1
√NcNT
N (0, σz2σ2u).
Using these results and the fact that limN →∞T θ2T
as (N, T →∞),
= σ2v /σ2u , we can show that
τm Nnt (βw - β) =⇒ N
(0, -σv-Y
p1,2q1
(13)
Tm√N(βb - β) =⇒ N
0, σ2u
(m+1)2
p1,2
P2,1 / ,
(14)
τm√NT(βg - β) = τm√NT(βw
β)
1 σv2 (p1,2
P2,1)
T σU p1,2q1 (m + 1)2
τ m√N (b
b - β)
+θp(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 ar(βbw) -Var(βbg)] =
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 trend. 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 (βw-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 Tm√NT, but not by the order of Tm√NT2.
Third, from (16) and (17), we can see that the Hausman statistic is asymptoti-
cally χ2 -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 (βw
that of (βb - β). From this, we can conjecture that the Hausman statistic is for
testing consistency of the between estimator βbb , not exactly for testing the RE
assumption. In fact, the RE assumption (3) is not a necessary condition for the
asymptotic unbiasedness of βb . For example, if the effect is correlated with zi ,
ъ ʌ ,
-β g ) depends on
10
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