used as a proper candidate for consistent testing H0 (Li, 1999, p. 104). Since
Е(^, I zt ) = Fy∖z {Qθ (xt ) I zt }- θ , we have
J = E {εt E(εt∣zt ) fz (zt )}.
(6)
The test is based on a sample analog of E{ε E(ε ∣ z) fz(z)}. Including the density
function fz (z) is to avoid the problem of trimming on the boundary of the density function,
see Powell, Stock, and Stoker (1989) for an analogue approach. The density weighted
conditional expectation E(ε ∣ z) fz(z) can be estimated by kernel methods
E(£t I zt )fz ( zt ) =
---i---Σ Ks'
(т -1) h ~ ts
(7)
where m=p+q is the dimension of z , Kts= K {(zt -zs)/h} is the kernel function and
h is a bandwidth. Then we have a sample analog of J as
1 TT
T ≡--------∑ ∑ Ksεtεs
Tm tsts
T(T- 1)h t=1 s≠t
1 TT
t,t 1,h∑ ∑ K [I{У, ≤Q(x)}-θ][I{у. ≤Qθ(χ.)}-θ] (8)
T(T- 1)h t=1 s≠t
The θ-th conditional quantile of yt given xt, Qθ (xt ), can also be estimated by the
nonparametric kernel method
Qθ ( χt ) = Fy∣x ^1(θ∣xt ),
where
(9)
(10)
∑ L.I (У. ≤ У. )
λ∙
F7y∣x(yt I xt ) = ^≠^ντ7------
∑ Lt.
.≠t
is the Nadaraya-Watson kernel estimator of Fy∣x (yt ∣ xt) with the kernel function of
[x4 - x I _
------I and the bandwidth parameter of a . The unknown error ε can be
a J
estimated as:
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