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

Since Vi is independent of wi, we have

^σT^vtr ^]1 X EDTwiw'0DT

≤ ^σv sup sup E ∖∖DtWik² → 0,
T N,T 1≤i≤N

as (N, T →∞) , where the last convergence holds because Part (b) warrants

sup sup E ∣∣Dtwi∣∣2 < ∞∙ ¥

N,T 1≤i≤N

Proof of Lemma 2

Part (a)

By definition, we have

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N^ √T^ GXχT ^(xit ^xi) ^(xit xi) GxT

— ^NX ʌTXGxT ^(xit Eixit + Eixit  Eixi + Eixi ^xi)

× ^(xit   Eixit + Eixit Eixi + Eixi   ^xi) GxT^,

— NXTX (IIIι,iT +III2,iT+III3,iT)(III1,iT +III2,iT+III3,iT)⁰, say.

First, we show that

N X T X IIIι,iτ IIII ,iT
it

1     1      D1T (x1,it - Ex1,it)    D1T (x1,it - Ex1,it)  ⁰

^— N∑T     x2,it - Ex2,it       JI x2,it - Ex2,it J

^{i t}    x3,it ^- Ex3,it            x3,it ^- Ex3,it

/ 0 0    0   ∖

→ p 0 Φ22 Φ23   ∙                                         (74)

0 Φ⁰23 Φ33

For this, set

1        D1T (x1,it - Ex1,it)

QiT — T X^? I x2,it - Ex2,it

^t     x3,it ^- Ex3,it

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