Introduction to Econometrics 3rd Edition by James Stock, Mark Watson

Exercise 9

Suppose that there are panel data for T = 2 time periods for a randomized controlled experiment, where the first observation (t = 1) is taken before the experiment and the second observation (t = 2) is for the post-treatment period. Suppose that the treatment is binary; that is, suppose that X it = 1 if the i th individual is in the treatment group and t = 2, and X it = 0 otherwise. Further suppose that the treatment effect can be modeled using the specification
Y it = t + ß i X it + u it
where , are individual-specific effects [see Equation (13.11)] with a mean of zero and a variance of 2 and u it is an error term, where u it is homoskedas-tic, cov( u it , u i1 ) = 0, and cov( u it , i ) for all i. Let
denote the differences estimator, that is, the OLS estimator in a regression of Y i2 on X i2 with an intercept, and let
denote the differences-in-differ-ences estimator, that is, the estimator of ß 1 based on the OLS regression of Y i = Y i2 - Y i1 and an intercept.
a. Show that nvar
.
b. Show that nvar

c. Based on your answers to (a) and (b), when would you prefer the differences-in-differences estimator over the differences estimator, based purely on efficiency considerations