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

Thus, (β^b_w -β^b_g) 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:

-N= ^aNT  =⇒ N(0^{, σ}V(q2,2 ^- ql,ι + ^σe));

√NbNT =⇒ ⁿ(^{0, σ}U(^p2,2 ^{- p}2,1));

√1NcNT =⇒ ^N(0, ^σU^σ2).

With these results, we can show

σ-,'λ    σ σ        σ²      ʌ            , _x

√^nt(bw - β) =⇒ N 0^, ----- v        ;          (22)

(q2,2 - q₂²_,1 + σ_e²)

√N(βb - β)=⇒ N ∣0, ---- ];              (23)

(p2,2 - p₂²_,1)

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