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



where G1 = G-,1TA1-1G-,1T .

Notice that

θT GR1G-,1T´ = θTG-T (A1 + θ2Fι)-1 G⅛
×Gx,TF1Gx,TG1Gx,TF1Gx,TG1.

Lemma 2(a) and Assumption 12 imply

G1 =Op(1).

In addition, by Lemma 2(c)

Gx,TF1Gx,T= Jx-,T1 Dx,T F1Dx,T Jx-,T1 =Op(1),           (102)

since Jx-,T1 = O (1) . Thus,

σ2θ2r (g-,1tRiG⅛´ = Op (θT) = op (1).             (103)

Now, consider

θτ √NT θT G-,1τ R2

=  θτT [√Tg-,1t (A-1F1A-1F2) + Tg-,1tRi (A2 + θTF2)]

n'2  -~ 1   - 1z-y -1            -   ʌf/ɑ-1   -1 -1        √~V

= θτ VT


θT Gx,T A1 Gx,T (Gx,TF1Gx,T) Gx,T A1 Gx,T Gx,T NF2

+θT (g-,1tRiG-,tX√NGχ,τA2 + θTNGχ,τF2)

Under the local alternatives (Assumption 11), we may deduce from Lemmas 2
and 3 that

Gι, Gχ,τFιGχ,τ, Gχ,τVNf2 = Op (1) ;

θT GRiG-,T ) , NGχ,τA2 = op (1).

Since θτ T = O (1) , we have

θτVntθTg,1tR2 = O (1) 2τOp (1) + op (1)] = op (1).      (104)

Using the results (103) and (104) , we now can approximate the Hausman
statistic as follows:

HMNT

= θτNT [(-T1DDχ,τFιDχ,τ J-1τ) (GιGχ,τA2) - J-TDχ,τF2 + op (1)] 0
σv

× ( Jχ-τDχ,τFτDχ,τ J-τ + op (1)) -l

×θτNT [ J-τDχ,τFτDχ,τ JχΓ (GτGχ,τA2) - J-τDχ,τF2 + op (1)]
σv

= θτNT (Dχ,τF2)0 (Dχ,τFτDχ,τ)-1 — √NT (Dχ,τF2)

σv                                  σv

+op (1) .

62



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