Consistency of the Least Squares Estimator
We shall obtain a useful set of conditions on X and X for the consistency of LS in Model 6. We shall use the following lemma in matrix analysis.
Lemma 6.1.1. Let A and В be nonnegative definite matrices of size n. Then tr (AB) ^ A, (A) tr B, where A, (A) denotes the largest characteristic root of A.
Proof. Let H be a matrix such that H’AH = D, diagonal, and H’H = I. Then, tr (AB) = tr (H AHH’BH) – tr DQ, where Q = H’BH. Let rf, be the ith diagonal element of D and qit be of Q. Then
tr (DQ) = 2 dfitt ^ max dt • 2?« = ^(A)trB-
i-i » i-i
Now we can prove the following theorem.
Theorem 6.1.2. In Model 6 assume
(A) АДХ) bounded for all t,
Then plim^^ fi=fi.
Proof. We have
tr Vp = lr [(X/X)-‘X, XX(X’X)-1]
S A,(X) tr [X(X’X)-2X’] by Lemma 6.1.1 = A,(X)tr(X’X)-‘
— Л A/X’X)
But the last term converges to 0 because of assumptions A and B.
Note that Theorem 3.5.1 is a special case of Theorem 6.1.2. One interesting implication of Theorems 6.1.2 and 5.2.3 is that LS is consistent if u follows a stationary time series satisfying (5.2.36).
We have not proved the asymptotic normality of LS or GLS in this section because the proof would require additional specific assumptions about the generation of u.