A researcher has two independent samples of observations on (Yi -, Xi). To be specific, suppose that Yi denotes earnings, Xi denotes years of schooling, and the independent samples are for men and women. Write the regression for men as Ymi = βm,0 + βm,l + Xm,j and the regression for women as Yw i = βW,0 + βw,1Xw,i + uwi. Let m,1 denote the OLS estimator constructed using the sample of men, w,1 denote the OLS estimator constructed from the sample of women, and SE(m,1) and SE(w,1) denote the corresponding standard errors. Show that the standard error of (m,1 - w,1)is given by

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- Bua) = V[SE(Bma)} + [SE(B). SE(Bm m.l