1. Consider a univariate regression model Y = XB + +e, with E[e | Z] =0 and Z is binary. (a) Show that EY Z - 1] - EY Z=0] B = EX Z = 1] -EX|Z=0] Define the group means: Y = 1_LY, Y1 = LAZY, Yo = ( 1-Z;)Y; Zizi Zi El=1 (1-Zi) X = 1CLIX;, X1 = Lil ZiXi EnZi , Xo = 2(1-Zi)Xi E1 (1-Zi) A natural moment estimator for B is B = Y1 - Yo X1 - Xo' which is called the 'Wald estimator'. The another estimator called 'iv estimator' for this regression is given by Y1 - Y Biv = X 1 - X (b) Show that Y = ZY1 + (1 - Z)Yo. Similarly, X = ZX1 + (1 -Z)Xo. (c) Show that the iv estimator is equal to the Wald estimator