13.6 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 posttreatment period. Suppose that the treatment is binary; that is, suppose that Xit = 1 if the ith individual is in the treatment group and t = 2, and Xit = 0 otherwise. Further suppose that the treatment effect can be modeled using the specification Yit = ai + b1Xit + uit, where ai are individual-specific effects [see Equation (13.11)] with a mean of zero and a variance of s2 a and uit is an error term, where uit is homoskedastic, cov(ui1, ui2) = 0, and cov (uit, ai) = 0 for all i. Let b n differences 1 denote the differences estimator—that is, the OLS estimator in a regression of Yi 2 on Xi2 with an intercept—and let b n diffs - in - diffs 1 denote the differences-in-differences estimator—that is, the estimator of b1 based on the OLS regression of Yi = Yi2 - Yi1 against Xi = Xi2 - Xi1 and an intercept.

a. Show that nvar(b ndifferences 1 )¡(s2 u + s2 a)>var(Xi2). (Hint: Use the homoskedasticity-only formulas for the variance of the OLS estimator in Appendix 5.1.)

b. Show that nvar (b ndiffs - in - diffs 1 )¡2s2 u>var(Xi2). (Hint: Note that Xi2 - Xi1 = Xi2. Why?)

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?