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



Therefore, the first term (69) is finite. The second term in (69) is also finite since
sup
i °μg32 i° , suPi °μg33 i° < ∞ as assumed in Assumption 6(v). Therefore,

sup sup ∖∖D3τ (Ex3,i - Ex3)∖∣ < ∞.

N,T 1iN


In addition, by Assumption 5,


sup sup E ∣∣zi — z∖∖4 .

N 1iN


(70)


Therefore, from (66) - (70) we have


sup sup E kI2,i,NTk4 < ∞.
N,T 1iN


Finally, since I3,NT = N ∑i Iι,i,NT,


sup E kI3,NT k4
N,T


sup E ( ^1 X kI1,i,NT k ) (by triangle inequality)

N,T N i

sup sup E kI1,i,N T k4 (by Holder’s inequality)

N,T 1iN

∞. ¥


Part (c)

Recall that


under Assumption 11, conditional on Fw ,ui is independently


distributed with mean wiDT λ, variance σ2, and κ4 = EF (ui — EF ui)4 < ∞,
N ,               u, u w             w             ,

where λ is a nonrandom vector in Rk+g. So, E (DtWbiUi) = √= E DTτWbiWboDTj λ.

Define Qi,T = DT wbi (ui — EFwui) ; and let ι ∈ Rk+g with kιk = 1. Then, we
can complete the proof by showing that as (N, T →∞) ,


N X l0Qi,T ⇒ N (0, σUι0X.
i


(71)


This is so because this condition, together with the Cramer-Wold device, As-
sumption 11 and Part (a), implies that


√N X DtWbiui = √N X DtWbiEFw ui +
ii

= N X DTw’iwiDT λ + √N X Qi,T

⇒ N (ξ^ σuξ .


47




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