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



1
√NcNT


N (0, σz2σ2u).


Using these results and the fact that limN →∞T θ2T
as (N, T →∞),


= σ2v 2u , we can show that


τm Nntw - β) = N


(0, -σv-Y
p1,2q1


(13)


Tm√N(βb - β) = N


0, σ2u


(m+1)2


p1,2


P2,1 / ,


(14)


τmNTg - β) = τm√NT(βw


β)


1 σv2 (p1,2


P2,1)


T σU p1,2q1 (m + 1)2


τ mN (b


b - β)


p(1∕√T );

(15)


τm Nττβ (βw


βbg)


σ2v  (p1,2


P1,1)


σ2u p1,2q1(m+1)2


Tm NN (βb - β) + Op (1)


N 0,


σ4v    (p1,2


Pι,ι)


σ2u (p1,2q1)2(m+1)2   ;


(16)


plimN,T→∞NT2m+2[V arbw) -Varbg)] =


σ4v    (p1,2


Pι,ι)


σ2u (p1,2q1)2(m + 1)2 .


(17)


Several remarks follow. First, not surprisingly, all of the within, between and
GLS estimators are superconsistent when the time-varying regressor
xit contains
a time tr
end. Second, from (15), we can see that the two estimators β w andβ g
are Tm√NT-equivalent in the sense that (βw - βg) is op(1∕Tm√NT). This is
so because the second term in the right-hand side of (15) is
Op(1∕y∕T). Nonethe-
less, from (16), we can see that
w-βg) is Op(1∕Tm√NT2) and asymptotically
normal. These results indicate that the within and GLS estimators are equiva-
lent to each other by the order of T
m√NT, but not by the order of Tm√NT2.
Third, from (16) and (17), we can see that the Hausman statistic is asymptoti-
cally χ
2 -distributed. Fourth, when the model is estimated without an intercept
term because ζ = 0, all of the results (14)-(17) are still valid with
p1,2 replacing

p21,1).


(p1,2


Finally, (16) provides some intuition about the power property of the Haus-
man test. Observe that the asymptotic distribution of (β
w
that of (βb - β). From this, we can conjecture that the Hausman statistic is for
testing consistency of the between estimator β
bb , not exactly for testing the RE
assumption. In fact, the RE assumption (3) is not a necessary condition for the
asymptotic unbiasedness of β
b . For example, if the effect is correlated with zi ,

ъ ʌ ,

-β g ) depends on


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