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

Since Vi is independent of wi, we have

X¹ X Dt WiVi

√N

-1= X DtτiiVi
√N

^σT^v^tr ^]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

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)

ID1T (x1,it - Ex1,it)
x2,it ^- Ex2,it
x3,it ^- Ex3,it

QiT — T X^? I x2,it - Ex2,it

^t x3,it ^- Ex3,it