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 {θT [(A4 + A5) - CB3-¹ (B4 + B5)] }

_θ2 √_tg_d-√ *^d Dχ^,τxi(ui₁+zi) Λ

T ^{x,τ x,}Tj— (NPiDχ,τ^χizi)(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

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

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

√^nt⁽bg ^-^β) = ^[^Gx,TA1^Gx,T ⁺ op^(1)]¹[√^NTGx,T^a2 ⁺ 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):

θ²√TGχ,T I

√1N Pi xi(zi ⁺ zi)

— (nN ∑i Xizi)(NN ∑i zizi)

¹( √1N Pi zi ⁽zi ⁺ zi

₎´ . (92)