is possible to show that
_1 S2
NSNT
σv2ι0E
11X1X
UγTγ
GxT (xit
- Xi) (Xit - xi)0 GxT
ιk
σv2ι0Ψxι > 0,
(80)
as (N, T →∞) . So, the asymptotic normality in (79) holds if
ι0Qi,τ
Snt
⇒ N(0,1),
(81)
as (N, T → ∞). Let Pi,Nτ = 1q,t. Then, E (Pi,Nτ) = 0 and Pi EPi2NT = 1.
Thus, by the central limit theorem of the double indexed process (e.g., see
Theorem 2 in Phillips and Moon, 1999), we can claim that (81) holds, if we can
show that
^X EPi,NT1 ©|Pi2NT I > ε} → 0 for all ε > 0, (82)
i
as (N, T →∞).
Now, in view of (80) , condition (82) holds if
supEkι0Qi2Tk4 ≤ supE kQi2T k4 < ∞. (83)
Note for some constant M1 that
sup E kQi2T k
iT
sup E
iT
sup E
iT
-1T X Gx,T (Xit
Tt
tr
≤ M1 sup
iT
Xi) vit
/1 X μ Gχ,τ (
τ2Λ ⅛Gx,τ∙
t2s2p2q
Xit
(Xip
xi) (Xis
Xi) (xiq
Xi)0 Gχ,τ ʌ
- Xxi ) Gx2T
+M1 sup
iT
+M1 sup
iT
×E (vitvisvipviq)
f'2 X tr (t2Gx∙τ (Xit — xi) (x
Gx,T (Xit Xi) (Xis
×E vi2tvi2s
is
Xi)0 Gχ,τ
Xi)0 Gχ,τ
(T2 Xtr (βC,.τ (x
Gx,T (Xit Xi) (Xis
is
Xi) (Xit
Xi)0 Gχ,τ
Xi)0 GxT
×E vi2tvi2s
(T2 Xtr (βC,.τ (x
Gχ,τ (Xit - Xi) (Xit
is
Xi) (Xis
Xi)0 Gχ,τ
Xi)0 GxT
. (84)
52
×E vi2tvi2s