Large-N and Large-T Properties of Panel Data Estimators and the Hausman Test

where we partition

xx

zx

xz

zz

conformably to the sizes of xit and zi , and set

)

Ξx = (Ξxx , Ξxz) . Similarly, by

Lemmas 2 and 4, under the fixed effect assumption, the second term of (92) is

θT √NTGχ,τ D-T

θT√NTGχ,τD^-,¹T {

Recall that it is assumed that NN → c < ∞. Also, recall that under the restric-
tions given in the theorem, Gx,T = diag (Ik22, Ik3) and Dχ,τ = diag √TTl_k22, D3τ) .
Then, letting λ_max (A) denote the maximum eigenvalue of matrix A, we can have

λmax (θT√NTGχ,TD-Т)

O ⁽¹⁾ ∖∕T ^λmax μ √TIk22 ^, D3T1^ → 0.

So, under the assumptions of Part (b), the probability limit of the numerator
of (88) is

1

NT

∑∑G∙,t ⅛⅛ + op (1).

(93)

Combining (91) and (93), we can obtain Part (b). ¥

Proof of Theorem 8

Using the notation in (87) , we can express the GLS estimator γ_g by

^

Yg ^- Y

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