Suppose you have the same data as in Exercise 13.7

(panel data with two periods, \(n\) observations), but ignore the \(W\) regressor. Consider the alternative regression model

\[ Y_{i t}=\beta_{0}+\beta_{1} X_{i t}+\beta_{2} G_{i}+\beta_{3} D_{t}+u_{i t} \]

where \(G_{i}=1\) if the individual is in the treatment group and \(G_{i}=0\) if the individual is in the control group. Show that the OLS estimator of \(\beta_{1}\) is the differences-in-differences estimator in Equation (13.4).

Equation (13.4)

Bdiffs-in-diffs = (treatment, after = AYtreatment - AY control Ytreatment, before)- - - (control, after control, before) (13.4)