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Thus, we must establish the limiting distribution of

where E [w] = 0. [See (5-3).] We can use the multivariate Lindberg-Feller version of the central limit theorem (D.19.A) to obtain the limiting distribution of V=w.4 Using that formulation, w is the average of n independent random vectors wi = xisi, with means 0 and variances

3White (2001) continues this line of analysis.

4Note that the Lindberg-Levy variant does not apply because Var[wi] is not necessarily constant.

TABLE 5.1 Grenander Conditions for Well Behaved Data

G1. For each column of X, xk, if d^k = x'kxk, then limnu« = +<. Hence, xk does not degenerate to a sequence of zeros. Sums of squares will continue to grow as the sample size increases. No variable will degenerate to a sequence of zeros.

G2. Limnu<xfk/4k = 0 for all i = 1, ..., n. This condition implies that no single observation will ever dominate xkxk, and as n u<, individual observations will become less important. G3. Let Rn be the sample correlation matrix of the columns of X, excluding the constant term if there is one. Then limnu« Rn = C, a positive definite matrix. This condition implies that the full rank condition will always be met. We have already assumed that X has full rank in a finite sample, so this assumption ensures that the condition will never be violated.

The variance of is a 2Q n = a > + Q2 +-----+ Qn]. (5-10)

As long as the sum is not dominated by any particular term and the regressors are well behaved, which in this case means that (5-1) holds, lim a2Qn = a2Q. (5-11)

Therefore, we may apply the Lindberg-Feller central limit theorem to the vector „Jn w, as we did in Section D.3 for the univariate case ^,/nxc. We now have the elements we need for a formal result. If [xi ei ], i = 1,...,n are independent vectors distributed with mean 0 and variance a2Qi < <, and if (5-1) holds, then

It then follows that

Combining terms,

-n(b - p) -U N[0, a2Q-1]. (5-14) Using the technique of Section D.3, we obtain the asymptotic distribution of b:

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