A Consistent Nonparametric Test for Causality in Quantile



Appendix

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

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

β(τ) = O(ρτ) for some 0 < ρ < 1.

(ii) For some integer v2, fy, fz, and fx all are bounded and belong to A v (see

D2).

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

<∞ and


E Иεii2∙∙∙εtl


1+ξ


<∞


for some arbitrarily small η> 0 and ξ> 0, where 2l4 is


an integer, 0ij4


and  ij 8 . σl (z) = E(ε21 z) ,  με4 (z) = E ' ε/ | zt = 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() = σ2ε (), με4 () .


(iv) Let fτ,K,τ() be the joint probability density function of (zτ ,K ,zτ


Then     fτ,K,τ( ) is bounded and satisfies a Lipschitz condition:


lfτ1,κ,τi ( z1 + u1, z 2 + u 2,∙∙∙ zl + U )-fι,,...,τi ( z1, z 2,k zl


τl(z1,K,zl


Dτ,K,τ() is integrable and satisfies the condition that Dτκτι( zι,κ, z )l Izl I2 ξ <M<∞,


Dτ1,K,τl(z1,K,zl


z1,K,zl) dz< M<∞ for some ξ> 1.


(v) For any y, x satisfying


0< Fy|x(y|x)< 1 and fx(x)> 0; for fixed y , the


conditional distribution function


Fy|x and the conditional density function fy|x belong to


A 3; fyx (Qθ (x)| x) 0 for all


x ; fy|x is uniformly bounded in x and y by cf , say.


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


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




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