Suppose 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 postreatment period. Suppose the treatment is binary; that is, suppose \(X_{i t}=1\) if the \(i^{\text {th }}\) individual is in the treatment group and \(t=2\), and \(X_{i t}=0\) otherwise. Further suppose the treatment effect can be modeled using the specification

\[ Y_{i t}=\alpha_{i}+\beta_{1} X_{i t}+u_{i t} \]

where \(\alpha_{i}\) are individual-specific effects with a mean of 0 and a variance of \(\sigma_{\alpha}^{2}\) and \(u_{i t}\) is an error term, where \(u_{i t}\) is homoskedastic, \(\operatorname{cov}\left(u_{i 1}, u_{i 2}\right)=0\), and \(\operatorname{cov}\left(u_{i t}, \alpha_{i}\right)=0\) for all \(i\). Let \(\hat{\beta}_{1}^{\text {differences }}\) denote the differences estimator that is, the OLS estimator in a regression of \(Y_{i 2}\) on \(X_{i 2}\) with an interceptand let \(\hat{\beta}_{1}^{\text {diffs-in-diffs }}\) denote the differences-in-differences estimator - that is, the estimator of \(\beta_{1}\) based on the OLS regression of \(\Delta Y_{i}=Y_{i 2}-Y_{i 1}\) against \(\Delta X_{i}=X_{i 2}-X_{i 1}\) and an intercept.

a. Show that \(n \operatorname{var}\left(\hat{\beta}_{1}^{\text {differences }}\right) \longrightarrow\left(\sigma_{u}^{2}+\sigma_{\alpha}^{2}\right) / \operatorname{var}\left(X_{i 2}\right)\). (Hint: Use the homoskedasticity-only formulas for the variance of the OLS estimator in Appendix 5.1.)

b. Show that \(n \operatorname{var}\left(\hat{\beta}_{1}^{\text {diffs -in-diffs }}\right) \longrightarrow 2 \sigma_{u}^{2} / \operatorname{var}\left(X_{i 2}\right)\). (Hint: Note that \(X_{i 2}-X_{i 1}=X_{i 2}\). 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?