13.6 Suppose there are panel data for T = 2 time periods for a randomized controlled experiment, where the first observation 1t = 12 is taken before the experiment and the second observation 1t = 22 is for the posttreatment period. Suppose the treatment is binary; that is, suppose Xit = 1 if the ith individual is in the treatment group and t = 2, and Xit = 0 otherwise. Further suppose the treatment effect can be modeled using the specification Yit = ai + b1Xit + uit, where ai are individual-specific effects with a mean of 0 and a variance of s2 a

and uit is an error term, where uit is homoskedastic, cov1ui1, ui22 = 0, and cov 1uit, ai2 = 0 for all i. Let b ndifferences 1 denote the differences estimator—

that is, the OLS estimator in a regression of Yi2 on Xi2 with an intercept—

and let b ndiffs@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 n var1b ndifferences 1 2 ¡ 1s2 u + s2 a2 >var1Xi22. (Hint: Use the homoskedasticity-only formulas for the variance of the OLS estimator in Appendix 5.1.)

b. Show that n var1b ndiffs - in - diffs 1 2 ¡2s2 u > var1Xi22. (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?