1T
= -∑ S(Qθ (xt, Zt))CtS(Qθ(xs))fz (zt)
T t=1
1T
+-∑ CrS(Qθ(xt))S(Qθ(xs, Zs))f(Zt)
T t=1
1r
- - ∑ CTS(Qθ (xt ))S(Qθ (Xs )) f z (Zt ), (A.36)
r t=1
where Qθ(xs,Zs) is between Qθ(xs) and Qθ(Zs). Then by using the same procedures as
in (A.27), we have
J3(Qθ)-J3(Qθ-CT)=O(CT). (A.37)
Now we have the result of Step 1 for the proof of Theorem (iii). □
Step 2: Show that JT=J+op(1) under the alternative hypothesis.
Using (7) and uniform convergence rate of kernel regression estimator under β-mixing
process, we have
1 TT
T =------∑ ∑ Kεεs
Tm tsts
T(T- 1)h t=1 s≠t
1
= -∑ E(st | z)fz(zε
T t=1
= ⅛ ∑ e(ε | z ) fz(z )ε
T t=1
+1 ∑ {EE:- | zt )f (zt )- e(ε | zt ) fz(zt )} εt
T t=1
= ⅛ ∑ e(ε | zt ) fz(zt )εt+op(1)
T t=1
=e [e(ε | zt ) fz(zt ε ]+op(1)
=J+op (1)
(A.38)
□
21
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