A Consistent Nonparametric Test for Causality in Quantile

compact neighborhood Θ₀ of 0, the set Θ = {v = Q_θ (x) + μ | x ∈ G, μ ∈ Θ₀} is compact
and for some constant c₀ > 0 , f_y|x(v|x) ≥ c₀ for all x∈ G, v∈Θ . (vii) There is an
increasing sequence s_T of positive integers such that for some finite A ,

TT

— β^sT^/(3^τ)(s_τ) ≤ A , 1 ≤ s_τ ≤ T for all T ≥ 1.

(A2) (i) we use product kernels for both L (∙) and K (∙) , let l and k be their
corresponding univariate kernel which is bounded and symmetric, then l(∙) is non-negative,

l(∙) ∈ Y_v, k(∙) is non-negative and k(∙) ∈ Y₂.

(ii) h = O(T ^α ) for some 0 < α ' < (7 / 8)m .

p1

(iii) a= o(1) and S_T=Ta(s_TlogT)→∞ for some s_T →∞

(iv) there exists a positive number δ such that for r = 2 +δ and a generic number M₀

h

(v) for some δ ' (0 < δ ' < δ), β(T) = O(T^-(^2+δ ')/^δ ).

The following definitions are due to Robinson (1988).

Definition (D1)   Y_λ, λ≥ 1 is the class of even functions k: R→ R satisfying

∫ uⁱk(u)du=δ_i0      (i =0,1,K,λ-1),

R

k(u) = O ( (1 + ∣uр^+1+г )^-1 ), for some ε > 0,

where δ_i is the Kronecker’s delta.
ij

Definition (D2)   A ^α_μ , α>0, μ>0 is the class of functions g: R^m → R satisfying that

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