σ=2E{σε(z,)f-.(z,)}∫Kuu)du and σ'ε<z,)=E(4∖z<)=θ(1-θ).
(ii) under the null hypothesis (3), Cf^ ≡ 2θ2(1 - θ)2------------∑ Ktis is a consistent
T(T- 1)hm s≠t
estimator of σ02= 2E {σε4(zt) fz(zt)} ∫ K2(u)du.
(iii) under the alternative hypothesis (4), JT →p E{[Fy,(Qθ(xt) ∖ zt)-θ]2 fz(zt)} > 0.
m/22
(iv) under the local alternatives (17), Th JT → N(μ,σ0) in distribution, where
μ = E [. ⅛, { Qθ (zt. )iɪt} 12(,ε ) f, (,t) )].
Theorem 1 generalizes the results of Zheng (1998) of independent data to the weakly
dependent data case. A detailed proof of Theorem is given in the Appendix. The main
difficulty in deriving the asymptotic distribution of the statistic defined in equation (12) is that
a nonparametric quantile estimator is included in the indicator function which is not
differentiable with respect to the quantile argument and thus prevents taking a Taylor
expansion around the true conditional quantile Qθ(xt). To circumvent the problem, Zheng
(1989) appealed to the work of Sherman (1994) on uniform convergence of U-statistics
indexed by parameters. In this paper, we bound the test statistic by the statistics in which the
nonparametric quantile estimator in the indicator function is replaced with sums of the true
conditional quantile and upper and lower bounds consistent with uniform convergence rate of
the nonparametric quantile estimator, 1(yt≤Qθ(xt)-CT) and 1(yt≤Qθ(xt)+CT).
An important further step is to show that the differences of the ideal test statistic JT given
in equation (8) and the statistics having the indicator functions obtained from the first step
stated above is asymptotically negligible. We may directly show that the second moments of
the differences are asymptotically negligible by using the result of Yoshihara (1976) on the
bound of moments of U-statistics for absolutely regular processes. However, it is tedious to
get bounds on the second moments with dependent data. In the proof we instead use the fact
that differences are second-order degenerate U-statistics. Thus by using the result on the
asymptotic normal distribution of the second-order degenerate U-statistic of Fan and Li