Introduction to Econometrics 3rd Edition by James Stock, Mark Watson

Exercise 20

A researcher has two independent samples of observations on ( Y i X i ,). To be specific, suppose that Y i denotes earnings, X i , denotes years of schooling, and the independent samples are for men and women. Write the regression for men as
and the regression for women as
denote the OLS estimator constructed using the sample of men,
denote the OLS estimator constructed from the sample of women, and
and
denote the corresponding standard errors. Show that the standard error of
is given by