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



By Lemma 2,

nN ∑i Dx,T XiXiDχ,τ

=Op(1),


nN Pi Dχ,τzizi) ( nN Pi zizi) -1 (nN Pi ziziDχ,T)
and by definition,

Gx,T Dx-,1T = Jx-,T1 = O(1) ,

as (N, T →∞) . Thus,

θTGχ,T ©A3 CB-1C0} Gχ,T = Op (θT) = Op (1).          (89)

Next, consider

√NTGχ,T T [(A4 + A5) - CB3-1 (B4 + B5)] }

θ2 tg  d-√           *d,τxi(ui1+zi)          Λ

T     x,τ x,Tj— (NPiDχ,τχizi)(N Pi ziz0)  NP Pizi (ui+zi))ʃ

By Lemma 2, under the local alternatives to random effects (Assumption 11),

1N Pi Dx,Txi(ui + vi)
nN Pi Dx,T xi¾i)( nN Pi ziz0)  N 1N

P          ´   = Op(1).

i zi (Ui + Vi) ) J


By definition,

Thus,


θT √ΤGχ,T D-,T = O


ntgxt ©θT £(A4 + A5)-CB-1 (B4 + B5)]}=Op{√0 = op(1).   (90)

Substituting (89) and (90) into (88), we have

nt(bg - β) = [Gx,TA1Gx,T + op(1)] 1[NTGx,Ta2 + op(1)]
= NT (bw - β) + Op(1).

The last equality results from Lemma 2(a), (b) and Theorem 5. ¥

Part (b)

Similarly to Part (a), we can easily show that under the assumptions given
in Part (b), the denominator in (88) is

NT X X √ Gx,T (xit   xi) (xit   xi) Gx,T + op (1) .          (91)

Consider the second term of the numerator of (88):

θ2TGχ,T I


1N Pi xi(zi + zi)


(nN ∑i Xizi)(NN ∑i zizi)


1( 1N Pi zi (zi + zi


)´    .      (92)


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