Suppose you have panel data from an experiment with 7 = 2 periods (so = = 1.2). Consider the panel data regression model with fixed individual and time effects and individual characteristics W; that do not change over time, such as gender. Let the treatment be binary, so X 1 for = 2 for the indi- viduals in the treatment group and let X = 0 otherwise. Consider the pop- ulation regression model Y = a + BX+B(D, x W) + BD, + Vit where

a, are individual fixed effects, D, is the binary variable that equals 1 if =2 and equals 0 it=1, D, x W, is the product of D, and W., and the a's and 's are unknown coefficients. Let AY, Y-Y. Derive Equation (13.5) (in the case of a single W regressor, so r = 1) from this population regression model.