Suppose that you have panel data from an experiment with T = 2 periods (so t = 1, 2). Consider the panel data regression model with fixed individual and time effects and individual characteristics Wi that do not change over time, such as gender. Let the treatment be binary, so that Xit = 1 for t = 2 for the individuals in the treatment group and let Xit = 0 otherwise. Consider the population regression model

where αi are individual fixed effects, Dt is the binary variable that equals 1 if t = 2 and equals 0 if t = 1, Dt × Wi is the product of Dt and Wi, and the α's and β's are unknown coefficients. Let ˆ†Yi = Yi2 - Yi1. Derive Equation (13.6) (in the case of a single W regressor, so r = 1) from this population regression model.

Y, = a, + B, X, + B2(D, x W) + B,D, + v %3D