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



Let Xh,2,it be the hth element of Xh,2,it. The required result follows if we can
show that

sup
i,T


T X (xh,2,it
T t


- Exh,2,it)


< M for all h.

4


(41)


The proof of (41) is similar to the proof of Lemma 1 of Andrews (1991). This
proof relies on the following
α- mixing inequality presented in Hall and Heyde
(1980, p. 278). Suppose that
Y and W are random variables that are G-
measurable and H- measurable, respectively, with EY p and EWq <
, where p,q > 1 with 1/p + 1/q < 1. Then,

E (Y - EY ) (W - EW ) ∣ ≤ 8 H Y ∣∣p ∣∣ W ∣∣g [α (G, H)]1-1/p-1/q ,     (42)

where α (G, H) is the α-mixing coefficient between the sigma fields G and H.
Now, let
Xit = xh,2,it - Exh,2,it. Notice that

4

p e .. Σ X«)

sup T χ x x xe (Xi,XisXipXik )

i,τ      t=1 s = 1 p=1 k= 1

T -tT-sT -p

⅛ 5⅛2χχχΣE (XitXi,t+sXi,t+s+pXi,t+s+p+k)
i,T      t s=0 p=0 k=0

4!siuτp T X


Σ

0p,ks
0p+k+sT-t


E (Xit (Xi,t+sXi,t+s+pXi,t+s+p+k))


+4!sup ɪ X X

i,T        t     0s,kp


0p+k+sT-t


+4!sup .=2 X X

i,i        t     0s,kp

0p+k+sT-t


1

+4!siup .χ


x

0s,pk
0p+k+sT-t


= I + II + III + IV, say.


E [(Xit

Xi,t+s) (Xi,t+s+pXi,t+s+p+k)]

-E (XitXi,t+s) E (Xi,t+s+pXi,t+s+p+k )


E (XitXi,t+s) E (Xi,t+s+pXi,t+s+p+k)


E ((Xit

Xi,t+sXi,t+s+p') Xi,t+s+p+k)


By applying the inequality of (42) to Xi,t+sXi,t+s+pXi,t+s+p+k and Xit and
then by the Holder inequality, we have

1T

i4!8sup .2                    HXitk4q HXi,t+sH4q Il Xi,t+s+p ∣∣ 4q

i,T     t=10p,ksT-t

×∣∣Xi,t+s+p+k∣∣4q αi (s)q-1


35




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