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       +^sup_i,τ ^e °^d3T (^e√^x3,i ^{- Ex}3,i) °

+ supn,t sup1<_i<N E ∣∣D3τ (Ef_zx3 - E⅝)∣∣

+ supN,_τ sup₁≤_i≤N ∣∣D3τ (Ex3,i - ES3)∣∣

∖         ^+supn ^sup1≤i≤N ^{e k}zi^- 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
M4,

∣∣D1τ (Ex_M - Ex1)∣∣⁴

≤   2 ∣∣H1∣∣ sup I∣θ1,i∏ < M4,

i

and

∣∣Ex21,i - Ex21∣⁴

(uniformly in i as T → ∞)

≤ 2sup ∣∣θ21,i∏⁴ < M4.
i

So,

sup sup ∣∣D1τ (Ex1,i - Ex1)∣∣⁴ , sup sup ∣∣Ex21,i - Ex21∣⁴ < ∞.   (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 (EFz_i ^x31,i ^{- Ex}31,i) °   →  θ^,

°D32τ (Ef₂.x31,i - Ex31_jj)°    →  ∣H32g32,i (zi)∣∣⁴ ,

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

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