A Consistent Nonparametric Test for Causality in Quantile

quantile as

(1) w_t does not cause y_t in quantile with respect to U_t-1 if

Qθ(y_t∖U-ι) = Qθ(y_t∖U_t-1 -Wt-ι) and

(2) w_t is a prima facie cause in quantile of y_t with respect to U_t-1 if

Qθ(yt∖Ut-1)≠Qθ(yt∖Ut-1-Wt-1),                                          (2)

where ^qθ (y_t ∖ ∙) ≡^inf {y_t ∖ F(y_t ∖ ∙) ≥ θ} is the θth(0< θ< 1) conditional quantile of y_t .
Denote   x_t≡(y_t-1,L,y_t-p), z_t≡(y_t-1,L,y_t-p,w_t-1,L,w_t-q), and the conditional

distribution function y given v by F_y∖v(y∖v), v=(x,z) . Denote Q_θ(v_t)≡Q_θ(y_t∖v_t).
In this paper, F_y∖v(y∖v) is assumed to be absolutely continuous in y for almost all
v= (x, z). Then we have

F_y∖v{Q_θ(v_t)∖v_t}=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 {y_t ≤ ^qθ (x_t)| z_t }] = θ or I {y_t ≤ ^qθ (^χ_t)} = θ + ^εt where E(^εt ∖ z_t ) = 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 f_z(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

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