A Consistent Nonparametric Test for Causality in Quantile

g is (d - 1) -times partially differentiable for d- 1 ≤μ≤d; for some ρ> 0 ,

sup У ∈φ_zρ g ( У ) - g (^z ) ^- Gg ( y^{, z} )|/l y ^{- z} Γ ≤ Dg (^z ) for all ^{z ,} where ^φ ρ ={ У | |У ^{- z}∖ < p}

; G_g = 0 when d = 1 ; G_g is a (d - 1)th degree homogeneous polynomial in y- z with
coefficients the partial derivatives of g at z of orders 1 through d - 1 when d > 1 ; and
g(z), its partial derivatives of order d -1 and less, and D_g(z), has finite αth moments.

Proof of Theorem (i)

In the proof, we use several approximations to JT . We define them now and recall a few
already defined statistics for convenience of reference.

where   ^εt = I {y_t ≤ Q_θ (x_t )}^- θ^,

ε_t=I{y_t≤Q_θ(x_t)}-θ,

ε_tU =I{y_t +C_T ≤Q_θ(x_t )} -θ,

ε_tL =I{y_t-C_T ≤Q_θ(x_t)}-θ and

C_T is an upper bound consistent with the uniform convergence rate of the nonparametric
estimator of conditional quantile given in equation (13). The proof of Theorem 1 (i) consists
of three steps.

Step 1. Asymptotic normality:

Th^m/2J_T → N(0,σ₀²),                                           (A.5)

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