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



Theorem 5 (asymptotic distribution of the within estimator): Under Assump-
tions 1-8 and Assumption 12, as
(N, T →∞),

NTg⅛(βw - β) N (0, σVΨ-1) .

Theorem 6 (asymptotic distribution of the between estimator): Suppose that
Assumption 1-8 and 12 hold. As
(N,T →∞),

(a) under Assumption 9 (random effects),

DT1N μ βb - β ) = D D-Tn (βb - β) ) N (0, σUΞ-4 ;

T V7b- γJN (^b - Y)      ) u 7

(b) under Assumption 11 (local alternatives to random effects),

DT1N μ β - β ) = D d-tN  ' - β)    N (Ξλ, σUΞ-^ .

T    ʊb - 7√   N (Yb - Y) J

Theorem 7 (asymptotic distribution of the GLS estimator of β): Suppose that
Assumptions 1-8 and 12 hold.

(a) Under Assumption 11 (local alternatives to random effects),

NTG-,1T (βg - β) = NTG-,1T (βw - β) + Op (1) ,

as (N, T →∞) .

(b) Suppose that Assumption 10 (fixed effects) holds. Partition λ =(λx , λz)0
conformably to the sizes of xit and zi. Assume that λx 6= 0k×1 .IfN/T c<
and the included regressors are only of the
x22,it- and x3,it -types (no trends and
no cross-sectional heteroskedasticity in
xit), then

NTG-,1T (βg - β) = NTG-,1T (βw - β) + Op (1) .

Theorem 8 (asymptotic distribution of the GLS estimator ofγ ): Suppose that
Assumptions 1-8 and 12 hold. Define
lz0 = 0g×k...Ig . Then, the following
statements hold as
(N, T →∞) .

(a) Under Assumption 11 (local alternatives to random effects),

√N (γg - γ)


(n X zizi


(√N X êiu! + op (1)


1 lz0 Ξλ, σ2u (lz0 Ξlz)-1 .

(b) Under Assumption10 (fixed effects),

(^g - γ) →p (lZξ1z)


1 l0 Ξλ.
z


27




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