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



Now we start the proof of (71) . Let si2,T = E (ι0Qi,T)2 and SN2,T = Pi si2,T.
Under Assumption 11, we have

2
si,T


E (ι0Qi,T)2
ι0E h(EFw (ui


EFw u»)2) DtWiWj'iDt ι


σ2uE [ι0Dtwiw0Dtι].

By Part (a),


N X ι0DτWJiWJ00Dtι ι0Ξι > 0,
i

as (N, T →∞) . By Part (b),

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

as M →∞, and so ∣ι0DTwJi 2 is uniformly integrable in N,T. Then, by Vitali’s
lemma, it follows that

1   2           20

NSN,T σuι ξ10,

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

X 1QT N (0,1),

(72)


SN,T

i,

as (N, T → ∞). Let Pi,Nτ = l-sQ*4 . Note that, under Assumption 9, E (P»,nt) =
0 and
Pi EPi2,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 ε0,
i

(73)


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

sup sup E Qi,T

N,T 1iN

≤ ∣ι8 sup sup E


DtWik4 (Efw (и.


N,T 1iN

κU sup sup E ∣∣DtW7i4 ,
N,T 1iN


EFwui)4


Part (d)

48



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