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 fi,T ,fi, and gi are integrable functions in probability space
, 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 supi gi , then Efi,T Efi
uniformly in i as T →∞.

Proof

Let hi,T = |fi,T


and hi,T 0 uniformly in i


fi | . Under given assumptions, 0 supi hi,T 2 supi gi


as T →∞. Then, by Fatou’s Lemma


2E sup gi
i


E liminf 2 sup gi

lim inf E 2 sup gi -


sup hi,T


sup hi,T


2E sup gi- lim sup E sup hi,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 (fi,T - fi)| ≤lim sup sup Ehi,T lim sup E sup hi,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 supi |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 tt0 implies that

sup |ait - ai | < ε.
i

Then, Part (a) follows because tt0 implies

sup 1 X (ait - ai)
iT


1 X sup |ait

Tt i


- ai | < ε.


31




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