Thus, (βbw -βbg) converges in probability to zero much faster in CASE 1.2 than in
CASE 1.1. Nonetheless, the Hausman statistic is asymptotically χ2-distributed
in both cases.
CASE 2: We now consider two simple examples in which the time-varying
regressor xit is stationary without trend. Assume:
xit = Θ2,i + Ψ2,t + eit, (21)
where Θ21,i and Ψ21,t are fixed individual-specific and time-specific effects, re-
spectively. Define
P2,1 = limN→∞ nN Pi θ2,i; p2,2 = limN→∞ N ^{ 02^;
q2,1 = limτ→∞ T Pt ψ2,t; q2,2 = limτ→∞ T Pt ψ2,t.
Notice that if the Θ2,i are allowed to vary across different i, the xit become
cross-sectionally heteroskedastic. Similarly to CASE 1, we will demonstrate
that the convergence rates of the between estimator and the Hausman statistic
depend on whether the xit is cross-sectionally heteroskedastic or homogeneous.
CASE 2.1: Assume that the Θ2,i vary across different i; that is, p2,1 -p22,2 6=
0. With this assumption, we can easily show:
|
plimN,τ →∞ NT Ant |
= |
q2,2 |
- q22,1 + σe2 ; |
|
ιl^ 1 R plvm∙N,τ →∞ NfBNT |
= |
p2,2 |
- p22,1; |
|
plimN,τ →∞ nCnt |
= |
0; | |
|
11^ 1 R plimN →∞ N Hn |
= |
2 |
-N= aNT =⇒ N(0, σV(q2,2 - ql,ι + σe));
√NbNT =⇒ n(0, σU(p2,2 - p2,1));
√1NcNT =⇒ N(0, σUσ2).
With these results, we can show
σ-,'λ σ σ σ2 ʌ , x
√nt(bw - β) =⇒ N 0, ----- v ; (22)
(q2,2 - q22,1 + σe2)
√N(βb - β)=⇒ N ∣0, ---- ]; (23)
(p2,2 - p22,1)