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

i


2

=E


-1= X DtτiiVi
√N

i


σTvtr ^]1 X EDTwiw'0DT

σv sup sup E ∖∖DtWik2 0,
T
N,T 1iN

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

sup sup E ∣∣Dtwi∣∣2 ¥

N,T 1iN

Proof of Lemma 2

Part (a)

By definition, we have

11

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

49



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