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

CASE 3.2: We now assume m = 1/2. For this case, define

P3,1 = limN→∞ N Pi ll,; p3,2 = limN→∞ N Pi ^π2^,

and q₃ = IimT→∞ T ₁-_m R01 r ^mdr = γ1m for m ≤ 2. With this notation, a
little algebra shows that as (N, T →∞),

both of which imply that the Hausman statistic is asymptotically χ²-distributed.

CASE 3.3: Finally, we consider the case in which m ∈ [0,1 ), where the
correlation between x_it and z_i decays over time slowly. Note that the correlation
remains constant over time if m = 0. We can show

λ∕ZN2m (βb — β) =⇒ N (0, 7------^σu ʌ 2 2 ) .

T ^{2m b}                   (p3,2 - p²_3,1)q₃²σ_z²

Observe that the asymptotic distribution of β^b_b no longer depends on σ_e² . This
implies that the term eit in xit dominates Πizi /t^m in the asymptotics forβ _b .
Furthermore, the convergency rate ofβ _b now depend on m. Specifically, so long
as m < 1, the convergence rate increases as m decreases. In particular, when
the correlation between x_it and z_i remains constant over time (m = 0), the
between estimator b_b is N-—consistent as in CASE 2.1. This is so because,
in this case, the term Πizi takes the role of the Θi term in CASE 2.1. Finally,
the following results indicate that the convergence rate of the Hausman statistic

HM_NT also depends on m:

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