A Consistent Nonparametric Test for Causality in Quantile

used as a proper candidate for consistent testing H₀ (Li, 1999, p. 104). Since

^Е(^, I z_t ) = Fy∖z {Qθ (xt ) I z_t }^{- θ ,} we have

J = E {ε_t E(ε_t∣z_t ) f_z (z_t )}.

The test is based on a sample analog of E{ε E(ε ∣ z) f_z(z)}. Including the density

function f_z (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) f_z(z) can be estimated by kernel methods

where m=p+q is the dimension of z , K_ts= K {(z_t -z_s)/h} is the kernel function and

h is a bandwidth. Then we have a sample analog of J as

1 TT

T ≡--------∑ ∑ K_sε_tε_s

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 y_t given x_t, Q_θ (x_t ), can also be estimated by the
nonparametric kernel method

Qθ ( χ_t ) = Fy∣x ^¹(θ∣xt ),
where

∑ L.I (У. ≤ У. )
^λ∙

F⁷y∣x⁽y_t I xt ) = ^≠^ν^τ7------

∑ L_t.

.≠t

is the Nadaraya-Watson kernel estimator of F_y∣x (y_t ∣ x_t) 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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