4.2 With a single explanatory variable, the equation used to obtain the between estimator is y¯i
0  1x¯i  ai  u¯i , where the overbar represents the average over time. We can assume that E(ai )
0 because we have included an intercept in the equation. Suppose that u¯i is uncorrelated with x¯i , but Cov(xit,ai )
xa for all t (and i because of random sampling in the cross section). (i) Letting ˜ 1 be the between estimator, that is, the OLS estimator using the time averages, show that plim ˜ 1
1  xa/Var(x¯i ), where the probability limit is defined as N → . [Hint: See equations (5.5) and (5.6).] (ii) Assume further that the xit, for all t
1,2, …, T, are uncorrelated with constant variance x 2. Show that plim ˜ 1
1  T (iii) If the explanatory variables are not very highly correlated across time, what does part (ii) suggest about whether the inconsistency in the between estimator is smaller when there are more time periods?