A Consistent Nonparametric Test for Causality in Quantile

^{+ 2} × Ks{¹(y_t ≤ g(^xt ))^- F(g(^xt)| Z)}{F(g(^xs)| ^zs)^{- θ}}

+ Kts {F (g ( x_t ) | z_t ) - θ} {F (g ( x_s ) | Zs ) - θ}

= H_1T(s,t,g)+2H_2T(s,t,g)+H_3T(s,t,g)

1 ^TT

Then let   J.[g] =-----------∑ ∑ H._τSs,t,g) , i = 1,2,3 . We will treat

^j      T(T- 1)h^{m            jT}

J_j[Q_θ]-J_j[Q_θ-C_T] for j=1,2,3 separately.

[1] Thm/2 [ Ji( Qθ ) - Ji( Qθ - Ct )] = Op (1):

By simple manipulation, we have

J1(Qθ)-J1(Qθ-CT)

=-----¹-----∑ ∑ [H_{1 t} (s, t, Q_θ ) - H_1T (s, t, Q_θ - C_t )]

m              1Tθ1TθT

T(T- 1)h _{t=1 s≠t}

1 TT

= τ,τ _nm∑∑ Ks{ [11(Qθ)-F(Qθ)][1 s(Q)-F(Qθ)]

T(T- 1)h _{t=1 s≠t}

-[1t(Qθ-CT)-Ft(Qθ-CT)][1s(Qθ-CT)-Fs(Qθ-CT)] }   (A.18)

To avoid tedious works to get bounds on the second moment of J₁(Q_θ)-J₁(Q_θ-C_T )
with dependent data, we note that the R.H.S. of (A.18) is a degenerate U-statistic of order 2.
Thus we can apply Lemma 2 and have

Thm/² [J₁(Q_θ) - J₁(Q_θ -C_t)] → N(0,c₁²) in distribution,             (A.19)

where the definition of the asymptotic variance σ₁² is based on the i.i.d. sequence having the
same marginal distributions as weakly dependent variables in (A.18). That is,

σ₁² =E^%[H_1T(s,t,Q_θ)-H_1T(s,t,Q_θ-C_T)],

where the notation E^% is expectation evaluated at an i.i.d. sequence having the same
marginal distribution as the mixing sequences in (A.18) (Fan and Li (1999), p. 248). Now, to
show that Thm/² [ J₁ (Qθ ) - J1(Qθ - Ct )] = Op (1) , we only need to show that the asymptotic

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