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
supi °μ_g32 i° , suPi °μ_g33 i° < ∞ as assumed in Assumption 6(v). Therefore,

sup sup ∖∖D3τ (Ex3,i - Ex3)∖∣ < ∞.

N,T 1≤i≤N

In addition, by Assumption 5,

sup sup E ∣∣zi — z∖∖4 < ∞.

N 1≤i≤N

(70)

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

sup sup E kI2,i,NTk⁴ < ∞.
N,T 1≤i≤N

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 1≤i≤N

∞. ¥

Part (c)

Recall that

under Assumption 11, conditional on F_w ,ui is independently

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

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

Define Q_i,T = D_T wb_i (u_i — E_Fwu_i) ; and let ι ∈ R^k+g with kιk = 1. Then, we
can complete the proof by showing that as (N, T →∞) ,

√N X ^l0Qi,T ⇒ N (^{0, σ}U^ι0X.
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 u_i +
ii

= N X DTw’iwiDT ^{λ +} √N X Qi,T

⇒ N (^ξ^ ^σu^ξ .

47