3. Dummy Variables! Suppose that one is interested in the gender wage gap. Specifically, they want to regress the weekly average earnings, Ei, on a dummy for male, Mi : Ei=0+1Mi+Ui a. Supposing that E(UiMi)=0, show that 0 is the mean earnings of women, while 1 is the difference in earnings between males and females. This is about the population regression, so consider E(EM). b. Now consider the OLS estimator. Show that, ^1=EMaleEFemale and ^0=EFemale. A few hints that all follow from M being a 0/1 variable: i. M is the fraction of males in the sample. ii. This implies, (1M) is the fraction of females. iii. One can decompose the mean of earnings as follows: E=NNmaleEMale+NNfemaleEFemale 3 iv. Another useful decomposition: i=1NMiEi=i:Mi=1Ei, where {i:Mi=1} means the set of i where Mi=1. v. Finally, M=M2. Which will come in handy for the numerator. 3. Dummy Variables! Suppose that one is interested in the gender wage gap. Specifically, they want to regress the weekly average earnings, Ei, on a dummy for male, Mi : Ei=0+1Mi+Ui a. Supposing that E(UiMi)=0, show that 0 is the mean earnings of women, while 1 is the difference in earnings between males and females. This is about the population regression, so consider E(EM). b. Now consider the OLS estimator. Show that, ^1=EMaleEFemale and ^0=EFemale. A few hints that all follow from M being a 0/1 variable: i. M is the fraction of males in the sample. ii. This implies, (1M) is the fraction of females. iii. One can decompose the mean of earnings as follows: E=NNmaleEMale+NNfemaleEFemale 3 iv. Another useful decomposition: i=1NMiEi=i:Mi=1Ei, where {i:Mi=1} means the set of i where Mi=1. v. Finally, M=M2. Which will come in handy for the numerator