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),
DT1√N μ βb - β ) = D D-T√n (βb - β) ) ⇒ N (0, σUΞ-4 ;
T V7b- γJ √N (^b - Y) ) u 7
(b) under Assumption 11 (local alternatives to random effects),
DT1√N μ β - β ) = D d-t√N ' - 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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