A Consistent Nonparametric Test for Causality in Quantile



where σ02=2E{θ2(1-θ)2f (zt)}{K2(u)du} under the null.

Step 2. Conditional asymptotic equivalence:

Suppose that both Thm/2(JT-JTU) =op(1) and Thm/2(JT-JTU) =op(1).

Then Thm/2(Jt - JT ) = op (1) .                                     (A.6)

Step 3. Asymptotic equivalence:
Thm/2(JT-JTU) =op(1) and Thm/2(JT-JTL) =op(1).            (A.7)

The combination of Steps 1-3 yields Theorem 1 (i).

Step 1: Asymptotic normality.

Since JT is a degenerate U-statistic of order 2, the result follows from Lemma 2.

Step 2: Conditional asymptotic equivalence.

The proof of Step 2 is motivated by the technique of Hardle and Stoker (1989) which was
used in treating trimming indicator function asymptotically. Suppose that the following two
statements hold.

Thm/2(JT-JTU)=op(1) and                                            (A.8)

Thm/2(JT-JTL)=op(1)                                                    (A.9)

Denote CT as an upper bound consistent with the uniform convergence rate of the
nonparametric estimator of conditional quantile given in equation (13). Suppose that

sup qθ (x) - qθ (x)lCt .                                                  (A.10)

If inequality (A.3) holds, then the following statements also hold:

{ Qθ(x) yt + CT } { (Qθ(x) yt } { Qθ(x) yt -CT },        (A.11-1)

1( Qθ (x) yt + CT ) 1( (Qθ (x) yt ) 1( Qθ (x) yt - CT ),       (A.11-2)

JtuJtJtl , and                                               (A.11-3)

| Thm/2(Jt - Jt)| max {Thm/2(Jt - Jtu)| , |Thm/2(Jt - Jtl)| } (A. 11-4)

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