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

6 Appendix A: Preliminary Results

We here provide some preliminary lemmas that are useful to prove the main
results in Section 4. From now on, we use the notation M to denote a generic
constant, if no explanation follows.

Lemma 10 Let f_i,T ,f_i, and g_i are integrable functions in probability space
(Ω, F,P). If fi,τ → fi a.s. uniformly in i as T → ∞, and there exists gi
such that |fi_,T | ≤ gi for all i and T with E sup_i gi < ∞, then Efi_,T → Efi
uniformly in i as T →∞.

2E sup g_i- lim sup E sup h_i,T ,
i      T→∞ i

from which we can deduce

lim sup E sup hi_,T ≤ 0.

T→∞    i

Then, since

0 ≤lim sup sup |E (f_i,T - f_i)| ≤lim sup sup Eh_i,T ≤lim sup E sup h_i,T ≤ 0,
T→∞ i              T→∞ i        T→∞    i

we have the required result: Efi_,T → Efi uniformly in i as T →∞. ¥

The following lemma is a uniform version of the Toeplitz lemma.

Lemma 11 Let ait be a sequence of real numbers such that ait → ai uniformly
in i as t → ∞ with sup_i |ai | < M. Then, (a) TT ɪ^t ait → ai uniformly in i, and
(b) TT Pt a2 → a2 uniformly in i.

Proof

From the uniform convergence of ait , for a given ε > 0, we can choose t0
such that t ≥ t0 implies that

sup |ait - ai | < ε.
i

Then, Part (a) follows because t ≥ t₀ implies

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