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

where G₁ = G^-_,¹_TA₁^-¹G^-_,¹_T .

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= J_x^-_,T¹ Dx,T F1Dx,T J_x^-_,T¹ =Op(1), (102)

since J_x^-_,T¹ = O (1) . Thus,

σ2θ²_r (g-,¹tRiG⅛´ = Op (θT) = op (1). (103)

Now, consider

θτ √NT θT G^-,¹_τ R2

= θτ√T [√NθTg^-,¹_t (A^-¹F1A^-¹F2) + √NθTg^-,¹_tRi (A2 + θTF2)]

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

= θ_τ 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 θ_τ √T = O (1) , we have

θτVntθTg^—,¹_tR2 = O (1) [θ²_τOp (1) + op (1)] = op (1). (104)

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

HMNT

= ^θ^τ√NT [(-T¹DDχ,τFιDχ,τ J-¹τ) (GιGχ,τA2) - J-TDχ,τF2 + op (1)] 0
σ_v

× ( Jχ^-τDχ,τFτDχ,τ J-τ + op (1)) -^l

×^θ^τ√NT [ J-τDχ,τFτDχ,τ JχΓ (GτGχ,τA2) - J-τDχ,τF2 + op (1)]
σv

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

σv σv

+op (1) .