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 1≤i≤N
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 ξ1 > 0,
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 ε > 0,
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 1≤i≤N
≤ ∣ι∣8 sup sup E ∣
DtWik4 (Efw (и.
N,T 1≤i≤N
≤ κU sup sup E ∣∣DtW7i∣4 < ∞,
N,T 1≤i≤N
EFwui)4
Part (d)
48
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