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

Two points may be worth mentioning here. First, Assumption 4(i) dictates
that the regressors in x_3,it are mixing conditional on F_zi . Alternately, we may
assume that x3_,it and zi have some common factors, say fi (i =1, ..., N), on
which, conditionally, the variables in x3_,it are mixing. This alternative assump-
tion does not generate any materially different asymptotic result. Second, as
discussed in Section 3, Assumption 5(ii) warrants that the conditional α-mixing
coefficient α_zi (d) is measurable.

Panel data estimators of individual coefficients have different convergence
rates depending on the types of the corresponding regressors. To address these
differences, we define:

D_x,T = diag (D_1T,D_2T,D_3T);

DT = diag (Dx,T,Ig) ,

where

D1T = diag T^-m1,...,T^-mk1 ;

D2T = diag (D21τ,D22τ) = diag(l^ ,VTlk22') ;

D3T = diag (D31T, D32T, D33T)

= diag (√TI_k31, √Tl_k32 ,T^m1³³ ,...,T^m    ´ .

Observe that D1T, D2T, and D3T are conformable to regressor vectors x1_,it,
x_2,it, and x_3,it , respectively, while D_T and ]_g are to x_it and z_i , respectively.
The diagonal matrix Dt is chosen so that plimN→∞ -N Pi DtWiWi0Dt is well
defined and finite. For future use, we also define

Gx,T = diag(D1T,]k21,]k22,]k3);

Jx,T = diag(]k1,]k21,D22T,D3T),

so that

Dx,T = Gx,T Jx,T.

Using this notation, we make the following regularity assumptions on the un-
conditional and conditional means of regressors:

Assumption 6 (convergence as T →∞): Defining t =[Tr], we assume that
the following restrictions hold as T →∞.

(i) Let τ1 (r) = diag (r^m1,1 , ..., r^mk1^,1 ) ,where mh_,1 is defined in Assumption

2. Then,

D1TE (x1,it) → τ1 (r) Θ1,i

uniformly in i and r ∈ [0, 1], for some Θ_1,i = (Θ_1,1,i, ..., Θ_k1,1,i)⁰ with
sup_i kΘ1,i k < ∞.

(ii) Θ21_,it → Θ21_,i and Θ3_,it → Θ3_,i uniformly in i with sup_i kΘ21_,i k < ∞ and
sup_i kΘ3,i k < ∞.

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