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

By the Cauchy-Schwarz inequality,

kQiT k² ≤
≤

T∑
t

(Dit (xι,it — Exι,it)
x2,it ^- Ex2,it
x3,it ^- Ex3,it

kD1T (x1,it

2²

- Ex1,it)k⁴ + kx2,it
+ kx3,it ^- Ex3,it k

^- Ex2,it k⁴

Notice that by Assumptions 2(i), 3(iii),

supE kx1,it - Ex1,it k⁴, supE kx2,it -Ex2,itk⁴ < ∞,
i,t i,t

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and by Assumption 4(iii) and Assumption 6(iii),

sup E kx3,it - Ex3,itk⁴i,t

≤ M1 sup EFz_i^°x3,it - EFz_i x3,it^°⁴ +M1E^°EFz_ix3,it

^- Ex3,it ^°

< ∞.

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Thus, we have

sup E kQiT k² < ∞.
i,T

From this,

supE ∣∣QiτII 1 {∣∣Qiτ∣∣ > M} ≤ s^up^i,τ E∣^lQⁱT^k → 0,
i,T M

as M →∞. By Corollary 14 and Assumption 8(ii), then,

NN Σ T Σ IIIι,iτ II¹⁰,iτ
it

1 ₁ ( 0 0 0 ʌ

= N Σ QiT →P^¹T N Σ EQiT = I ^{0 φ}2²^φ2³ ) .

i ^, i 0 Φ⁰₂₃ Φ33

Next, by Assumption 6(i) and (ii), Lemma 11, and Lemma 12, we have

N Σ T ΣIII2,iTIII2,iT
it

1 1 DDT (Exι,it - Exι,i)∖DD_it (Exι,it - Exι,i)∖

N∑T∑ EX2,it — EX2,i EX2,it — EX2,i

i ^t ∖Exi,it — EX3,i J∖Ex3,it — EX3,i )

[ Ro¹ (^τ 1 — R ^τ il i∣ⁱmN ¹ Pi ^θ1,i^θ1,i∏^τ 1 — R ^τ 1/dr ^{0 0} ʌ

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