1. With a single explanatory variable, the equation used to obtain the between estimator is
yi
5 b0 1 b1xi
1
ai
1
ui, where the overbar represents the average over time. We can assume that E1ai
2 5 0 because we have included an intercept in the equation. Suppose that
ui
is uncorrelated with
xi, but Cov1xit,
ai
2 5 sxa
for all
t
(and
i
because of random sampling in the cross section).
(i) Letting b| 1 be the between estimator, that is, the OLS estimator using the time averages, show that plim b| 1 5 b1 1 sxa
/Var 1xi
2, where the probability limit is defined as
N
S `. [Hint:
See equations (5.5) and (5.6).]
(ii) Assume further that the
xit, for all
t
5 1, 2, p ,
T, are uncorrelated with constant variance s2x
. Show that plim b| 1 5 b1 1
T1sxa
/s2x
2.
(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?