A Consistent Nonparametric Test for Causality in Quantile

Appendix

Here we collect all required assumptions to establish the results of Theorem 1.

(A1) (i) {y_t,w_t} is strictly stationary and absolutely regular with mixing coefficients

β(τ) = O(ρ^τ) for some 0 < ρ < 1.

(ii) For some integer v ≥ 2, f_y, f_z, and f_x all are bounded and belong to A _v^∞ (see

D2).

(iii) with probability one, E[ε_t | μ-∞ (z), μ-∞¹(z)] = 0

<∞ and

E И^εiⁱ2∙∙∙^εtl∣

1+ξ

<∞

for some arbitrarily small η> 0 and ξ> 0, where 2≤l ≤4 is

an integer, 0≤i_j ≤4

and ∑ ij ≤ 8 . σl (z) = E(ε²1 z) , με4 (z) = E ' ε/ | z_t = z ] all

j=1

satisfy some Lipschitz

conditions: ∣α(u + v) - a(u)∣ ≤ D(u)∣∣v∣∣ with E ∣D(z)∣²⁺^η

<∞

for some small η ' > 0, where a(∙) = σ²_ε (∙), μ_ε4 (∙) .

(iv) Let f_τ_,_K_,_τ() be the joint probability density function of (z_τ ,K ,z_τ

Then f_τ_,_K_,_τ( ) is bounded and satisfies a Lipschitz condition:

l^fτ₁,κ,τ_i ( z1 ⁺ u1^, z 2 ⁺ u 2^,∙∙∙ zl ⁺ U )-^fι,,...,τ_i ( z1^, z 2^,k zl

_τ_l(z₁,K,z_l

D_τ_,_K_,_τ() is integrable and satisfies the condition that ∫D_τκτι⁽ zι,κ, z )l Izl I²^ξ <M<∞,

∫Dτ1,K,τl(z1,^K,zl

z₁,K,z_l) dz< M<∞ for some ξ> 1.

(v) For any y, x satisfying

0< F_y_|_x(y|x)< 1 and f_x(x)> 0; for fixed y , the

conditional distribution function

F_y_|_x and the conditional density function f_y_|_x belong to

A 3∞ ; ^f_y∖x ⁽Qθ (^x)| ^x) > ⁰ for all

x ; f_y_|_x is uniformly bounded in x and y by c_f , say.

(vi) For some compact set G , there are ε> 0,γ >0 such that f_x ≥γ for all x in the

ε -neighborhood {x 11∣x - u∣∣ < ε, u ∈ G } of G ; For the compact set G and some