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
= θτ√T [√NθTg-,1t (A-1F1A-1F2) + √NθTg-,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 + θT√NGχ,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) [θ2τOp (1) + op (1)] = op (1). (104)
Using the results (103) and (104) , we now can approximate the Hausman
statistic as follows:
HMNT
= θτ√NT [(-T1DDχ,τFιDχ,τ J-1τ) (GιGχ,τA2) - J-TDχ,τF2 + op (1)] 0
σv
× ( Jχ-τDχ,τFτDχ,τ J-τ + op (1)) -l
×θτ√NT [ J-τDχ,τFτDχ,τ JχΓ (GτGχ,τA2) - J-τDχ,τF2 + op (1)]
σv
= θτ√NT (Dχ,τF2)0 (Dχ,τFτDχ,τ)-1 — √NT (Dχ,τF2)
σv σv
+op (1) .
62
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