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

as (N, T → ∞). Now, recall that

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

0

=    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 ^{- EX}33,i) - (Efz^x33 ^{- EX}33) ^{+ (EX}33,i ^{- EX}33))

∖                              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

0

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 ^{g33,i (zi)}] ⁺ H^{33 μ}g33,i ^- N Pi ^μg32,i

V                      ^zi ^- N Pi ^zi                      /

40

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