A Consistent Nonparametric Test for Causality in Quantile



=K2(u)dufz2(z)dz+op(1)

(A.31)


The result of Theorem (ii) follows from (A.31).

Proof of Theorem 1 (iii)

The proof of Theorem (iii) consists of the two steps.

Step 1. Show that JT = Jt + op (1) under the alternative hypothesis (4).

Step 2. Show that JT=J+op(1) under the alternative hypothesis (4),

where J = E{[Fy^z(Qθ(xt) zt)-θ]2fz(zt)} . The combination of Steps 1 and 2 yields
Theorem (iii).

Step 1: Show that Jt = Jt + op (1) under the alternative hypothesis.

We need to show that the results of Step 2 and Step 3 in the proof of Theorem (i) hold under
the alternative hypothesis. First, we show that the result of Step 2 in the proof of Theorem (i)
still holds under the alternative hypothesis. We can show that
J2(Qθ - CT)= op(1) by the
same procedures as in (A.24). Thus we focus on showing that
J2 (Qθ) = op (1). As in the proof
of Theorem (i), denote
S(g) ≡ ∂F[g] / g . By taking a Taylor expansion of

Fyz(qθ (xs )| zs ) around qθ (zs ), we have

J2(Qθ)=-


1

T(T-1)


TT

∑ ∑

t=1 st


z I

—s-{1(yt Qθ(xt)) -Fyz(Qθ(xt)zt)}
h )

× S (Qθ (xs . zs ))

1T

= -∑ {1(yt Qθ(xt)) Fyz (Qθ (xt ))}S(Qθ (xs, zs)) f z(zt)

T t=1

1T

(A.32)


7∑ u,S(Qθ(x,,z,)).f (zt).

T t=1

where Qθ(xs,zs) is between Qθ(xs) and Qθ(zs). By using the same procedures as in

19



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