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




More intriguing information

1. WP RR 17 - Industrial relations in the transport sector in the Netherlands
2. MATHEMATICS AS AN EXACT AND PRECISE LANGUAGE OF NATURE
3. The Functions of Postpartum Depression
4. The purpose of this paper is to report on the 2008 inaugural Equal Opportunities Conference held at the University of East Anglia, Norwich
5. ARE VOLATILITY EXPECTATIONS CHARACTERIZED BY REGIME SHIFTS? EVIDENCE FROM IMPLIED VOLATILITY INDICES
6. Dual Inflation Under the Currency Board: The Challenges of Bulgarian EU Accession
7. The Advantage of Cooperatives under Asymmetric Cost Information
8. HACCP AND MEAT AND POULTRY INSPECTION
9. Nach der Einführung von Arbeitslosengeld II: deutlich mehr Verlierer als Gewinner unter den Hilfeempfängern
10. Strengthening civil society from the outside? Donor driven consultation and participation processes in Poverty Reduction Strategies (PRSP): the Bolivian case