Partial Correlations

Valid instruments are supposed to be correlated with the endogenous regressor. However, an important determinant of the statistical properties of the IV estimator is the degree of correlation between the instrument and the endogenous regressor. Furthermore, it is the independent corre­lation between the instrument and the endogenous regressor that is important. The higher, the better.

One way to get at this in a multiple regression model is to partial out the correlation in variables measured with error that is due to the exogenous regressors. Whatever common variation that remains will measure the independent correlation between the variable measured with error and the instrument. This sounds complicated, but it is not. It is simple to do in gretl.

1 ols educ const exper sq_exper

2 series e1 = $uhat

3 ols mothereduc const exper sq_exper

4 series e2 = $uhat

5 ols e1 e2

6 corr e1 e2

The first statement regresses const, exper, and sq_exper on educ and saves the residuals, e1. The residuals contain all variation in educ not accounted for by the regressors. In effect, the variation in const, exper, and sq_exper has been partialled out of the variable measured with error, educ. The second regression does the same for the instrument, mothereduc. The residuals, e2, have the correlation with const, exper, and sq_exper partialled out. Regressing e2 onto e1 yields, 0.26769.

This turns out to be exactly the coefficient on mothereduc in the first-stage regression. This is no coincidence since regression coefficients are the effect of one variable on another, holding the remaining regressors constant.[73]

coefficient

std. error

t-ratio

p-value

const

9.77510

0.423889

23.06

7.57e-077

exper

0.0488615

0.0416693

1.173

0.2416

sq_exper

-0.00128106

0.00124491

-1.029

0.3040

mothereduc

0.267691

0.0311298

8.599

1.57e-016

First Stage Regression: OLS, using observations 1-428

Dependent variable: educ

***

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The correlation between the two sets of residuals yields what is called a partial correlation. This is a correlation where the common effects of const, exper, and sq_exper have been removed. The partial correlation between e1 and e2 is 0.3854. Partial correlations play a key role in testing for weak instruments.

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