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

but not with x_it , β^b_b could be asymptotically unbiased, as we find in section 4.⁸
Thus, the Hausman test does not have power to detect the violations of the RE
assumption in the direction in which β_b remains asymptotically unbiased. This
issue will be further explored later.

CASE 1.2: We now assume that Θ_1,i = Θ₁ for all i; that is, p_1,2 - p²_1,1 =0.
As we have discussed above, the terms B_NT, C_NT, and b_NT do not depend
on T ^m . Furthermore, the asymptotic distribution of the between estimator β_b
depends on the eit instead of the Θ1,i . Specifically, we can easily show that
while other asymptotics are essentially the same as those obtained for CASE
1.1. The asymptotic distributions of the within and GLS estimators are the
same under both CASE 1.1 and CASE 1.2, but those of the between and the
Hausman statistic are different. For CASE 1.2,

Nβ(bb - β)=⇒ N (o, σ2) ;                (18)

T^2m√NT3(bw - .)  =  -σ²θ^ʌ/r(bb - β) + Op(1)

=⇒ ⁿ (⁰∙ Θ⅛≈ );              ⁽¹⁹⁾

plimN,τ→∞NT^4m+3[Var(βw) - Var(βg)] =        .        (20)

Θ₁q₁σ²_u

Several comments follow. First, the between estimator is no longer supercon-
sistent if the time trend in xit is common to every individual (i.e., the parameters
Θ1_,i are the same for i). An interesting result is obtained when N/T → c<∞.
For this case, the between estimator is inconsistent, although it is still asymp-
totically unbiased. This implies that the between estimator is an inconsistent
estimator for the analysis of cross-sectionally homogeneous panel data unless
N is substantially larger than T. Second, the convergence rate of (β _w -β _g)
is quite different between CASE 1.1 and 1.2. Notice that the convergence rate
of (β_w - β_g) is √NT^4m+3 for CASE 1.2, while it is √NT^2m+2 for CASE 1.1.

⁸In contrast, the between estimator of γ, γb_b , is inconsistent whenever the RE assumption
is violated.

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