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

Now we start the proof of (71) . Let s_i²_,T = E (ι⁰Q_i,T)² and S_N²_,T = ^P_i s_i²_,T.
Under Assumption 11, we have

σ²_uE [ι⁰Dtwiw⁰Dtι].

N X ι⁰DτWJiWJ⁰⁰Dtι → ι⁰Ξι > 0,
i

as (N, T →∞) . By Part (b),

sup sup E ∣∣ι⁰DτW7ik2 1 {∣∣ι⁰Dτw»|| >M} → 0,
N,T 1≤i≤N

as M →∞, and so ∣ι⁰D_TwJ_i ∣² is uniformly integrable in N,T. Then, by Vitali’s
lemma, it follows that

1   2           20

NSN,T → ^σu^{ι ξ1} > ⁰,

as (N,T →∞) . Thus, for our required result of (71), it is sufficient to show

X ¹QT ⇒ N (0,1),

SN,T

i^,

as (N, T → ∞). Let Pi,Nτ = ^l-sQ*⁴ . Note that, under Assumption 9, E (P»,nt) =
0 and ^P_i EP_i²_{,N T} =1. According to Theorem 2 of Phillips and Moon (1999),
the weak convergence in (72) follows if we can show that

^2EPi2NT¹ ©IPi2NT I > ^ε} → ⁰ for all ^ε > ^0,
i

as (N, T →∞). Since
the Lindeberg-Feller condition (73) follows, and we have all the desired results.
¥

Part (d)

48

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