With a single explanatory variable, the equation used to obtain the between estimator iswh...

With a single explanatory variable, the equation used to obtain the between estimator is

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 ūi. is uncorrelated with but Cov(xit, ai) = σxa for all t (and i because of random sampling in the cross section).

(i) Letting be the between estimator, that is, the OLS estimator using the time aver¬ages, show that

where the probability limit is defined as N → ∞. [Hint: See equations]

(ii) Assume further that the xit, for all t = 1, 2,T, are uncorrelated with constant variance a2. Show that plim = β1 + T (σxaσ2x).

(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?

Equation