1. Consider the simple regression model y=0+1D+u, where D is a binary variable. a) Let yi1 be yi with Di=1,yi0 be yi with Di=0, and i=1Nyi=i=1N1yi1+i=1N0yi0, where N1+N0=N. Show that y=NN1N1i=1N1yi1+NN0N0i=1N0yi0=Dy1+(1D)y0. b) Show that the OLS estimator: 1=i=1N(DiD)2i=1N(DiD)(yiy) can be written as y1y0. [Hint: Note that Diyyi=yi1 and Di2=Di. Therefore 1= i=1NDi2ND2i=1NDiyii=1NDyii=1NyDi+i=1NDy=i=1NDiND2i=1Nyi1Di=1Nyiyi=1NDi+NDy. c) Consider another simple model y=0+1x+u, and let z be a binary instrument variable for x. Show that the IV estimator: ^1=i=1N(ziz)(xix)i=1N(ziz)(yiy) can be written as the Wald estimator: x1x0y1y0. [Hint: Write down 1 as the ratio of the reduced form estimate and the first stage estimate.] 1. Consider the simple regression model y=0+1D+u, where D is a binary variable. a) Let yi1 be yi with Di=1,yi0 be yi with Di=0, and i=1Nyi=i=1N1yi1+i=1N0yi0, where N1+N0=N. Show that y=NN1N1i=1N1yi1+NN0N0i=1N0yi0=Dy1+(1D)y0. b) Show that the OLS estimator: 1=i=1N(DiD)2i=1N(DiD)(yiy) can be written as y1y0. [Hint: Note that Diyyi=yi1 and Di2=Di. Therefore 1= i=1NDi2ND2i=1NDiyii=1NDyii=1NyDi+i=1NDy=i=1NDiND2i=1Nyi1Di=1Nyiyi=1NDi+NDy. c) Consider another simple model y=0+1x+u, and let z be a binary instrument variable for x. Show that the IV estimator: ^1=i=1N(ziz)(xix)i=1N(ziz)(yiy) can be written as the Wald estimator: x1x0y1y0. [Hint: Write down 1 as the ratio of the reduced form estimate and the first stage estimate.]