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

as (N, T → ∞). Now, recall that

^d2T

^D3T

diag (D21τ,D22τ) = diag ∣I_k2,, √T⅛₂} ;

diag (D31τ,D32τ,D33τ) = diag √TIl_k3γ, VTl_k32, (Tmh)_ft) ,

where 0 ≤ mh,33 < ɪ for all h = 1, ...,k₃₃. Therefore, as (N, T → ∞) ,

Finally, by the Cauchy-Schwarz inequality, the off-diagonal block component

I1,21 I1,22 I1,31 I1,32 I1,33 ) →p 0,

as (N, T → ∞).

Proof of (49): Recallthat

E^t_iuj_i = (Ep^E^EF-zi x3,i ^,z'i)' ;

E*w = (Ex1, Ex2, Ef_zX3, z^,).

Write

I2,i,NT

/ D1τ (EX1,_l - EX1) ∖

EX21,_l - EX21

= D31τ ((EFz_i X31,i - EX31,i) - (EFz5⅛1 - EX31) + (EX31,i - EX31))

D32τ ((EFz_i X32,i - EX32,i) - (EfzX32 - EX32) + (EX32,i - EX32))
D33τ ((Eri^X33,i ^-^EX33,i) - (Efz^x33 ^-^EX33) ^{+ (}^EX33,i ^-^EX33))

∖ Zi - z /

To obtain the required result, we use Lemma 12. By Assumption 6, as T → ∞,
we have

I2,i,NT → I2,i,N a.s. uniformly in i, (52)

where

/ H1Θ 1,i ∖

Θ 21,i

I - 0

I21,i,N = 1 г ₁ 1 .

H32 L^g32,i ⁽zi) ^- N ∑i ^g32,i (zi)J ⁺ H32 ^μg32,i ^- N ∑i ^μg32,i

H³³ [^g³³^,i⁽^zⁱ⁾^- N Pi ^g^33,i⁽^zⁱ⁾] ⁺ H³³^μg33,i ^- N Pi ^μg32,i

V ^zi ^- N Pi ^zi /