control variable(s).

Description

The aim in three-way deletions is to retain only the largest number of available data triplets with all non-missing data.

Usage

naTriplet(x, y, ctrl)

Arguments

x

Vector of x data
y

Vector of y data
ctrl

Data matrix on the control variable(s) kept beyond causal path determinations

Value

newx

A new vector x after removing triplet-wise missing data
newy

A new vector or matrix y after removing triplet-wise missing data
newctrl

A new vector or matrix ctrl after removing triplet-wise missing data

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

See Also

See napair.

Examples

## Not run: 
x=sample(1:10);y=sample(1:10);x[2]=NA; y[3]=NA
w=sample(2:11)
naTriplet(x,y,w)
## End(Not run)

Compute fitted values from kernel regression of x on y and y on x

Description

This is an auxiliary function for ‘gmcmtxBlk.’ It uses two numerical vectors (x, y) of same length to create two vectors (xhat, yhat) of fitted values using nonlinear kernel regressions. It uses package ‘np’ called by kern function to kernel regress x on y, and conversely y on x. It uses the option ‘residuals=TRUE’ of ‘kern’

Usage

NLhat(x, y)

Arguments

x

A column vector of x data
y

A column vector of y data

Value

two vectors named xhat and yhat for fitted values

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

See Also

See Also as gmcmtxBlk.

Examples

## Not run: 
set.seed(34);x=sample(1:15);y=sample(1:15)
NLhat(x,y)
## End(Not run)

internal out1

Description

intended for internal use only

Usage

out1

Compare out-of-sample portfolio choice algorithms by a leave-percent-out method.

Description

This function randomly leaves out 5 percent (‘pctOut’=5 by default) data and finds portfolio choice by seven different portfolio selection algorithms using the data on the remaining 95 percent (say). The randomization removes any bias in time series definitions of ‘out-of-sample’ data. For example, the input to outOFsamp(.) named ‘mtx’ is a matrix with p columns for p stocks and n returns. Also, let the maximum number of stocks admitted to belong in the portfolio be four, or ‘maxChosen=4’. Now outOFsamp function computes the returns earned by the seven portfolio selection algorithms, called "SD1", "SD2", "SD3", "SD4", "SDAll4", "decile," and "moment," where SDAll4 refers to a weighted sum of SD1 to SD4 algorithms. Each algorithm provides a choice ranking of p stocks with choice values 1,2,3,..,p where stock ranked 1 should get the highest portfolio weight. The outOFsamp function then calls the function ‘rank2return,’ which uses these rank choice numbers to the selected ‘maxChosen’ stocks. The allocation is linearly declining. For example, it is 4/10, 3/10, 2/10, and 1/10, with the top choice stock receiving 4/10 of the capital. Each choice of ‘pctOut’ rows of the ‘mtx’ data yields an outOFsamp return for each of the seven portfolio selection algorithms. These outOFsamp return computations are repeated reps times. A new random selection of ‘pctOut’ rows (must be 2 or more) of data is made for each repetition. We set reps=20 by default. The low default is set to save processing time in early phases, but we recommend reps=100+. The final choice of stock-picking algorithm out of seven is suggested by the one yielding the largest average out-of-sample return over the ‘reps’ repetitions.'Its standard deviation measures the variability of performance over the ‘reps’ repetitions.

Usage

outOFsamp(mtx, pctOut = 5, reps = 10, seed = 23, maxChosen = 2, verbo = FALSE)

Arguments

mtx

matrix size n by p of data on n returns from p stocks
pctOut

percent of n randomly chosen rows left out as out-of-sample, default=5 percent. One must leave out at least two rows of data
reps

number of random repetitions of left-out rows over which we average the out-of-sample performance of a stock-picking algorithm, default reps=20
seed

seed for random number generation, default =23
maxChosen

number of stocks (out of p) with nonzero weights in the portfolio
verbo

logical, TRUE means print details, default=FALSE

Value

a matrix called ‘avgRet’ with seven columns for seven stock-picking algorithms "SD1","SD2","SD3","SD4","SDAll4","decile",and "moment," containing out-of-sample average returns for linearly declining allocation in a portfolio. The user needs to change rank2return() for alternate portfolio allocations.

Note

The traditional time-series out-of-sample leaves out the last few time periods, and estimates the stock-picking model using part of the data time periods. The pandemic of 2019 has revealed that the traditional out-of-sample would have a severe bias in favor of pessimistic stock-picking algorithms. The traditional method is fundamentally flawed since it is sensitive to the trends (ups and downs) in the out-of-sample period. The method proposed here is free from such biases. The stock-picking algorithm recommended by our outOFsamp() is claimed to be robust against such biases.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

See Also

rank2return

Examples

## Not run: 
x1=c(2,5,6,9,13,18,21,5,11,14,4,7,12,13,6,3,8,1,15,2,10,9)
x2=c(3,6,9,12,14,19,27,9,11,2,3,8,1,6,15,10,13,14,5,7,4,12)
x3=c(2,6,NA,11,13,25,25,11,9,10,12,6,4,3,2,1,7,8,5,15,14,13)
mtx=cbind(x1,x2,x3)
mtx=mtx[complete.cases(mtx),]
os=outOFsamp(mtx,verbo=FALSE,maxChosen=2, reps=3)
apply(os,2,mean)
## End(Not run)

Compare out-of-sample (short) selling algorithms by a leave-percent-out method.

Description

This function randomly leaves out 5 percent (‘pctOut’=5 by default) data and finds portfolio choice to sell by seven different portfolio selection algorithms using the data on the remaining 95 percent (say). The randomization removes any bias in time series definitions of ‘out-of-sample’ data. For example, the input to outOFsamp(.) named ‘mtx’ is a matrix with p columns for p stocks and n returns. Also, let the maximum number of stocks admitted to belong in the sell portfolio be four, or ‘maxChosen=4’. Now outOFsamp function computes the returns earned by the seven portfolio selection algorithms, called "SD1", "SD2", "SD3", "SD4", "SDAll4", "decile," and "moment," where SDAll4 refers to a weighted sum of SD1 to SD4 algorithms. Each algorithm provides a choice ranking of p stocks with choice values 1,2,3,..,p where stock ranked p should get the highest portfolio weight. (worst is sold) The outOFsamp function then calls the function ‘rank2sell,’ which uses these rank choice numbers to the selected ‘maxChosen’ stocks. The allocation is linearly declining. For example, it is 1/10, 2/10, 3/10, and 4/10, with the worst return stock (top choice for selling) receiving highest proportion of the capital designated for selling. Each choice of ‘pctOut’ rows of the ‘mtx’ data yields an outOFsamp return for each of the seven portfolio selection algorithms. These outOFsamp return computations are repeated reps times. A new random selection of ‘pctOut’ rows (must be 2 or more) of data is made for each repetition. We set reps=20 by default. The low default is set to save processing time in early phases, but we recommend reps=100+. The final choice of stock-selling algorithm out of seven is suggested by the average out-of-sample return over the ‘reps’ repetitions. This function is sell version of outOFsamp().

Usage

outOFsell(mtx, pctOut = 5, reps = 10, seed = 23, maxChosen = 2, verbo = FALSE)

Arguments

mtx

matrix size n by p of data on n returns from p stocks
pctOut

percent of n randomly chosen rows left out as out-of-sample, default=5 percent. One must leave out at least two rows of data
reps

number of random repetitions of left-out rows over which we average the out-of-sample performance of a stock-picking algorithm, default reps=20
seed

seed for random number generation, default =23
maxChosen

number of stocks (out of p) with nonzero weights in the portfolio
verbo

logical, TRUE means print details, default=FALSE

Value

a matrix called ‘avgRet’ with seven columns for seven stock-picking algorithms "SD1","SD2","SD3","SD4","SDAll4","decile",and "moment," containing out-of-sample average returns for linearly declining allocation in a portfolio. User needs to change rank2sell() for alternate portfolio allocations.

Note

The traditional time-series out-of-sample leaves out the last few time periods, and estimates the stock-picking model using part of the data time periods. The pandemic of 2019 has revealed that the traditional out-of-sample would have a severe bias in favor of pessimistic stock-picking algorithms. The traditional method is fundamentally flawed since it is sensitive to the trends (ups and downs) in the out-of-sample period. The method proposed here is free from such biases. The stock-picking algorithm recommended by our outOFsamp() is claimed to be robust against such biases.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

See Also

rank2sell

Examples

## Not run: 
x1=c(2,5,6,9,13,18,21,5,11,14,4,7,12,13,6,3,8,1,15,2,10,9)
x2=c(3,6,9,12,14,19,27,9,11,2,3,8,1,6,15,10,13,14,5,7,4,12)
x3=c(2,6,NA,11,13,25,25,11,9,10,12,6,4,3,2,1,7,8,5,15,14,13)
mtx=cbind(x1,x2,x3)
mtx=mtx[complete.cases(mtx),]
os=outOFsell(mtx,verbo=FALSE,maxChosen=2, reps=3)
apply(os,2,mean)
## End(Not run)

internal p1

Description

intended for internal use only

Usage

p1

Function to compute a vector of 2 lagged values of a variable from panel data.

Description

The panel data have a set of time series for each entity (e.g. country) arranged such that all time series data for one entity is together. The data for the second entity should be below the entire data for first entity. When a variable is lagged twice, special care is needed to insert NA's for the first two time points (e.g. weeks) for each entity (country).

Usage

Panel2Lag(ID, xj)

Arguments

ID

Location of the column having time identities (e.g. the week number)
xj

Data on variable to be lagged linked to ID

Value

Vector containing 2 lagged values of xj.

Note

This function is provided for convenient user modifications.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

See Also

A more general function PanelLag has examples.

Function for computing a vector of one-lagged values of xj, a variable from panel data.

Description

Panel data have a set of time series for each entity (e.g. country) arranged such that all time series data for one entity is together, and the data for the second entity should be below the entire data for first entity and so on for entities. In such a data setup, When a variable is lagged once, special care is needed to insert an NA for the first time point in the data (e.g. week) for each entity.

Usage

PanelLag(ID, xj, lag = 1)

Arguments

ID

Location of the column having time identities (e.g. week number).
xj

Data vector of variable to be lagged and is linked with the ID.
lag

Number of lags desired (lag=1 is the default).

Value

Vector containing one-lagged values of variable xj.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

## Not run: 
indiv=gl(6,12,labels=LETTERS[1:6])  
#creates A,A,A 12 times B B B also 12 times etc.
set.seed(99);cost=sample(30:90, 72, replace=TRUE)
revenu=sample(50:110, 72, replace=TRUE); month=rep(1:12,6)
df=data.frame(indiv,month,cost,revenu);head(df);tail(df)
L2cost=PanelLag(ID=month,xj=df[,'cost'], lag=2)
head(L2cost)
tail(L2cost)

gmcmtx0(cbind(revenu,cost,L2cost))

gmcxy_np(revenu,cost)

## End(Not run)

Generalized partial correlation coefficients between Xi and Xj, after removing the effect of xk, via nonparametric regression residuals.

Description

This function uses data on two column vectors, xi, xj and a third xk which can be a vector or a matrix, usually of the remaining variables in the model, including control variables, if any. It first removes missing data from all input variables. Then, it computes residuals of kernel regression (xi on xk) and (xj on xk). The function reports the generalized correlation between two kernel residuals. This version avoids ridge type adjustment present in an older version.

Usage

parcor_ijk(xi, xj, xk)

Arguments

xi

Input vector of data for variable xi
xj

Input vector of data for variable xj
xk

Input data for variables in xk, usually control variables

Value

ouij

Generalized partial correlation Xi with Xj (=cause) after removing xk
ouji

Generalized partial correlation Xj with Xi (=cause) after removing xk

allowing for control variables.

Note

This function calls kern,

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

See Also

See parcor_linear.

Examples

## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
options(np.messages=FALSE)
parcor_ijk(x[,1], x[,2], x[,3])

## End(Not run)#'

Generalized partial correlation coefficient between Xi and Xj after removing the effect of all others. (older version, deprecated)

Description

This function uses a generalized correlation matrix R* as input to compute generalized partial correlations between XiX_i and XjX_j where j can be any one of the remaining variables. Computation removes the effect of all other variables in the matrix. The user is encouraged to remove all known irrelevant rows and columns from the R* matrix before submitting it to this function.

Usage

parcor_ijkOLD(x, i, j)

Arguments

x

Input a p by p matrix R* of generalized correlation coefficients.
i

A column number identifying the first variable.
j

A column number identifying the second variable.

Value

ouij

Partial correlation Xi with Xj (=cause) after removing all other X's
ouji

Partial correlation Xj with Xi (=cause) after removing all other X's
myk

A list of column numbers whose effect has been removed

Note

This function calls minor, and cofactor and is called by parcor_ridge.

Examples

## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
colnames(x)=c('V1', 'v2', 'V3')
gm1=gmcmtx0(x)
parcor_ijkOLD(gm1, 2,3)

## End(Not run)#'

Partial correlation coefficient between Xi and Xj after removing the linear effect of all others.

Description

This function uses a symmetric correlation matrix R as input to compute usual partial correlations between XiX_i and XjX_j where j can be any one of the remaining variables. Computation removes the effect of all other variables in the matrix. The user is encouraged to remove all known irrelevant rows and columns from the R matrix before submitting it to this function.

Usage

parcor_linear(x, i, j)

Arguments

x

Input a p by p matrix R of symmetric correlation coefficients.
i

A column number identifying the first variable.
j

A column number identifying the second variable.

Value

ouij

Partial correlation Xi with Xj after removing all other X's
ouji

Partial correlation Xj with Xi after removing all other X's
myk

A list of column numbers whose effect has been removed

Note

This function calls minor, and cofactor

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

See Also

See parcor_ijk for generalized partial correlation coefficients useful for causal path determinations.

Examples

## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
colnames(x)=c('V1', 'v2', 'V3')
c1=cor(x)
parcor_linear(c1, 2,3)

## End(Not run)

Compute generalized (ridge-adjusted) partial correlation coefficients from matrix R*. (deprecated)

Description

This function calls parcor_ijkOLD function which uses a generalized correlation matrix R* as input to compute generalized partial correlations between XiX_i and XjX_j where j can be any one of the remaining variables. Computation removes the effect of all other variables in the matrix. It further adjusts the resulting partial correlation coefficients to be in the appropriate [-1,1] range by using an additive constant in the fashion of ridge regression.

Usage

parcor_ridg(gmc0, dig = 4, idep = 1, verbo = FALSE, incr = 3)

Arguments

gmc0

This must be a p by p matrix R* of generalized correlation coefficients.
dig

The number of digits for reporting (=4, default)
idep

The column number of the first variable (=1, default)
verbo

Make this TRUE for detailed printing of computational steps
incr

incremental constant for iteratively adjusting ‘ridgek’ where ridgek is the constant times the identity matrix used to make sure that the gmc0 matrix is positive definite. If not iteratively increas the incr till all partial correlations are within the [-1,1] interval.

Value

A five column ‘out’ matrix containing partials. The first column has the name of the idep variable. The second column has the name of the j variable, while the third column has r*(i,j | k). The 4-th column has r*(j,i | k) (denoted partji), and the 5-th column has rijMrji, that is the difference in absolute values (abs(partij) - abs(partji)).

Note

The ridgek constant created by the function during the first round may not be large enough to make sure that that other pairs of r*(i,j | k) are within the [-1,1] interval. The user may have to choose a suitably larger input incr to get all relevant partial correlation coefficients in the correct [-1,1] interval.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlations and Instantaneous Causality for Data Pairs Benchmark,' (March 8, 2015) https://www.ssrn.com/abstract=2574891

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Vinod, H. D. "A Survey of Ridge Regression and Related Techniques for Improvements over Ordinary Least Squares," Review of Economics and Statistics, Vol. 60, February 1978, pp. 121-131.

See Also

See Also parcor_ijkOLD.

Examples

set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10  #x is partly indep and partly affected by z
y=1+2*x+3*z+rnorm(10)# y depends on x and z not vice versa
mtx=cbind(x,y,z)
g1=gmcmtx0(mtx)
parcor_ijkOLD(g1,1,2) # ouji> ouij implies i=x is the cause of j=y
parcor_ridg(g1,idep=1)
 
   
## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
colnames(x)=c('V1', 'v2', 'V3')
gm1=gmcmtx0(x)
parcor_ridg(gm1, idep=1)

## End(Not run)

Block version of generalized partial correlation coefficients between Xi and Xj, after removing the effect of xk, via nonparametric regression residuals.

Description

This function uses data on two column vectors, xi, xj and a third xk which can be a vector or a matrix, usually of the remaining variables in the model, including control variables, if any. It first removes missing data from all input variables. Then, it computes residuals of kernel regression (xi on xk) and (xj on xk). This is a block version of parcor_ijk.

Usage

parcorBijk(xi, xj, xk, blksiz = 10)

Arguments

xi

Input vector of data for variable xi
xj

Input vector of data for variable xj
xk

Input data for variables in xk, usually control variables
blksiz

block size, default=10, if chosen blksiz >n, where n=rows in matrix then blksiz=n. That is, no blocking is done

Value

ouij

Generalized partial correlation Xi with Xj (=cause) after removing xk
ouji

Generalized partial correlation Xj with Xi (=cause) after removing xk

allowing for control variables.

Note

This function calls kern,

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

See Also

See parcor_ijk.

Examples

## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
options(np.messages=FALSE)
parcorBijk(x[,1], x[,2], x[,3], blksi=10)

## End(Not run)#'

Block version reports many generalized partial correlation coefficients allowing control variables.

Description

This function calls a block version parcorBijk of the function which uses original data to compute generalized partial correlations between XidepX_{idep} and XjX_j where j can be any one of the remaining variables in the input matrix mtx. Partial correlations remove the effect of variables XkX_k other than XiX_i and XjX_j. Calculation further allows for the presence of control variable(s) (if any) to remain always outside the input matrix and whose effect is also removed in computing partial correlations.

Usage

parcorBMany(mtx, ctrl = 0, dig = 4, idep = 1, blksiz = 10, verbo = FALSE)

Arguments

mtx

Input data matrix with at least 3 columns.
ctrl

Input vector or matrix of data for control variable(s), default is ctrl=0 when control variables are absent
dig

The number of digits for reporting (=4, default)
idep

The column number of the dependent variable (=1, default)
blksiz

block size, default=10, if chosen blksiz >n, where n=rows in matrix then blksiz=n. That is, no blocking is done
verbo

Make this TRUE for detailed printing of computational steps

Value

A five column ‘out’ matrix containing partials. The first column has the name of the idep variable. The second column has the name of the j variable, while the third column has partial correlation coefficients r*(i,j | k).The last column reports the absolute difference between two partial correlations.

Note

This function reports all partial correlation coefficients, while avoiding ridge type adjustment.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlations and Instantaneous Causality for Data Pairs Benchmark,' (March 8, 2015) https://www.ssrn.com/abstract=2574891

Vinod, H. D. (2021) 'Generalized, Partial and Canonical Correlation Coefficients' Computational Economics, 59(1), 1–28.

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

See Also

See Also parcor_ijk, parcorMany.

Examples

set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10  #x is partly indep and partly affected by z
y=1+2*x+3*z+rnorm(10)# y depends on x and z not vice versa
mtx=cbind(x,y,z)
parcorBMany(mtx, blksiz=10)
 
   
## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
colnames(x)=c('V1', 'v2', 'V3')
parcorBMany(x, idep=1)

## End(Not run)

Generalized partial correlation coefficients between Xi and Xj, after removing the effect of Xk, via OLS regression residuals.

Description

This function uses data on two column vectors, xi, xj, and a third set xk, which can be a vector or a matrix. xk usually has the remaining variables in the model, including control variables, if any. This function first removes missing data from all input variables. Then, it computes residuals of OLS (no kernel) regression (xi on xk) and (xj on xk). This hybrid version uses both OLS and then generalized correlation among OLS residuals. This solves the potential problem of having too little information content in kernel regression residuals, since kernel fits are sometimes too close, especially when there are many variables in xk.

Usage

parcorHijk(xi, xj, xk)

Arguments

xi

Input vector of data for variable xi
xj

Input vector of data for variable xj
xk

Input data for all variables in xk, usually control variables

Value

ouij

Generalized partial correlation Xi with Xj (=cause) after removing xk
ouji

Generalized partial correlation Xj with Xi (=cause) after removing xk

allowing for control variables.

Note

This function calls kern,

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

See Also

See parcor_ijk.

Examples

## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
options(np.messages=FALSE)
parcorHijk(x[,1], x[,2], x[,3])

## End(Not run)#'

Generalized partial correlation coefficients between Xi and Xj,

Description

The 2 in the name of the function means second version. The H in the function name means hybrid. This removes the effect of Xk, via OLS regression residuals. This function uses data on two column vectors, xi, xj, and a third set xk, which can be a vector or a matrix, usually of the remaining variables in the model, including control variables, if any. It first removes missing data from all input variables. Then, it computes residuals of OLS regression (xi on xk) and (xj on xk). The function reports the generalized correlation between two OLS residuals. This hybrid version uses both OLS and then generalized correlation among OLS residuals. This second version works when 'parcorVecH' fails. It is called by the function ‘parcorVecH2’.

Usage

parcorHijk2(xi, xj, xk)

Arguments

xi

Input vector of data for variable xi
xj

Input vector of data for variable xj
xk

Input data for variables in xk, usually control variables

Value

ouij

Generalized partial correlation Xi with Xj (=cause) after removing xk
ouji

Generalized partial correlation Xj with Xi (=cause) after removing xk

allowing for control variables.

Note

This function calls kern,

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

See Also

See parcor_ijk.

Examples

## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
options(np.messages=FALSE)
parcorHijk2(x[,1], x[,2], x[,3])

## End(Not run)#'

Report many generalized partial correlation coefficients allowing control variables.

Description

This function calls parcor_ijk function which uses original data to compute generalized partial correlations between XidepX_{idep} and XjX_j where j can be any one of the remaining variables in the input matrix mtx. Partial correlations remove the effect of variables xkx_k other than XiX_i and XjX_j. Calculation further allows for the presence of control variable(s) (if any) to remain always outside the input matrix and whose effect is also removed in computing partial correlations.

Usage

parcorMany(mtx, ctrl = 0, dig = 4, idep = 1, verbo = FALSE)

Arguments

mtx

Input data matrix with at least 3 columns.
ctrl

Input vector or matrix of data for control variable(s), default is ctrl=0 when control variables are absent
dig

The number of digits for reporting (=4, default)
idep

The column number of the first variable (=1, default)
verbo

Make this TRUE for detailed printing of computational steps

Value

A five column ‘out’ matrix containing partials. The first column has the name of the idep variable. The second column has the name of the j variable, while the third column has partial correlation coefficients r*(i,j | k). The last column reports the absolute difference between two partial correlations.

Note

This function reports all partial correlation coefficients, while avoiding ridge type adjustment.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlations and Instantaneous Causality for Data Pairs Benchmark,' (March 8, 2015) https://www.ssrn.com/abstract=2574891

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

See Also

See Also parcor_ijk.

Examples

set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10  #x is partly indep and partly affected by z
y=1+2*x+3*z+rnorm(10)# y depends on x and z not vice versa
mtx=cbind(x,y,z)
parcorMany(mtx)
 
   
## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
colnames(x)=c('V1', 'v2', 'V3')
parcorMany(x, idep=1)

## End(Not run)

Matrix of generalized partial correlation coefficients, always leaving out control variables, if any.

Description

This function calls parcor_ijk function which uses original data to compute generalized partial correlations between XiX_i and XjX_j where j can be any one of the remaining variables in the input matrix mtx. Partial correlations remove the effect of variables xkx_k other than XiX_i and XjX_j. Calculation further allows for the presence of control variable(s) (if any) to remain always outside the input matrix and whose effect is also removed in computing partial correlations.

Usage

parcorMtx(mtx, ctrl = 0, dig = 4, verbo = FALSE)

Arguments

mtx

Input data matrix with p columns. p is at least 3 columns.
ctrl

Input vector or matrix of data for control variable(s), default is ctrl=0 when control variables are absent
dig

The number of digits for reporting (=4, default)
verbo

Make this TRUE for detailed printing of computational steps

Value

A p by p ‘out’ matrix containing partials r*(i,j | k). and r*(j,i | k).

Note

We want to get all partial correlation coefficient pairs removing other column effects. Vinod (2018) shows why one needs more than one criterion to decide the causal paths or exogeneity.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlations and Instantaneous Causality for Data Pairs Benchmark,' (March 8, 2015) https://www.ssrn.com/abstract=2574891

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Vinod, H. D. 'New Exogeneity Tests and Causal Paths,' (June 30, 2018). Available at SSRN: https://www.ssrn.com/abstract=3206096

See Also

See Also parcor_ijk.

Examples

set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10  #x is partly indep and partly affected by z
y=1+2*x+3*z+rnorm(10)# y depends on x and z not vice versa
mtx=cbind(x,y,z)
parcorMtx(mtx)
 
   
## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
colnames(x)=c('V1', 'v2', 'V3')
parcorMtx(x)

## End(Not run)

Silently compute generalized (ridge-adjusted) partial correlation coefficients from matrix R*.

Description

This function calls parcor_ijkOLD function which uses a generalized correlation matrix R* as input to compute generalized partial correlations between XiX_i and XjX_j where j can be any one of the remaining variables. Computation removes the effect of all other variables in the matrix. It further adjusts the resulting partial correlation coefficients to be in the appropriate [-1,1] range by using an additive constant in the fashion of ridge regression.

Usage

parcorSilent(gmc0, dig = 4, idep = 1, verbo = FALSE, incr = 3)

Arguments

gmc0

This must be a p by p matrix R* of generalized correlation coefficients.
dig

The number of digits for reporting (=4, default)
idep

The column number of the first variable (=1, default)
verbo

Make this TRUE for detailed printing of computational steps
incr

incremental constant for iteratively adjusting ‘ridgek’ where ridgek is the constant times the identity matrix used to make sure that the gmc0 matrix is positive definite. If not, this function iteratively increases the incr till relevant partial correlations are within the [-1,1] interval.

Value

A five column ‘out’ matrix containing partials. The first column has the name of the idep variable. The second column has the name of the j variable, while the third column has r*(i,j | k). The 4-th column has r*(j,i | k) (denoted partji), and the 5-th column has rijMrji, that is the difference in absolute values (abs(partij) - abs(partji)).

Note

The ridgek constant created by the function during the first round may not be large enough to make sure that that other pairs of r*(i,j | k) are within the [-1,1] interval. The user may have to choose a suitably larger input incr to get all relevant partial correlation coefficients in the correct [-1,1] interval.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlations and Instantaneous Causality for Data Pairs Benchmark,' (March 8, 2015) https://www.ssrn.com/abstract=2574891

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Vinod, H. D. "A Survey of Ridge Regression and Related Techniques for Improvements over Ordinary Least Squares," Review of Economics and Statistics, Vol. 60, February 1978, pp. 121-131.

See Also

See Also parcor_ijk for a better version using original data as input.

Examples

set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10  #x is partly indep and partly affected by z
y=1+2*x+3*z+rnorm(10)# y depends on x and z not vice versa
mtx=cbind(x,y,z)
g1=gmcmtx0(mtx)
parcor_ijkOLD(g1,1,2) # ouji> ouij implies i=x is the cause of j=y
parcor_ridg(g1,idep=1)
parcorSilent(g1,idep=1)
 
   
## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
colnames(x)=c('V1', 'v2', 'V3')
gm1=gmcmtx0(x)
parcorSilent(gm1, idep=1)

## End(Not run)

Vector of generalized partial correlation coefficients (GPCC), always leaving out control variables, if any.

Description

This function calls parcor_ijk function which uses original data to compute generalized partial correlations between XiX_i, the dependent variable, and XjX_j which is the current regressor of interest. Note that j can be any one of the remaining variables in the input matrix mtx. Partial correlations remove the effect of variables XkX_k other than XiX_i and XjX_j. Calculation merges control variable(s) (if any) into XkX_k. Let the remainder effect from kernel regressions of XiX_i on XkX_k equal the residuals u*(i,k). Analogously define u*(j,k). (asterisk for kernel regressions) Now partial correlation is generalized correlation between u*(i,k) and u*(j,k). Calculation merges control variable(s) (if any) into XkX_k.

Usage

parcorVec(mtx, ctrl = 0, verbo = FALSE, idep = 1)

Arguments

mtx

Input data matrix with p (> or = 3) columns
ctrl

Input vector or matrix of data for control variable(s), default is ctrl=0 when control variables are absent
verbo

Make this TRUE for detailed printing of computational steps
idep

The column number of the dependent variable (=1, default)

Value

A p by 1 ‘out’ vector containing partials r*(i,j | k).

Note

Generalized Partial Correlation Coefficients (GPCC) allow comparison of the relative contribution of each XjX_j to the explanation of XiX_i, because GPCC are scale-free pure numbers

We want to get all partial correlation coefficient pairs removing other column effects. Vinod (2018) shows why one needs more than one criterion to decide the causal paths or exogeneity.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlations and Instantaneous Causality for Data Pairs Benchmark,' (March 8, 2015) https://www.ssrn.com/abstract=2574891

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Vinod, H. D. 'New Exogeneity Tests and Causal Paths,' (June 30, 2018). Available at SSRN: https://www.ssrn.com/abstract=3206096

Vinod, H. D. (2021) 'Generalized, Partial and Canonical Correlation Coefficients' Computational Economics, 59(1), 1–28.

See Also

See Also parcor_ijk.

See Also a hybrid version parcorVecH.

Examples

set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10  #x is partly indep and partly affected by z
y=1+2*x+3*z+rnorm(10)# y depends on x and z not vice versa
mtx=cbind(x,y,z)
parcorVec(mtx)
 
   
## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
colnames(x)=c('V1', 'v2', 'V3')#some names needed
parcorVec(x)

## End(Not run)

Vector of hybrid generalized partial correlation coefficients.

Description

This is a hybrid version of parcorVec subtracting only the linear effects (OLS residuals instead of kernel regression residuals), but using the generalized correlation between the OLS residuals for the last stage of the generalized partial correlation.

Usage

parcorVecH(mtx, ctrl = 0, dig = 4, verbo = FALSE, idep = 1)

Arguments

mtx

Input data matrix with p (> or = 3) columns, the first column must have the dependent variable
ctrl

Input vector or matrix of data for control variable(s), default is ctrl=0 when control variables are absent
dig

The number of digits for reporting (=4, default)
verbo

Make this TRUE for detailed printing of computational steps
idep

The column number of the dependent variable (=1, default)

Details

This function calls parcor_ijk function, which uses original data to compute generalized partial correlations between XiX_i, the dependent variable, and XjX_j, which is the current regressor of interest. Note that j can be any one of the remaining variables in the input matrix mtx. Partial correlations remove the effect of variables XkX_k other than XiX_i and XjX_j. Calculation merges control variable(s) (if any) into XkX_k. Let the remainder effect from OLS regressions of XiX_i on XkX_k equal the residuals u(i,k). Analogously define u(j,k). It is a hybrid of OLS and generalized. Finally, partial correlation is generalized (kernel) correlation between u(i,k) and u(j,k).

Value

A p by 1 ‘out’ vector containing hybrid partials r*(i,j | k).

Note

Hybrid Generalized Partial Correlation Coefficients (HGPCC) allow comparison of the relative contribution of each XjX_j to the explanation of XiX_i, because HGPCC has scale-free pure numbers.

We want to get all partial correlation coefficient pairs removing other column effects. Vinod (2018) shows why one needs more than one criterion to decide the causal paths or exogeneity.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlations and Instantaneous Causality for Data Pairs Benchmark,' (March 8, 2015) https://www.ssrn.com/abstract=2574891

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Vinod, H. D. 'New Exogeneity Tests and Causal Paths,' (June 30, 2018). Available at SSRN: https://www.ssrn.com/abstract=3206096

Vinod, H. D. (2021) 'Generalized, Partial and Canonical Correlation Coefficients' Computational Economics, 59(1), 1–28.

See Also

See Also parcor_ijk.

See Also parcorVec.

Examples

set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10  #x is partly indep and partly affected by z
y=1+2*x+3*z+rnorm(10)# y depends on x and z not vice versa
mtx=cbind(x,y,z)
parcorVecH(mtx)
 
   
## Not run: 
set.seed(34);mtx=matrix(sample(1:600)[1:80],ncol=4)
colnames(mtx)=c('V1', 'v2', 'V3', 'V4')
parcorVecH(mtx,verbo=TRUE, idep=2)

## End(Not run)

Vector of hybrid generalized partial correlation coefficients.

Description

This is a second version to be used when ‘parcorVecH’ fails. (H=hybrid). This hybrid version of parcorVec subtracting only linear effects but using generlized correlation between OLS residuals

Usage

parcorVecH2(mtx, dig = 4, verbo = FALSE, idep = 1)

Arguments

mtx

Input data matrix with p (> or = 3) columns, first column must have the dependent variable
dig

The number of digits for reporting (=4, default)
verbo

Make this TRUE for detailed printing of computational steps
idep

The column number of the dependent variable (=1, default)

Details

This function calls parcorHijk2 function which uses original data to compute generalized partial correlations between XiX_i, the dependent variable, and XjX_j which is the current regressor of interest. Note that j can be any one of the remaining variables in the input matrix mtx. Partial correlations remove the effect of variables XkX_k other than XiX_i and XjX_j. Calculation merges control variable(s) (if any) into XkX_k. Let the remainder effect from OLS regressions of XiX_i on XkX_k equal the residuals u(i,k). Analogously define u(j,k). It is a hybrid of OLS and generalized. Finally, partial correlation is generalized (kernel) correlation between u(i,k) and u(j,k).

Value

A p by 1 ‘out’ vector containing hybrid partials r*(i,j | k).

Note

Hybrid Generalized Partial Correlation Coefficients (HGPCC) allow comparison of the relative contribution of each XjX_j to the explanation of XiX_i, because HGPCC are scale-free pure numbers.

We want to get all partial correlation coefficient pairs removing other column effects. Vinod (2018) shows why one needs more than one criterion to decide the causal paths or exogeneity.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlations and Instantaneous Causality for Data Pairs Benchmark,' (March 8, 2015) https://www.ssrn.com/abstract=2574891

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Vinod, H. D. 'New Exogeneity Tests and Causal Paths,' (June 30, 2018). Available at SSRN: https://www.ssrn.com/abstract=3206096

Vinod, H. D. (2021) 'Generalized, Partial and Canonical Correlation Coefficients' Computational Economics, 59(1), 1–28.

See Also

See Also parcor_ijk.

See Also parcorVec.

Examples

set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10  #x is partly indep and partly affected by z
y=1+2*x+3*z+rnorm(10)# y depends on x and z not vice versa
mtx=cbind(x,y,z)
parcorVecH2(mtx)
 
   
## Not run: 
set.seed(34);mtx=matrix(sample(1:600)[1:80],ncol=4)
colnames(mtx)=c('V1', 'v2', 'V3', 'V4')
parcorVecH2(mtx,verbo=TRUE, idep=2)

## End(Not run)

Compute the bootstrap probability of correct causal direction.

Description

Maximum entropy bootstrap (‘meboot’) package is used for statistical inference regarding δ\delta which equals GMC(X|Y)-GMC(Y|X) defined by Zheng et al (2012). The bootstrap provides an approximation to chances of correct determination of the causal direction.

Usage

pcause(x, y, n999 = 999)

Arguments

x

Vector of x data
y

Vector of y data
n999

Number of bootstrap replications (default=999)

Value

P(cause) the bootstrap proportion of correct causal determinations.

Note

'pcause' is computer intensive and generally slow. It is better to use it at a later stage in the investigation when a preliminary causal determination is already made. Its use may slow the exploratory phase. In my experience, if P(cause) is less than 0.55, there is a cause for concern.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Zheng, S., Shi, N.-Z., and Zhang, Z. (2012). Generalized measures of correlation for asymmetry, nonlinearity, and beyond. Journal of the American Statistical Association, vol. 107, pp. 1239-1252.

Vinod, H. D. and Lopez-de-Lacalle, J. (2009). 'Maximum entropy bootstrap for time series: The meboot R package.' Journal of Statistical Software, Vol. 29(5), pp. 1-19.

Examples

## Not run: 
set.seed(34);x=sample(1:10);y=sample(2:11)
pcause(x,y,n999=29)

data('EuroCrime')
attach(EuroCrime)
pcause(crim,off,n999=29)

## End(Not run)

Create a 3D pillar chart to display (x, y, z) data coordinate surface.

Description

Given data on (x, y, z) coordinate values of a 3D surface, one can directly plot a 3D plot with pins of the height z. By contrast, this function fattens each pin by creating pillars near each z value by adding and subtracting small amounts of dz. By eliminating the pins of the height z, this depicts pillars that better resemble a surface. It uses the wireframe() function of the ‘lattice’ package to do the plotting.

Usage

pillar3D(
  z = c(657, 936, 1111, 1201),
  x = c(280, 542, 722, 1168),
  y = c(162, 214, 186, 246),
  drape = TRUE,
  xlab = "y",
  ylab = "x",
  zlab = "z",
  mymain = "Pillar Chart"
)

Arguments

z

z-coordinate values
x

x-coordinate values
y

y-coordinate values
drape

logical value, default drape=TRUE to give color to heights
xlab

default "x" label on the x axis
ylab

default "y" label on the y axis
zlab

default "z" label on the z axis
mymain

default "Pillar Chart" main label on the plot

Details

For additional plotting features the user should type ‘pillar3D()’ on the R console to get my code and adjust my wireframe() function defaults.

Value

A 3D plot

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

## Not run: 
pillar3D())
## End(Not run)

Intermediate weighting function giving Non-Expected Utility theory weights.

Description

Computes cumulative probabilities and difference between consecutive cumulative probabilities described in Vinod (2008) textbook. This is a simpler version of the version in the book without mapping to non-expected utility theory weights as explained in Vinod (2008).

Usage

prelec2(n)

Arguments

n

A (usually small) integer.

Value

x

sequence 1:n
p

probabilities p= x[i]/n
pdif

consecutive differences p[i] - p[i - 1]

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Hands-On Intermediate Econometrics Using R' (2008) World Scientific Publishers: Hackensack, NJ. https://www.worldscientific.com/worldscibooks/10.1142/12831

Examples

## Not run: prelec2(10)

Compute probability of positive or negative sign from bootPairs output

Description

If there are p columns of data, probSign produces a p-1 by 1 vector of probabilities of correct signs assuming that the mean of n999 values has the correct sign and assuming that m of the 'sum' index values inside the range [-tau, tau] are neither positive nor negative but indeterminate or ambiguous (being too close to zero). That is, the denominator of P(+1) or P(-1) is (n999-m) if m signs are too close to zero.

Usage

probSign(out, tau = 0.476)

Arguments

out

output from bootPairs with p-1 columns and n999 rows
tau

threshold to determine what value is too close to zero, default tau=0.476 is equivalent to 15 percent threshold for the unanimity index ui

Value

sgn When mtx has p columns, sgn reports pairwise p-1 signs representing (fixing the first column in each pair) the average sign after averaging the output of of bootPairs(mtx) (a n999 by p-1 matrix) each containing resampled ‘sum’ values summarizing the weighted sums associated with all three criteria from the function silentPairs(mtx) applied to each bootstrap sample separately. #'

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. and Lopez-de-Lacalle, J. (2009). 'Maximum entropy bootstrap for time series: The meboot R package.' Journal of Statistical Software, Vol. 29(5), pp. 1-19.

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See Also silentPairs.

Examples

## Not run: 
options(np.messages = FALSE)
set.seed(34);x=sample(1:10);y=sample(2:11)
bb=bootPairs(cbind(x,y),n999=29)
probSign(bb,tau=0.476) #gives summary stats for n999 bootstrap sum computations

bb=bootPairs(airquality,n999=999);options(np.messages=FALSE)
probSign(bb,tau=0.476)#signs for n999 bootstrap sum computations

data('EuroCrime')
attach(EuroCrime)
bb=bootPairs(cbind(crim,off),n999=29) #col.1= crim causes off 
#hence positive signs are more intuitively meaningful.
#note that n999=29 is too small for real problems, chosen for quickness here.
probSign(bb,tau=0.476)#signs for n999 bootstrap sum computations

## End(Not run)

Compute the portfolio return knowing the rank of a stock in the input ‘mtx’.

Description

This function computes the return earned knowing the rank of a stock in the input mtx of stock returns. For example, mtx has p=28 Dow Jones stocks over n=169 monthly returns. Portfolio weights are assumed to be linearly declining. If maxChosen=4, the weights are 4/10, 3/10, 2/10 and 1/10, which add up to unity. These portfolio weights are assigned in reverse order in the sense that first chosen stock (choice rank =1) gets portfolio weight=4/10. The function computes return from the stocks using the ‘myrank’ argument.

Usage

rank2return(mtx, myrank, maxChosen = 0, pctChoose = 20, verbo = FALSE)

Arguments

mtx

a matrix with n rows (number of returns) p columns (number of stocks)
myrank

vector of p integers listing the rank of each stock, 1=best
maxChosen

number of stocks in the portfolio (with nonzero weights) default=0. When maxChosen=0, we let pctChoose determine the maxChosen
pctChoose

percent of p stocks chosen inside the portfolio, default=20
verbo

logical if TRUE, print, default=TRUE

Value

average return from the linearly declining portfolio implied by the myrank vector.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

See Also

outOFsamp

Compute the portfolio return knowing the rank of a stock in the input ‘mtx’. This function computes the return earned knowing the rank of a stock computed elsewhere and named myrank associate with the data columns in the input mtx of stock returns. For example, mtx has p=28 Dow Jones stocks over n=169 monthly returns. Portfolio weights are assumed to be linearly declining. If maxChosen=4, the weights are 1/10, 2/10, 3/10 and 4/10, which add up to unity. These portfolio weights are assigned in their order in the sense that first chosen stock (choice rank =p) gets portfolio weight=4/10. The function computes return from the stocks using the ‘myrank’ argument. This helps in assessing out-of-sample performance of (short) the strategy of selling lowest ranking stocks. It is mostly for internal use by outOFsell(). This is a sell version of rank2return().

Description

Compute the portfolio return knowing the rank of a stock in the input ‘mtx’.

This function computes the return earned knowing the rank of a stock computed elsewhere and named myrank associate with the data columns in the input mtx of stock returns. For example, mtx has p=28 Dow Jones stocks over n=169 monthly returns. Portfolio weights are assumed to be linearly declining. If maxChosen=4, the weights are 1/10, 2/10, 3/10 and 4/10, which add up to unity. These portfolio weights are assigned in their order in the sense that first chosen stock (choice rank =p) gets portfolio weight=4/10. The function computes return from the stocks using the ‘myrank’ argument. This helps in assessing out-of-sample performance of (short) the strategy of selling lowest ranking stocks. It is mostly for internal use by outOFsell(). This is a sell version of rank2return().

Usage

rank2sell(mtx, myrank, maxChosen = 0, pctChoose = 20, verbo = FALSE)

Arguments

mtx

a matrix with n rows (number of returns) p columns (number of stocks)
myrank

vector of p integers listing the rank of each stock, 1=best
maxChosen

number of stocks in the portfolio (with nonzero weights) default=0. When maxChosen=0, we let pctChoose determine the maxChosen
pctChoose

percent of p stocks chosen inside the portfolio, default=20
verbo

logical if TRUE, print, default=TRUE

Value

average return from the linearly declining portfolio implied by the myrank vector.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

See Also

outOFsell

internal rhs.lag2

Description

intended for internal use only

Usage

rhs.lag2

internal rhs1

Description

intended for internal use only

Usage

rhs1

internal ridgek

Description

intended for internal use only

Usage

ridgek

internal rij

Description

intended for internal use only

Usage

rij

internal rijMrji

Description

intended for internal use only

Usage

rijMrji

internal rji

Description

intended for internal use only

Usage

rji

internal rrij

Description

intended for internal use only

Usage

rrij

internal rrji

Description

intended for internal use only

Usage

rrji

Function to compute generalized correlation coefficients r*(x,y).

Description

Uses Vinod (2015) definition of generalized (asymmetric) correlation coefficients. It requires kernel regression of x on y obtained by using the ‘np’ package. It also reports usual Pearson correlation coefficient r and p-value for testing the null hypothesis that (population r)=0.

Usage

rstar(x, y)

Arguments

x

Vector of data on the dependent variable
y

Vector of data on the regressor

Value

Four objects created by this function are:

corxy

r*x|y or regressing x on y
coryx

r*y|x or regressing y on x
pearson.r

Pearson's product moment correlation coefficient
pv

The p-value for testing the Pearson r

Note

This function needs the kern function which in turn needs the np package.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

See Also

See Also gmcmtx0 and gmcmtxBlk.

Examples

x=sample(1:30);y=sample(1:30); rstar(x,y)

internal sales2Lag

Description

intended for internal use only

Usage

sales2Lag

internal salesLag

Description

intended for internal use only

Usage

salesLag

internal seed

Description

intended for internal use only

Usage

seed

internal sgn.e0

Description

intended for internal use only

Usage

sgn.e0

No-print kernel-causality unanimity score matrix with optional control variables

Description

Allowing input matrix of control variables and missing data, this function produces a p by p matrix summarizing the results, where the estimated signs of stochastic dominance order values (+1, 0, –1) are weighted by wt=c(1.2, 1.1, 1.05, 1) to compute an overall result for all orders of stochastic dominance by a weighted sum for the criteria Cr1 and Cr2 and added to the Cr3 estimate as: (+1, 0, –1). Final weighted index is always in the range [–3.175, 3.175]. It is converted to the more intuitive range [–100, 100].

Usage

silentMtx(mtx, ctrl = 0, dig = 6, wt = c(1.2, 1.1, 1.05, 1), sumwt = 4)

Arguments

mtx

The data matrix with p columns. Denote x1 as the first column which is fixed and then paired with all other columns, say: x2, x3, .., xp, one by one for the purpose of flipping with x1. p must be 2 or more
ctrl

data matrix for designated control variable(s) outside causal paths
dig

Number of digits for reporting (default dig=6).
wt

Allows user to choose a vector of four alternative weights for SD1 to SD4.
sumwt

Sum of weights can be changed here =4(default).

Details

The reason for slightly declining weights on the signs from SD1 to SD4 is simply that the local mean comparisons implicit in SD1 are known to be more reliable than local variance implicit in SD2, local skewness implicit in SD3 and local kurtosis implicit in SD4. Why are higher moment estimates less reliable? The higher power of the deviations from the mean needed in their computations lead to greater sampling variability. The summary results for all three criteria are reported in a vector of numbers internally called crall:

Value

With p columns in mtx argument to this function, x1 can be paired with a total of p-1 columns (x2, x3, .., xp). Note we never flip any of the control variables with x1. This function produces i=1,2,..,p-1 numbers representing the summary sign, or ‘sum’ from the signs sg1 to sg3 associated with the three criteria: Cr1, Cr2 and Cr3. Note that sg1 and sg2 themselves are weighted signs using weighted sum of signs from four orders of stochastic dominance. In general, a positive sign in the i-th location of the ‘sum’ output of this function means that x1 is the kernel cause while the variable in (i+1)-th column of mtx is the ‘effect’ or ‘response’ or ‘endogenous.’ The magnitude represents the strength (unanimity) of the evidence for a particular sign. Conversely a negative sign in the i-th location of the ‘sum’ output of this function means that that the first variable listed as the input to this function is the ‘effect,’ while the variable in (i+1)-th column of mtx is the exogenous kernel cause. This function is a summary of someCPairs allowing for control variables.

Note

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers. The command attach(EuroCrime); silentPairs(cbind(crim,off)) returns only one number: 3.175, implying a high unanimity strength. The index 3.175 is the highest. The positive sign of the index suggests that ‘crim’ variable in the first column of the matrix input to this function kernel causes ‘off’ in the second column of the matrix argument mtx to this function.

Interpretation of the output matrix produced by this function is as follows. A negative index means the variable named in the column kernel-causes the variable named in the row. A positive index means the row name variable kernel-causes the column name variable. The abs(index) measures unanimity by three criteria, Cr1 to Cr3 representing the strength of evidence for the identified causal path.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

H. D. Vinod 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See silentPairs.

See someCPairs, some0Pairs

Examples

## Not run: 
options(np.messages=FALSE)
colnames(mtcars[2:ncol(mtcars)])
silentMtx(mtcars[,1:3],ctrl=mtcars[,4:5]) # mpg paired with others

## End(Not run)

options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
silentMtx(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

Older kernel-causality unanimity score matrix with optional control variables

Description

Allowing input matrix of control variables and missing data, this function produces a p by p matrix summarizing the results, where the estimated signs of stochastic dominance order values (+1, 0, –1) are weighted by wt=c(1.2, 1.1, 1.05, 1) to compute an overall result for all orders of stochastic dominance by a weighted sum for the criteria Cr1 and Cr2 and added to the Cr3 estimate as: (+1, 0, –1). Final weighted index is always in the range [–3.175, 3.175]. It is converted to the more intuitive range [–100, 100].

Usage

silentMtx0(mtx, ctrl = 0, dig = 6, wt = c(1.2, 1.1, 1.05, 1), sumwt = 4)

Arguments

mtx

The data matrix with p columns. Denote x1 as the first column which is fixed and then paired with all other columns, say: x2, x3, .., xp, one by one for the purpose of flipping with x1. p must be 2 or more
ctrl

data matrix for designated control variable(s) outside causal paths
dig

Number of digits for reporting (default dig=6).
wt

Allows user to choose a vector of four alternative weights for SD1 to SD4.
sumwt

Sum of weights can be changed here =4(default).

Details

The reason for slightly declining weights on the signs from SD1 to SD4 is simply that the local mean comparisons implicit in SD1 are known to be more reliable than local variance implicit in SD2, local skewness implicit in SD3 and local kurtosis implicit in SD4. Why are higher moment estimates less reliable? The higher power of the deviations from the mean needed in their computations lead to greater sampling variability. The summary results for all three criteria are reported in a vector of numbers internally called crall:

Value

With p columns in mtx argument to this function, x1 can be paired with a total of p-1 columns (x2, x3, .., xp). Note we never flip any of the control variables with x1. This function produces i=1,2,..,p-1 numbers representing the summary sign, or ‘sum’ from the signs sg1 to sg3 associated with the three criteria: Cr1, Cr2 and Cr3. Note that sg1 and sg2 themselves are weighted signs using weighted sum of signs from four orders of stochastic dominance. In general, a positive sign in the i-th location of the ‘sum’ output of this function means that x1 is the kernel cause while the variable in (i+1)-th column of mtx is the ‘effect’ or ‘response’ or ‘endogenous.’ The magnitude represents the strength (unanimity) of the evidence for a particular sign. Conversely a negative sign in the i-th location of the ‘sum’ output of this function means that that the first variable listed as the input to this function is the ‘effect,’ while the variable in (i+1)-th column of mtx is the exogenous kernel cause. This function allows for control variables.

Note

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers. The command attach(EuroCrime); silentPairs(cbind(crim,off)) returns only one number: 3.175, implying a high unanimity strength. The index 3.175 is the highest. The positive sign of the index suggests that ‘crim’ variable in the first column of the matrix input to this function kernel causes ‘off’ in the second column of the matrix argument mtx to this function.

Interpretation of the output matrix produced by this function is as follows. A negative index means the variable named in the column kernel-causes the variable named in the row. A positive index means the row name variable kernel-causes the column name variable. The abs(index) measures unanimity by three criteria, Cr1 to Cr3 representing the strength of evidence for the identified causal path.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

H. D. Vinod 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See silentPairs0 using older Cr1 criterion based on kernel regression local gradients.

See someCPairs, some0Pairs

Examples

## Not run: 
options(np.messages=FALSE)
colnames(mtcars[2:ncol(mtcars)])
silentMtx0(mtcars[,1:3],ctrl=mtcars[,4:5]) # mpg paired with others

## End(Not run)

## Not run: 
options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
silentMtx0(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

## End(Not run)

kernel causality (version 2) scores with control variables

Description

This function uses flipped kernel regressions to decide causal directions. This version 2 avoids Anderson's trapezoidal approximation used in ‘silenPairs.’ It calls functions: decileVote, momentVote, exactSdMtx, and summaryRank after stochastic dominance is computed. It computes an average of ranks used. The column with the “choice” rank value helps in choosing the flip having the lowest Hausman-Wu (residual times RHS regressor) and secondly the lowest absolute residual. The chosen flipped regression defines the “cause" based on the variable on its right-hand side. In portfolio selection, choice rank 1 has the highest return. Here we want low residuals and low Hausman-Wu value, hence we choose choice=2 as the desirable flip.

The function develops a unanimity index regarding the particular flip (y on xi) or (xi on y) is best. A summary of all relevant signs determines the causal direction and unanimity index among three criteria. The ‘2’ in the name of the function suggests a second implementation where exact stochastic dominance, decileVote, and momentVote algorithms are used.

Usage

silentPair2(mtx, ctrl = 0, dig = 6)

Arguments

mtx

The data matrix with p columns. Denote x1 as the first column, which is fixed in all rows of the output and then it is paired with all other columns, say: x2, x3, .., xp, one by one for the purpose of flipping with x1. p must be 2 or more
ctrl

data matrix for designated control variable(s) outside causal paths, default is ctrl=0, which means that there are no control variables used.
dig

Number of digits for reporting (default dig=6).

Value

A matrix with p columns in mtx argument to this function, x1 can be paired with a total of p-1 columns (x2, x3, .., xp). Note we never flip any of the control variables with x1. This function produces i=1,2,..,p-1 numbers representing the summary sign, or ‘sum’ from the signs sg1 to sg3 associated with the three criteria: Cr1, Cr2, and Cr3. Note that sg1 and sg2 themselves are weighted signs using a weighted sum of signs from four orders of stochastic dominance. In general, a positive sign in the i-th location of the ‘sum’ output of this function means that x1 is the kernel cause while the variable in (i+1)-th column of mtx is the ‘effect’ or ‘response’ or ‘endogenous.’ The magnitude represents the strength (unanimity) of the evidence for a particular sign. Conversely, a negative sign in the i-th location of the ‘sum’ output of this function means that the first variable listed as the input to this function is the ‘effect,’ while the variable in (i+1)-th column of mtx is the exogenous kernel cause.

Note

The European Crime data has all three criteria correctly suggesting that a high crime rate kernel causes the deployment of a large number of police officers. The command attach(EuroCrime); silentPairs(cbind(crim,off)) returns only one number: 3.175, implying the highest unanimity strength index, with the positive sign suggesting ‘crim’ in the first column kernel causes ‘off’ in the second column of the argument mtx to this function.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

H. D. Vinod 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See summaryRank, decileVote

See momentVote, exactSdMtx

Examples

## Not run: 
options(np.messages=FALSE)
colnames(mtcars[2:ncol(mtcars)])
silentPair2(mtcars[,1:3],ctrl=mtcars[,4:5]) # mpg paired with others

## End(Not run)

options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
silentPair2(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

No-print kernel causality scores with control variables Hausman-Wu Criterion 1

Description

Allowing input matrix of control variables and missing data, this function produces a 3 column matrix summarizing the results where the estimated signs of stochastic dominance order values (+1, 0, -1) are weighted by wt=c(1.2,1.1, 1.05, 1) to compute an overall result for all orders of stochastic dominance by a weighted sum for the criteria Cr1 and Cr2 and added to the Cr3 estimate as: (+1, 0, -1), always in the range [–3.175, 3.175].

Usage

silentPairs(mtx, ctrl = 0, dig = 6, wt = c(1.2, 1.1, 1.05, 1), sumwt = 4)

Arguments

mtx

The data matrix with p columns. Denote x1 as the first column which is fixed and then paired with all other columns, say: x2, x3, .., xp, one by one for the purpose of flipping with x1. p must be 2 or more
ctrl

data matrix for designated control variable(s) outside causal paths default ctrl=0 which means that there are no control variables used.
dig

Number of digits for reporting (default dig=6).
wt

Allows user to choose a vector of four alternative weights for SD1 to SD4.
sumwt

Sum of weights can be changed here =4(default).

Details

The reason for slightly declining weights on the signs from SD1 to SD4 is simply that the local mean comparisons implicit in SD1 are known to be more reliable than local variance implicit in SD2, local skewness implicit in SD3 and local kurtosis implicit in SD4. The source of slightly declining sampling unreliability of higher moments is the higher power of the deviations from the mean needed in their computations. The summary results for all three criteria are reported in a vector of numbers internally called crall:

Value

With p columns in mtx argument to this function, x1 can be paired with a total of p-1 columns (x2, x3, .., xp). Note we never flip any of the control variables with x1. This function produces i=1,2,..,p-1 numbers representing the summary sign, or ‘sum’ from the signs sg1 to sg3 associated with the three criteria: Cr1, Cr2 and Cr3. Note that sg1 and sg2 themselves are weighted signs using weighted sum of signs from four orders of stochastic dominance. In general, a positive sign in the i-th location of the ‘sum’ output of this function means that x1 is the kernel cause while the variable in (i+1)-th column of mtx is the ‘effect’ or ‘response’ or ‘endogenous.’ The magnitude represents the strength (unanimity) of the evidence for a particular sign. Conversely a negative sign in the i-th location of the ‘sum’ output of this function means that that the first variable listed as the input to this function is the ‘effect,’ while the variable in (i+1)-th column of mtx is the exogenous kernel cause.

Note

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers. The command attach(EuroCrime); silentPairs(cbind(crim,off)) returns only one number: 3.175, implying the highest unanimity strength index, with the positive sign suggesting ‘crim’ in the first column kernel causes ‘off’ in the second column of the argument mtx to this function.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

H. D. Vinod 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See bootPairs, silentMtx

See someCPairs, some0Pairs

Examples

## Not run: 
options(np.messages=FALSE)
colnames(mtcars[2:ncol(mtcars)])
silentPairs(mtcars[,1:3],ctrl=mtcars[,4:5]) # mpg paired with others

## End(Not run)

options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
silentPairs(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

Older version, kernel causality weighted sum allowing control variables

Description

Allowing input matrix of control variables and missing data, this function produces a 3 column matrix summarizing the results where the estimated signs of stochastic dominance order values (+1, 0, -1) are weighted by wt=c(1.2,1.1, 1.05, 1) to compute an overall result for all orders of stochastic dominance by a weighted sum for the criteria Cr1 and Cr2 and added to the Cr3 estimate as: (+1, 0, -1), always in the range [–3.175, 3.175].

Usage

silentPairs0(mtx, ctrl = 0, dig = 6, wt = c(1.2, 1.1, 1.05, 1), sumwt = 4)

Arguments

mtx

The data matrix with p columns. Denote x1 as the first column which is fixed and then paired with all other columns, say: x2, x3, .., xp, one by one for the purpose of flipping with x1. p must be 2 or more
ctrl

data matrix for designated control variable(s) outside causal paths default ctrl=0 which means that there are no control variables used.
dig

Number of digits for reporting (default dig=6).
wt

Allows user to choose a vector of four alternative weights for SD1 to SD4.
sumwt

Sum of weights can be changed here =4(default).

Details

This uses an older version of the first criterion Cr1 based on absolute values of local gradients of kernel regressions, not absolute Hausman-Wu statistic (RHS variable times kernel residuals). It calls abs_stdapd and abs_stdapdC The reason for slightly declining weights on the signs from SD1 to SD4 is simply that the local mean comparisons implicit in SD1 are known to be more reliable than local variance implicit in SD2, local skewness implicit in SD3 and local kurtosis implicit in SD4. The source of slightly declining sampling unreliability of higher moments is the higher power of the deviations from the mean needed in their computations. The summary results for all three criteria are reported in a vector of numbers internally called crall:

Value

With p columns in mtx argument to this function, x1 can be paired with a total of p-1 columns (x2, x3, .., xp). Note we never flip any of the control variables with x1. This function produces i=1,2,..,p-1 numbers representing the summary sign, or ‘sum’ from the signs sg1 to sg3 associated with the three criteria: Cr1, Cr2 and Cr3. Note that sg1 and sg2 themselves are weighted signs using weighted sum of signs from four orders of stochastic dominance. In general, a positive sign in the i-th location of the ‘sum’ output of this function means that x1 is the kernel cause while the variable in (i+1)-th column of mtx is the ‘effect’ or ‘response’ or ‘endogenous.’ The magnitude represents the strength (unanimity) of the evidence for a particular sign. Conversely a negative sign in the i-th location of the ‘sum’ output of this function means that that the first variable listed as the input to this function is the ‘effect,’ while the variable in (i+1)-th column of mtx is the exogenous kernel cause. This function is a summary of someCPairs allowing for control variables.

Note

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers. The command attach(EuroCrime); silentPairs(cbind(crim,off)) returns only one number: 3.175, implying the highest unanimity strength index, with the positive sign suggesting ‘crim’ in the first column kernel causes ‘off’ in the second column of the argument mtx to this function.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

H. D. Vinod 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See bootPairs, silentMtx

See someCPairs, some0Pairs

See silentPairs for newer version using more direct Hausman-Wu exogeneity test statistic.

Examples

## Not run: 
options(np.messages=FALSE)
colnames(mtcars[2:ncol(mtcars)])
silentPairs0(mtcars[,1:3],ctrl=mtcars[,4:5]) # mpg paired with others

## End(Not run)

options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
silentPairs0(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

Block Version of silentPair2 for causality scores with control variables

Description

Block version allows a new bandwidth (chosen by the np package) while fitting kernel regressions for each block of data. This may not be appropriate in all situations. Block size is flexible. The function develops a unanimity index regarding the particular flip (y on xi) or (xi on y) is best. Relevant signs determine the causal direction and unanimity index among three criteria. The ‘2’ in the name of the function suggests a second implementation where exact stochastic dominance, decileVote, and momentVote are used. It avoids Anderson's trapezoidal approximation. The summary results for all three criteria are reported in a vector of numbers internally called crall.

Usage

siPair2Blk(mtx, ctrl = 0, dig = 6, blksiz = 10)

Arguments

mtx

The data matrix with p columns. Denote x1 as the first column, which is fixed and then paired with all other columns, say: x2, x3, .., xp, one by one flipping with x1.The number of columns, p, must be 2 or more
ctrl

data matrix for designated control variable(s) outside causal paths. The default ctrl=0 means that there are no control variables used.
dig

Number of digits for reporting (default dig=6).
blksiz

block size, default=10, if chosen blksiz >n, where n=rows in the matrix, then blksiz=n. That is, no blocking is done

Value

With p columns in mtx argument to this function, x1 can be paired with a total of p-1 columns (x2, x3, .., xp). Note we never flip any of the control variables with x1. This function produces i=1,2,..,p-1 numbers representing the summary sign, or ‘sum’ from the signs sg1 to sg3 associated with the three criteria: Cr1, Cr2 and Cr3. Note that sg1 and sg2 themselves are weighted signs using the weighted sum of signs from four orders of stochastic dominance. In general, a positive sign in the i-th location of the ‘sum’ output of this function means that x1 is the kernel cause while the variable in (i+1)-th column of mtx is the ‘effect’ or ‘response’ or ‘endogenous.’ The magnitude represents the strength (unanimity) of the evidence for a particular sign. Conversely, a negative sign in the i-th location of the ‘sum’ output of this function means that that the first variable listed as the input to this function is the ‘effect,’ while the variable in (i+1)-th column of mtx is the exogenous kernel cause.

Note

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers. The command attach(EuroCrime); silentPairs(cbind(crim,off)) returns only one number: 3.175, implying the highest unanimity strength index, with the positive sign suggesting ‘crim’ in the first column kernel causes ‘off’ in the second column of the argument mtx to this function.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

H. D. Vinod 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See bootPairs, silentMtx

See someCPairs, compPortfo

Examples

## Not run: 
options(np.messages=FALSE)
colnames(mtcars[2:ncol(mtcars)])
siPair2Blk(mtcars[,1:3],ctrl=mtcars[,4:5]) # mpg paired with others

## End(Not run)

options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
siPair2Blk(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

Block Version of silentPairs for causality scores with control variables

Description

Allowing input matrix of control variables and missing data, this function produces a 3 column matrix summarizing the results where the estimated signs of stochastic dominance order values (+1, 0, -1) are weighted by wt=c(1.2,1.1, 1.05, 1) to compute an overall result for all orders of stochastic dominance by a weighted sum for the criteria Cr1 and Cr2 and added to the Cr3 estimate as: (+1, 0, -1), always in the range [–3.175, 3.175].

Usage

siPairsBlk(
  mtx,
  ctrl = 0,
  dig = 6,
  blksiz = 10,
  wt = c(1.2, 1.1, 1.05, 1),
  sumwt = 4
)

Arguments

mtx

The data matrix with p columns. Denote x1 as the first column which is fixed and then paired with all other columns, say: x2, x3, .., xp, one by one for the purpose of flipping with x1. p must be 2 or more
ctrl

data matrix for designated control variable(s) outside causal paths default ctrl=0 which means that there are no control variables used.
dig

Number of digits for reporting (default dig=6).
blksiz

block size, default=10, if chosen blksiz >n, where n=rows in matrix then blksiz=n. That is, no blocking is done
wt

Allows user to choose a vector of four alternative weights for SD1 to SD4.
sumwt

Sum of weights can be changed here =4(default).

Details

The reason for slightly declining weights on the signs from SD1 to SD4 is simply that the local mean comparisons implicit in SD1 are known to be more reliable than local variance implicit in SD2, local skewness implicit in SD3 and local kurtosis implicit in SD4. The source of slightly declining sampling unreliability of higher moments is the higher power of the deviations from the mean needed in their computations. The summary results for all three criteria are reported in a vector of numbers internally called crall:

Value

With p columns in mtx argument to this function, x1 can be paired with a total of p-1 columns (x2, x3, .., xp). Note we never flip any of the control variables with x1. This function produces i=1,2,..,p-1 numbers representing the summary sign, or ‘sum’ from the signs sg1 to sg3 associated with the three criteria: Cr1, Cr2 and Cr3. Note that sg1 and sg2 themselves are weighted signs using weighted sum of signs from four orders of stochastic dominance. In general, a positive sign in the i-th location of the ‘sum’ output of this function means that x1 is the kernel cause while the variable in (i+1)-th column of mtx is the ‘effect’ or ‘response’ or ‘endogenous.’ The magnitude represents the strength (unanimity) of the evidence for a particular sign. Conversely a negative sign in the i-th location of the ‘sum’ output of this function means that that the first variable listed as the input to this function is the ‘effect,’ while the variable in (i+1)-th column of mtx is the exogenous kernel cause.

Note

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers. The command attach(EuroCrime); silentPairs(cbind(crim,off)) returns only one number: 3.175, implying the highest unanimity strength index, with the positive sign suggesting ‘crim’ in the first column kernel causes ‘off’ in the second column of the argument mtx to this function.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

H. D. Vinod 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. Causal Paths and Exogeneity Tests in Generalcorr Package for Air Pollution and Monetary Policy (June 6, 2017). Available at SSRN: https://www.ssrn.com/abstract=2982128

See Also

See bootPairs, silentMtx

See someCPairs, some0Pairs

Examples

## Not run: 
options(np.messages=FALSE)
colnames(mtcars[2:ncol(mtcars)])
siPairsBlk(mtcars[,1:3],ctrl=mtcars[,4:5]) # mpg paired with others

## End(Not run)

options(np.messages=FALSE)
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10 #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
siPairsBlk(mtx=cbind(x2,y2), ctrl=cbind(z,w2))

Function reporting detailed kernel causality results in a 7-column matrix (uses deprecated criterion 1, no longer recommended but may be useful for second and third criterion typ=2,3)

Description

The seven columns produced by this function summarize the results where the signs of stochastic dominance order values (+1 or -1) are weighted by wt=c(1.2,1.1, 1.05, 1) to compute an overall result for all orders of stochastic dominance by a weighted sum for the criteria Cr1 and Cr2. The weighting is obviously not needed for the third criterion Cr3.

Usage

some0Pairs(
  mtx,
  dig = 6,
  verbo = TRUE,
  rnam = FALSE,
  wt = c(1.2, 1.1, 1.05, 1),
  sumwt = 4
)

Arguments

mtx

The data matrix in the first column is paired with all others.
dig

Number of digits for reporting (default dig=6).
verbo

Make verbo= TRUE for printing detailed steps.
rnam

Make rnam= TRUE if cleverly created row-names are desired.
wt

Allows user to choose a vector of four alternative weights for SD1 to SD4.
sumwt

Sum of weights can be changed here =4(default).

Details

The reason for slightly declining weights on the signs from SD1 to SD4 is simply that the local mean comparisons implicit in SD1 are known to be more reliable than local variance implicit in SD2, local skewness implicit in SD3 and local kurtosis implicit in SD4. The source of slightly declining sampling unreliability of higher moments is the higher power of the deviations from the mean needed in their computations. The summary results for all three criteria are reported in one matrix called outVote:

typ=1 reports ('Y', 'X', 'Cause', 'SD1apd', 'SD2apd', 'SD3apd', 'SD4apd') naming variables identifying 'cause' and measures of stochastic dominance using absolute values of kernel regression gradients (or amorphous partial derivatives, apd-s) being minimized by the kernel regression algorithm while comparing the kernel regression of X on Y with that of Y on X.

typ=2 reports ('Y', 'X', 'Cause', 'SD1res', 'SD2res', 'SD3res', 'SD4res') and measures of stochastic dominance using absolute values of kernel regression residuals comparing regression of X on Y with that of Y on X.

typ=3 reports ('Y', 'X', 'Cause', 'r*x|y', 'r*y|x', 'r', 'p-val') containing generalized correlation coefficients r*, 'r' refers to. Pearson correlation coefficient p-val is the p-value for testing the significance of 'r'

Value

Prints three matrices detailing results for Cr1, Cr2 and Cr3. It also returns a grand summary matrix called ‘outVote’ which summarizes all three criteria. In general, a positive sign for weighted sum reported in the column ‘sum’ means that the first variable listed as the input to this function is the ‘kernel cause.’ For example, crime ‘kernel causes’ police officer deployment (not vice versa) is indicated by the positive sign of ‘sum’ (=3.175) reported for that example included in this package.

Note

The output matrix last column for ‘mtcars’ example has the sum of the scores by the three criteria combined. If ‘sum’ is positive, then variable X (mpg) is more likely to have been engineered to kernel cause the response variable Y, rather than vice versa.

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

See Also

See Also somePairs

Examples

## Not run: 
some0Pairs(mtcars) # first variable is mpg and effect on mpg is of interest

## End(Not run)

## Not run: 
data(EuroCrime)
attach(EuroCrime)
some0Pairs(cbind(crim,off))

## End(Not run)

Kernel causality computations admitting control variables.

Description

This function reports a 7-column matrix (has the older version of criterion Cr1). It allows an additional input matrix having control variables. It produces a 7-column matrix summarizing the results, where the signs of stochastic dominance order values (+1 or -1) are weighted by wt=c(1.2,1.1, 1.05, 1) to compute an overall result for all orders of stochastic dominance by a weighted sum for the criteria Cr1 and Cr2. The weighting is obviously not needed for the third criterion Cr3 which compares asymmetric correlation coefficients.

Usage

someCPairs(
  mtx,
  ctrl,
  dig = 6,
  verbo = TRUE,
  rnam = FALSE,
  wt = c(1.2, 1.1, 1.05, 1),
  sumwt = 4
)

Arguments

mtx

The data matrix with many columns where the first column is fixed and then paired with all other columns, one by one.
ctrl

data matrix for designated control variable(s) outside causal paths
dig

Number of digits for reporting (default dig=6).
verbo

Make verbo= TRUE for printing detailed steps.
rnam

Make rnam= TRUE if cleverly created rownames are desired.
wt

Allows user to choose a vector of four alternative weights for SD1 to SD4.
sumwt

Sum of weights can be changed here =4(default).

Details

The reason for slightly declining weights on the signs from SD1 to SD4 is somewhat arbitrary. The summary results for all three criteria are reported in one matrix called outVote:

typ=1 reports ('Y', 'X', 'Cause', 'SD1apdC', 'SD2apdC', 'SD3apdC', 'SD4apdC') naming variables identifying 'cause' and measures of stochastic dominance using absolute values of kernel regression gradients (or amorphous partial derivatives, apd-s) being minimized by the kernel regression algorithm while comparing the kernel regression of X on Y with that of Y on X. The letter C in the titles reminds presence of control variable(s).

typ=2 reports ('Y', 'X', 'Cause', 'SD1resC', 'SD2resC', 'SD3resC', 'SD4resC') and measures of stochastic dominance using absolute values of kernel regression residuals comparing regression of X on Y with that of Y on X.

typ=3 reports ('Y', 'X', 'Cause', 'r*x|yC', 'r*y|xC', 'r', 'p-val') containing generalized correlation coefficients r*, 'r' refers to. Pearson correlation coefficient p-val is the p-value for testing the significance of 'r'. The letter C in the titles reminds the presence of control variable(s).

Value

Prints three matrices detailing results for Cr1, Cr2 and Cr3. It also returns a grand summary matrix called ‘outVote’ which summarizes all three criteria. In general, a positive sign for weighted sum reported in the column ‘sum’ means that the first variable listed as the input to this function is the ‘kernel cause.’ This function is an extension of some0Pairs to allow for control variables. For example, crime ‘kernel causes’ police officer deployment (not vice versa) is indicated by the positive sign of ‘sum’ (=3.175) reported for that example included in this package.

Note

The output matrix last column for ‘mtcars’ example has the sum of the scores by the three criteria combined. If ‘sum’ is positive, then variable X (mpg) is more likely to have been engineerd to kernel cause the response variable Y, rather than vice versa.

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

See Also

See Also somePairs, some0Pairs

Examples

## Not run: 
someCPairs(mtcars[,1:3],ctrl=mtcars[4:5]) # first variable is mpg and effect on mpg is of interest

## End(Not run)

## Not run: 
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10  #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
someCPairs(cbind(x2,y2), cbind(z,w2)) #yields x2 as correct cause

## End(Not run)

Kernel causality computations admitting control variables reporting a 7-column matrix, version 2.

Description

Second version of someCPairs also allows input matrix of control variables, produce 7 column matrix summarizing the results where the signs of stochastic dominance order values (+1 or -1) are weighted by wt=c(1.2,1.1, 1.05, 1) to compute an overall result for all orders of stochastic dominance by a weighted sum for the criteria Cr1 and Cr2. The weighting is obviously not needed for the third criterion Cr3.

Usage

someCPairs2(
  mtx,
  ctrl,
  dig = 6,
  verbo = TRUE,
  rnam = FALSE,
  wt = c(1.2, 1.1, 1.05, 1),
  sumwt = 4
)

Arguments

mtx

The data matrix with many columns where the first column is fixed and then paired with all other columns, one by one.
ctrl

data matrix for designated control variable(s) outside causal paths
dig

Number of digits for reporting (default dig=6).
verbo

Make verbo= TRUE for printing detailed steps.
rnam

Make rnam= TRUE if cleverly created rownames are desired.
wt

Allows user to choose a vector of four alternative weights for SD1 to SD4.
sumwt

Sum of weights can be changed here =4(default).

Details

The reason for slightly declining weights on the signs from SD1 to SD4 is simply that the local mean comparisons implicit in SD1 are known to be more reliable than local variance implicit in SD2, local skewness implicit in SD3 and local kurtosis implicit in SD4. The source of slightly declining sampling unreliability of higher moments is the higher power of the deviations from the mean needed in their computations. The summary results for all three criteria are reported in one matrix called outVote:

(typ=1) reports ('Y', 'X', 'Cause', 'SD1.rhserr', 'SD2.rhserr', 'SD3.rhserr', 'SD4.rhserr') naming variables identifying the 'cause' and measures of stochastic dominance using absolute values of kernel regression abs(RHS first regressor*residual) values comparing flipped regressions X on Y versus Y on X. The letter C in the titles reminds presence of control variable(s).

typ=2 reports ('Y', 'X', 'Cause', 'SD1resC', 'SD2resC', 'SD3resC', 'SD4resC') and measures of stochastic dominance using absolute values of kernel regression residuals comparing regression of X on Y with that of Y on X.

typ=3 reports ('Y', 'X', 'Cause', 'r*x|yC', 'r*y|xC', 'r', 'p-val') containing generalized correlation coefficients r*, 'r' refers to. Pearson correlation coefficient p-val is the p-value for testing the significance of 'r'. The letter C in the titles reminds the presence of control variable(s).

Value

Prints three matrices detailing results for Cr1, Cr2 and Cr3. It also returns a grand summary matrix called ‘outVote’ which summarizes all three criteria. In general, a positive sign for weighted sum reported in the column ‘sum’ means that the first variable listed as the input to this function is the ‘kernel cause.’ This function is an extension of some0Pairs to allow for control variables. For example, crime ‘kernel causes’ police officer deployment (not vice versa) is indicated by the positive sign of ‘sum’ (=3.175) reported for that example included in this package.

Note

The output matrix last column for ‘mtcars’ example has the sum of the scores by the three criteria combined. If ‘sum’ is positive, then variable X (mpg) is more likely to have been engineered to kernel cause the response variable Y, rather than vice versa.

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

See Also

See Also somePairs, some0Pairs

Examples

## Not run: 
someCPairs2(mtcars[,1:3],ctrl=mtcars[4:5]) # first variable is mpg and effect on mpg is of interest

## End(Not run)

## Not run: 
set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10  #x is somewhat indep and affected by z
y=1+2*x+3*z+rnorm(10)
w=runif(10)
x2=x;x2[4]=NA;y2=y;y2[8]=NA;w2=w;w2[4]=NA
someCPairs2(cbind(x2,y2), cbind(z,w2)) #yields x2 as correct cause

## End(Not run)

Summary magnitudes after removing control variables in several pairs where dependent variable is fixed.

Description

This builds on the function mag_ctrl, where the input matrix mtx has p columns. The first column is present in each of the (p-1) pairs. Its output is a matrix with four columns containing the names of variables and approximate overall estimates of the magnitudes of partial derivatives (dy/dx) and (dx/dy) for a distinct (x,y) pair in a row. The estimated overall derivatives are not always well-defined, because the real partial derivatives of nonlinear functions are generally distinct for each observation point.

Usage

someMagPairs(mtx, ctrl, dig = 6, verbo = TRUE)

Arguments

mtx

The data matrix with many columns where the first column is fixed and then paired with all other columns, one by one.
ctrl

data matrix for designated control variable(s) outside causal paths. A constant vector is not allowed as a control variable.
dig

Number of digits for reporting (default dig=6).
verbo

Make verbo= TRUE for printing detailed steps.

Details

The function mag_ctrl has kernel regressions: x~ y + ctrl and x~ ctrl to evaluate the‘incremental change’ in R-squares. Let (rxy;ctrl) denote the square root of that ‘incremental change’ after its sign is made the same as that of the Pearson correlation coefficient from cor(x,y)). One can interpret (rxy;ctrl) as a generalized partial correlation coefficient when x is regressed on y after removing the effect of control variable(s) in ctrl. It is more general than the usual partial correlation coefficient, since this one allows for nonlinear relations among variables. Next, the function computes ‘dxdy’ obtained by multiplying (rxy;ctrl) by the ratio of standard deviations, sd(x)/sd(y). Now our ‘dxdy’ approximates the magnitude of the partial derivative (dx/dy) in a causal model where y is the cause and x is the effect. The function also reports entirely analogous ‘dydx’ obtained by interchanging x and y.

someMegPairs function runs the function mag_ctrl on several column pairs in a matrix input mtx where the first column is held fixed and all others are changed one by one, reporting two partial derivatives for each row.

Value

Table containing names of Xi and Xj and two magnitudes: (dXidXj, dXjdXi). dXidXj is the magnitude of the effect on Xi when Xi is regressed on Xj (i.e., when Xj is the cause). The analogous dXjdXi is the magnitude when Xj is regressed on Xi.

Note

This function is intended for use only after the causal path direction is already determined by various functions in this package (e.g. someCPairs). That is, after the researcher knows whether Xi causes Xj or vice versa. The output of this function is a matrix of 4 columns, where first columns list the names of Xi and Xj and the next two numbers in each row are dXidXj, dXjdXi, respectively, representing the magnitude of effect of one variable on the other.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C. R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

See Also

See mag_ctrl, someCPairs

Examples

set.seed(34);x=sample(1:10);y=1+2*x+rnorm(10);z=sample(2:11)
  w=runif(10)
  ss=someMagPairs(cbind(y,x,z),ctrl=w)

Function reporting kernel causality results as a 7-column matrix.(deprecated)

Description

This function lets the user choose one of three criteria to determine causal direction by setting typ as 1, 2 or 3. This function reports results for only one criterion at a time unlike the function some0Pairs which summarizes the resulting causal directions for all criteria with suitable weights. If some variables are ‘control’ variables, use someCPairs, C=control.

Usage

somePairs(mtx, dig = 6, verbo = FALSE, typ = 1, rnam = FALSE)

Arguments

mtx

The data matrix in the first column is paired with all others.
dig

Number of digits for reporting (default dig=6).
verbo

Make verbo= TRUE for printing detailed steps.
typ

Must be 1 (default), 2 or 3 for the three criteria.
rnam

Make rnam= TRUE if cleverly created rownames are desired.

Details

(typ=1) reports ('Y', 'X', 'Cause', 'SD1apd', 'SD2apd', 'SD3apd', 'SD4apd') nameing variables identifying 'cause' and measures of stochastic dominance using absolute values of kernel regression gradients comparing regresson of X on Y with that of Y on X.

(typ=2) reports ('Y', 'X', 'Cause', 'SD1res', 'SD2res', 'SD3res', 'SD4res') and measures of stochastic dominance using absolute values of kernel regression residuals comparing regresson of X on Y with that of Y on X.

(typ=3) reports ('Y', 'X', 'Cause', 'r*X|Y', 'r*Y|X', 'r', 'p-val') containing generalized correlation coefficients r*, 'r' refers to the Pearson correlation coefficient and p-val column has the p-values for testing the significance of Pearson's 'r'.

Value

A matrix containing causal identification results for one criterion. The first column of the input mtx having p columns is paired with (p-1) other columns The output matrix headings are self-explanatory and distinct for each criterion Cr1 to Cr3.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

H. D. Vinod 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

See Also

The related function some0Pairs may be more useful, since it reports on all three criteria (by choosing typ=1,2,3) and further summarizes their results by weighting to help choose causal paths.

Examples

## Not run: 
data(mtcars)
somePairs(mtcars)

## End(Not run)

Function reporting kernel causality results as a 7-column matrix, version 2.

Description

This function is an alternative implementation of somePairs which also lets the user choose one of three criteria to determine causal direction by setting typ as 1, 2 or 3. This function reports results for only one criterion at a time unlike the function some0Pairs which summarizes the resulting causal directions for all criteria with suitable weights. If some variables are ‘control’ variables, use someCPairs, where notation C=control.

Usage

somePairs2(mtx, dig = 6, verbo = FALSE, typ = 1, rnam = FALSE)

Arguments

mtx

The data matrix in the first column is paired with all others.
dig

Number of digits for reporting (default dig=6).
verbo

Make verbo= TRUE for printing detailed steps.
typ

Must be 1 (default), 2 or 3 for the three criteria.
rnam

Make rnam= TRUE if cleverly created rownames are desired.

Details

(typ=1) reports ('Y', 'X', 'Cause', 'SD1.rhserr', 'SD2.rhserr', 'SD3.rhserr', 'SD4.rhserr') naming variables identifying the 'cause,' using Hausman-Wu criterion. It measures of stochastic dominance using absolute values of kernel regression abs(RHS first regressor*residual), comparing flipped regressions X on Y versus Y on X.

(typ=2) reports ('Y', 'X', 'Cause', 'SD1res', 'SD2res', 'SD3res', 'SD4res') and measures of stochastic dominance using absolute values of kernel regression residuals comparing regression of X on Y with that of Y on X.

(typ=3) reports ('Y', 'X', 'Cause', 'r*X|Y', 'r*Y|X', 'r', 'p-val') containing generalized correlation coefficients r*, 'r' refers to the Pearson correlation coefficient and p-val column has the p-values for testing the significance of Pearson's 'r'.

Value

A matrix containing causal identification results for one criterion. The first column of the input mtx having p columns is paired with (p-1) other columns The output matrix headings are self-explanatory and distinct for each criterion Cr1 to Cr3.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

H. D. Vinod 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

See Also

The related function some0Pairs may be more useful, since it reports on all three criteria (by choosing typ=1,2,3) and further summarizes their results by weighting to help choose causal paths.

Alternative and revised function somePairs2 implements the Cr1 (first criterion) with a direct estimate of the Hausman-Wu statistic for testing exogeneity.

Examples

## Not run: 
data(mtcars)
somePairs2(mtcars)

## End(Not run)

Sort all columns of matrix x with respect to the j-th column.

Description

This function can use the sort.list function in R. The reason for using it is that one wants the sort to carry along all columns.

Usage

sort_matrix(x, j)

Arguments

x

An input matrix with several columns
j

The column number with reference to which one wants to sort

Value

A sorted matrix

Examples

set.seed(30)
x=matrix(sample(1:50),ncol=5)
y=sort_matrix(x,3);y

internal sort.abse0

Description

intended for internal use only

Usage

sort.abse0

internal sort.e0

Description

intended for internal use only

Usage

sort.e0

Residuals of kernel regressions of x on y when both x and y are standardized.

Description

1) Standardize the data to force mean zero and variance unity, 2) kernel regress x on y, with the option ‘residuals = TRUE’, and finally 3) compute the residuals. The standardization yields comparable residuals.

Usage

stdres(x, y)

Arguments

x

vector of data on the dependent variable
y

data on the regressors which can be a matrix

Details

The first argument is assumed to be the dependent variable. If stdres(x,y) is used, you are regressing x on y (not the usual y on x). The regressors can be a matrix with 2 or more columns. The missing values are suitably ignored by the standardization.

Value

kernel regression residuals are returned after standardizing the data on both sides so that the magnitudes of residuals are comparable between regression of x on y on the one hand, and the flipped regression of y on x on the other.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D. 'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, doi:10.1080/03610918.2015.1122048

Examples

## Not run: 
set.seed(330)
x=sample(20:50)
y=sample(20:50)
stdres(x,y)

## End(Not run)

Standardize x and y vectors to achieve zero mean and unit variance.

Description

Standardize x and y vectors to achieve zero mean and unit variance.

Usage

stdz_xy(x, y)

Arguments

x

Vector of data which can have NA's
y

Vector of data which can have NA's

Value

stdx

standardized values of x
stdy

standardized values of y

Note

This works even if there are missing x or y values.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

Examples

## Not run: 
set.seed(30)
x=sample(20:30)
y=sample(21:31)
stdz_xy(x,y) 
## End(Not run)

Compute vectors measuring stochastic dominance of four orders.

Description

Stochastic dominance originated as a sophisticated comparison of two distributions of stock market returns. The dominating distribution is superior in terms of local mean, variance, skewness, and kurtosis, respectively. However, stochastic dominance orders 1 to 4 are really not related to the four moments. Some details are in Vinod (2022, sec. 4.3) and vignettes. Nevertheless, this function uses the output of ‘wtdpapb.’ and Anderson's algorithm. Of course, Anderson's method remains subject to the trapezoidal approximation avoided by exact stochastic dominance methods.

Usage

stochdom2(dj, wpa, wpb)

Arguments

dj

Vector of (unequal) distances of consecutive intervals defined on common support of two probability distributions being compared
wpa

Vector of the first set of (weighted) probabilities
wpb

Vector of the second set of (weighted) probabilities

Value

sd1b

Vector measuring stochastic dominance of order 1, SD1
sd2b

Vector measuring stochastic dominance of order 2, SD2
sd3b

Vector measuring stochastic dominance of order 3, SD3
sd4b

Vector measuring stochastic dominance of order 4, SD4

Note

The input to this function is the output of the function wtdpapb.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D.', 'Hands-On Intermediate Econometrics Using R' (2008) World Scientific Publishers: Hackensack, NJ. https://www.worldscientific.com/worldscibooks/10.1142/12831

Vinod, H. D. 'Ranking Mutual Funds Using Unconventional Utility Theory and Stochastic Dominance,' Journal of Empirical Finance Vol. 11(3) 2004, pp. 353-377.

See Also

See Also wtdpapb

Examples

## Not run: 
 set.seed(234);x=sample(1:30);y=sample(5:34)
 w1=wtdpapb(x,y) #y should dominate x with mostly positive SDs
 stochdom2(w1$dj, w1$wpa, w1$wpb) 
## End(Not run)

Pseudo regression coefficients from generalized partial correlation coefficients, (GPCC).

Description

This function gets the GPCCs by calling the parcorVec function. The pseudo regression coefficient of a kernel regression is then obtained by [GPCC*(sd dep.var)/(sd regressor)], that is, by multiplying the GPCC by the standard deviation (sd) of the dependent variable, and dividing by the sd of the regressor.

Usage

sudoCoefParcor(mtx, ctrl = 0, verbo = FALSE, idep = 1)

Arguments

mtx

Input data matrix with p (> or = 3) columns,
ctrl

Input vector or matrix of data for control variable(s), default is ctrl=0, when control variables are absent
verbo

Make this TRUE for detailed printing of computational steps
idep

The column number of the dependent variable (=1, default)

Value

A p by 1 ‘out’ vector pseudo partial derivatives.

Note

Generalized Partial Correlation Coefficients (GPCC) allow comparison of the relative contribution of each XjX_j to the explanation of XiX_i, because GPCC are scale-free. The pseudo regression coefficient are not scale-free since they equal GPCC*(sd dep.var)/(sd regressor)

We want to get all partial correlation coefficient pairs removing other column effects. Vinod (2018) shows why one needs more than one criterion to decide the causal paths or exogeneity.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlations and Instantaneous Causality for Data Pairs Benchmark,' (March 8, 2015) https://www.ssrn.com/abstract=2574891

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Vinod, H. D. 'New Exogeneity Tests and Causal Paths,' (June 30, 2018). Available at SSRN: https://www.ssrn.com/abstract=3206096

Vinod, H. D. (2021) 'Generalized, Partial and Canonical Correlation Coefficients' Computational Economics, 59(1), 1–28.

See Also

See Also parcor_ijk.

See Also a hybrid version parcorVecH.

Examples

set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10  #x is partly indep and partly affected by z
y=1+2*x+3*z+rnorm(10)# y depends on x and z not vice versa
mtx=cbind(x,y,z)
sudoCoefParcor(mtx, idep=2)
 
   
## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
colnames(x)=c('V1', 'v2', 'V3')#some names needed
sudoCoefParcor(x)

## End(Not run)

Peudo regression coefficients from hybrid generalized partial correlation coefficients (HGPCC).

Description

This function gets HGPCCs by calling parcorVecH function. Pseudo regression coefficient of a kernel regression is obtained by HGPCC*(sd dep.var)/(sd regressor), that is multiplying the HGPCC by the standard deviation (sd) of the dependent variable and dividing by the sd of the regressor.

Usage

sudoCoefParcorH(mtx, ctrl = 0, verbo = FALSE, idep = 1)

Arguments

mtx

Input data matrix with p (> or = 3) columns,
ctrl

Input vector or matrix of data for control variable(s), default is ctrl=0 when control variables are absent
verbo

Make this TRUE for detailed printing of computational steps
idep

The column number of the dependent variable (=1, default)

Value

A p by 1 ‘out’ vector pseudo partial derivatives

Note

Hybrid Generalized Partial Correlation Coefficients (HGPCC) allow comparison of the relative contribution of each XjX_j to the explanation of XiX_i, because GPCC are scale-free. Hybrid refers to use of OLS residuals. Now pseudo hybrid regr coeff are HGPCC*(sd dep.var)/(sd regressor)

We want to get all partial correlation coefficient pairs removing other column effects. Vinod (2018) shows why one needs more than one criterion to decide the causal paths or exogeneity.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

References

Vinod, H. D. 'Generalized Correlations and Instantaneous Causality for Data Pairs Benchmark,' (March 8, 2015) https://www.ssrn.com/abstract=2574891

Vinod, H. D. 'Matrix Algebra Topics in Statistics and Economics Using R', Chapter 4 in Handbook of Statistics: Computational Statistics with R, Vol.32, co-editors: M. B. Rao and C.R. Rao. New York: North Holland, Elsevier Science Publishers, 2014, pp. 143-176.

Vinod, H. D. 'New Exogeneity Tests and Causal Paths,' (June 30, 2018). Available at SSRN: https://www.ssrn.com/abstract=3206096

Vinod, H. D. (2021) 'Generalized, Partial and Canonical Correlation Coefficients' Computational Economics, 59(1), 1–28.

See Also

See Also parcor_ijk.

See Also a hybrid version parcorVecH.

Examples

set.seed(234)
z=runif(10,2,11)# z is independently created
x=sample(1:10)+z/10  #x is partly indep and partly affected by z
y=1+2*x+3*z+rnorm(10)# y depends on x and z not vice versa
mtx=cbind(x,y,z)
sudoCoefParcor(mtx, idep=2)
 
   
## Not run: 
set.seed(34);x=matrix(sample(1:600)[1:99],ncol=3)
colnames(x)=c('V1', 'v2', 'V3')#some names needed
sudoCoefParcorH(x)

## End(Not run)

Compute ranks of rows of matrix and summarize them into a choice suggestion.

Description

This function allows getting out the choice (of a column representing a stock) from four rows of numbers quantifying the four orders of exact stochastic dominance comparisons. If the last or 10-th row for “choice" has 1 then the stock representing that column is to be chosen. That is it should get the largest (portfolio) weight. If the original matrix row names are SD1 to SD4, the same names are repeated for the extra rows representing their ranks. The row name for “sum of ranks" is sumRanks. Finally, the ranks associated with sumRanks provide the row named choice along the bottom (10-th) row of the output matrix called "out."

Usage

summaryRank(mtx)

Arguments

mtx

matrix to be ranked by row and summarized

Value

a matrix called ‘out’ having 10 rows and p columns (p=No.of stocks). Row Numbers 1 to 4 have SD1 to SD4 evaluation of areas over ECDFs. There are 6 more rows. Row No.5= SD1 ranks, Row No.6= SD2 ranks, Row No.7= SD3 ranks, Row No.8= SD4 ranks Row No.9= sum of the ranks in earlier four rows for ranks of SD1 to SD4 Row No.10= choice rank based on all four (SD1 to SD4) added together Thus, the tenth row yields choice priority number for each stock (asset) after combining the all four criteria.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

See Also

exactSdMtx

Replace asymmetric matrix by max of abs values of [i,j] or [j,i] elements.

Description

It is useful in symmetrizing the gmcmtx0 matrix containing a non-symmetric generalized correlation matrix.

Usage

symmze(mtx)

Arguments

mtx

non-symmetric matrix

Value

mtx2

replace [i,j] and [j,i] by the max of absolute values with common sign

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

Examples

## Not run: 
example 
mtx=matrix(1:16,nrow=4)
symmze(mtx)

## End(Not run)#'

Creates input for the stochastic dominance function stochdom2

Description

Stochastic dominance is a sophisticated comparison of two distributions of stock market returns. The dominating distribution is superior in terms of mean, variance, skewness and kurtosis respectively, representing dominance orders 1 to 4, without directly computing four moments. Vinod(2008) sec. 4.3 explains the details. The ‘wtdpapb’ function creates the input for stochdom2 which in turn computes the stochastic dominance. See Vinod (2004) for details about quantitative stochastic dominance.

Usage

wtdpapb(xa, xb)

Arguments

xa

Vector of (excess) returns for the first investment option A or values of any random variable being compared to another.
xb

Vector of returns for the second option B

Value

wpa

Weighted vector of probabilities for option A
wpb

Weighted vector of probabilities for option B
dj

Vector of interval widths (distances) when both sets of data are forced on a common support

Note

Function is needed before using stochastic dominance

In Vinod (2008) where the purpose of wtdpapb is to map from standard ‘expected utility theory’ weights to more sophisticated 'non-expected utility theory' weights using Prelec's (1998, Econometrica, p. 497) method. These weights are not needed here. Hence we provide the function prelec2 which does not use Prelec weights at all, thereby simplifying and speeding up the R code provided in Vinod (2008). This function avoids sophisticated ‘non-expected’ utility theory which incorporates commonly observed human behavior favoring loss aversion and other anomalies inconsistent with precepts of the expected utility theory. Such weighting is not needed for our application.

Author(s)

Prof. H. D. Vinod, Economics Dept., Fordham University, NY

References

Vinod, H. D.', 'Hands-On Intermediate Econometrics Using R' (2008) World Scientific Publishers: Hackensack, NJ. https://www.worldscientific.com/worldscibooks/10.1142/12831

Vinod, H. D. 'Ranking Mutual Funds Using Unconventional Utility Theory and Stochastic Dominance,' Journal of Empirical Finance Vol. 11(3) 2004, pp. 353-377.

See Also

See Also stochdom2

Examples

## Not run: 
 set.seed(234);x=sample(1:30);y=sample(5:34)
 wtdpapb(x,y)
## End(Not run)