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

For Part (b), notice that

sup ∣a²_t — a²∣ ≤ sup ∣αi_t + αi∣ sup ∣αi_t — αi∣
≤ M sup |ait - ai | → 0

as t →∞, where the last inequality holds because sup_i ∣ai ∣ < ∞ and sup ∣ait — ai ∣
→ 0. Then, by Part (a), we can obtain the desired result. ¥

Lemma 12 Suppose that X_i,T and X_i are sequences of random vectors. Sup-
pose that Xi_,T → Xi in probability (or almost surely) uniformly in i as T →∞,
and NN Pi Xi → X in probability (or almost surely) as N → ∞. Then, as
(N,T→∞),

-1 X x_iT → x

N ʌ i^,T

i

in probability (or almost surely).

Proof

We only prove the lemma for the case of convergence in probability, because
the almost sure convergence case can be proven by the similar fashion. Since
NI Pi Xi →_p X as N → ∞ and X →_p Xi uniformly in i as T → ∞, for given
ε, δ > 0, we can choose N0 and T0 such that

P I 1 X Xi — X > ^εj ≤ ^δ ;

I Nγ ⁱ       2 ∫ ≤ 2;

P {^sup ∣∣Xi,T ^- Xik > 2 j∙ ≤ 2^,

whenever N ≥ N₀ and T ≥ T₀ . Now, suppose that N ≥ N₀ and T ≥ T₀ . Then,

P {^{sup k (X}i,T ^- Xi)^k > 2J^, + 2= ^δ ¥

Lemma 13 Suppose that a sequence of random vectors Z_i is independently dis-
tributed across i. Let F_zi = σ (Zi) . Assume that QiT (k × k) is a sequence of
independent random matrices across i satisfying

sup EF_zi kQiT k 1 {kQiT k >M} → 0 a.s.,              (36)

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