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
More intriguing information
1. What Lessons for Economic Development Can We Draw from the Champagne Fairs?2. A Computational Model of Children's Semantic Memory
3. The name is absent
4. The name is absent
5. Federal Tax-Transfer Policy and Intergovernmental Pre-Commitment
6. Picture recognition in animals and humans
7. Moi individuel et moi cosmique Dans la pensee de Romain Rolland
8. The name is absent
9. The name is absent
10. The name is absent