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.

1 | ```
someMagPairs(mtx, ctrl, dig = 6, verbo = TRUE)
``` |

`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 |

`verbo` |
Make |

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.

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.

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.

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

Vinod, H. D.'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, http://dx.doi.org/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 `mag_ctrl`

, `someCPairs`

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