Introductory Econometrics: A Modern Approach 6th Edition by Jeffrey M Wooldridge

Exercise 4

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 ?i. is uncorrelated with but Cov(xit, ai) = ?xa 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 = ?1 + T (?xa?2x).

(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

Step 1 of 3

Consider the single explanatory variable

The equation to obtain the between estimator is:

The overbar in the equation represents the average over time

Assumeand consider that is uncorrelated with

Also consider for all

Thus, it can be concluded that:

(i)

Given that is the between estimator

In the simple regression case, the inconsistency in (also called asymptotic bias) is given by:

Since,

This implies

Hence,