quantile as
(1) wt does not cause yt in quantile with respect to Ut-1 if
(1)
Qθ(yt∖U-ι) = Qθ(yt∖Ut-1 -Wt-ι) and
(2) wt is a prima facie cause in quantile of yt with respect to Ut-1 if
Qθ(yt∖Ut-1)≠Qθ(yt∖Ut-1-Wt-1), (2)
where qθ (yt ∖ ∙) ≡inf {yt ∖ F(yt ∖ ∙) ≥ θ} is the θth(0< θ< 1) conditional quantile of yt .
Denote xt≡(yt-1,L,yt-p), zt≡(yt-1,L,yt-p,wt-1,L,wt-q), and the conditional
distribution function y given v by Fy∖v(y∖v), v=(x,z) . Denote Qθ(vt)≡Qθ(yt∖vt).
In this paper, Fy∖v(y∖v) is assumed to be absolutely continuous in y for almost all
v= (x, z). Then we have
Fy∖v{Qθ(vt)∖vt}=θ, v=(x,z)
and from the definitions (1) and (2), the hypotheses to be tested are
H0: Pr{Fy∖z(Qθ(xt)∖zt)=θ}=1 (3)
H1: Pr{Fy∖z(Qθ(xt)∖zt)=θ}<1. (4)
Zheng (1998) proposed an idea to reduce the problem of testing a quantile restriction to a
problem of testing a particular type of mean restriction. The null hypothesis (3) is true if and
only if E [I {yt ≤ qθ (xt)| zt }] = θ or I {yt ≤ qθ (χt)} = θ + εt where E(εt ∖ zt ) = 0
and I(∙) is the indicator function. There is a rich literature on constructing nonparametric
tests for conditional mean restrictions. Refer to Li (1998) and Zheng (1998) for the list of
related works. While various distance measures can be used to consistently test the hypothesis
(3), we consider the following distance measure,
J ≡ E [{Fyz (Qθ (x, ) ∖ z, ) - θ}2 f: (z. )], (5)
where fz(z) be the marginal density function of z . Note that J ≥ 0 and the equality
holds if and only if H0 is true, with strict inequality holding under H1. Thus J can be