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

× (B4 + B5^{) -} C0 μθ2A1 ⁺ A3)   μθ2^A2 + (A4 + A5))

Part (a)

Notice that

C0Dx,T Tθ2p √T∙X-T'Gx,TA1Gx,τ Jχ,T√^t + Dx,TA3Dχ,Tj

Jx,T²

Op I τ I = op⁽¹⁾.

The third equality holds because the limit of G_x,TA₁G_x,T is positive definite
(by Lemma 2(a) and Assumption 12), Tθ²_T = O(1), and

Dx,T C, Gx,TA1Gx,T,Dx,TA3Dx,T = Op (1)

(by Lemma 2(a), (c)). The last equality results from the fact that

J ^-T¹
√J = o (1).

Thus, as (N, T →∞) , the denominator of (94) is

Next, under both the random effects assumption (Assumption 9) and the
local alternatives (Assumption 11), it follows from Lemmas 2 and 3 that the
second term in the numerator of (94) is

C0 μθ2Ai+A3)  μθ2√NA2+√N (A4+A5))

C0Dχ,τ Tθθ2√√TJχ,τGχ,τAiGχ,τ Jχ,τ√T + Dχ,τA3Dχ,τ^
× TTJ ^jχ,τ√NTG_xτA2 + √NDχ,τ (A4 + A5)^

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