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



B3 - C0


μθT A1+A3)  C


-1


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

. (94)


Part (a)

Notice that

C0


μθTA1+A3)  C


C0Dx,T 2p √T∙X-T'Gx,TA1Gx,τ Jχ,Tt + Dx,TA3Dχ,Tj

1

Dx,T C


2T(C0Dx,T


-1Gx,TA1Gx,T

+ jt Dx,T A3Dx,τJ


-1
x,T


1 JxT1

T (Dχ,τC)


Jx,T2

Op I τ I = op(1).

The third equality holds because the limit of Gx,TA1Gx,T is positive definite
(by Lemma 2(a) and Assumption 12),
2T = 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

B3 - C0


(⅛ A1+A3)


-1

C=B3+op(1).


(95)


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)  μθ2NA2+N (A4+A5))

-1


C0Dχ,τ Tθθ2√√TJχ,τGχ,τAiGχ,τ Jχ,τT + Dχ,τA3Dχ,τ^
× TTJ jχ,τNTGxτA2 + NDχ,τ (A4 + A5)^

58



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