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



By Lemma 15 and the definitions of Dit , D2τ, and D3τ, each term in (64) is
finite. Thus,

sup sup E ∣∣I1,i,NTIl4 < ∞.

N,T 1≤i≤N

Next, we consider the second term in the right-hand side of the inequality
(63). For some constant
M3,

sup sup E ∣∣I2,i,NTIl4

N,T 1<i<N

/    supN,Tsupι≤i≤N ∣∣D1T (Exι,i - EX1)∣∣4    ʌ

+ supn,t sup1<i<N ∣∣EX21,i - EX21k4

M3       +supie °d3T (ex3,i - Ex3,i) °

(65)


+ supn,t sup1<i<N E ∣∣D(Efzx3 - E⅝)∣∣

+ supN,τ sup1i≤N ∣∣D(Ex3,i - ES3)∣∣

∖         +supn sup1≤i≤N e kzi- zl∣4         /

For the required result, we need to show that all of the terms in the right-hand
side of the inequality (65) are bounded. Notice that for some finite constant
M
4,

∣∣D (ExM - Ex1)∣∣4

H1


1i - N X θ1


(uniformly in i as T → ∞)


≤   2 ∣∣H1∣∣ sup I∣θ1,i∏ < M4,

i

and

∣∣Ex21,i - Ex214

θ21,i - (N X θ21,i

i


(uniformly in i as T → ∞)

≤ 2sup ∣∣θ21,i4 < M4.
i

So,

sup sup ∣∣D1τ (Ex1,i - Ex1)∣∣4 , sup sup ∣∣Ex21,i - Ex214 < ∞.   (66)

N,T 1≤i≤N            ,               N,T 1≤i≤N       ,

By Assumption 6(iii), the followings hold uniformly in i almost surely as T →
∞ :

°d31T (EFzi x31,i - Ex31,i) °   →  θ,

°D32τ (Ef2.x31,i - Ex31jj)°    →  H32g32,i (zi)∣∣4 ,

°D33τ (Efz.x33,i - Ex33,i) °    →   ∣∣H33g33,j (zi)∣∣4.

45



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