A Consistent Nonparametric Test for Causality in Quantile

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

Step 2. Conditional asymptotic equivalence:

Suppose that both Th^m/2(J_T-J_TU) =o_p(1) and Th^m/2(J_T-J_TU) =o_p(1).

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

Step 3. Asymptotic equivalence:
Th^m/2(J_T-J_TU) =o_p(1) and Th^m/2(J_T-J_TL) =o_p(1).            (A.7)

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

Step 1: Asymptotic normality.

Since J_T 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.

Th^m/2(J_T-J_TU)=o_p(1) and                                            (A.8)

Th^m/2(JT-JTL)=op(1)                                                    (A.9)

Denote C_T 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) ^{- q}θ ^(x)^l≤ Ct .                                                  ^(A.10)

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

{ Qθ^(x) > y_t ⁺ CT } ⊂ { (Qθ^(x) > y_t } ⊂ { Qθ^(x) > y_t ^-CT }^{,        (A.11-1)}

¹⁽ Qθ ^(x) > y_t ⁺ CT ) ≤ ¹⁽ (Qθ ^(x) > y_t ) ≤ ¹⁽ Qθ ^(x) > y_t ^- CT )^{,       (A.11-2)}

J_tu ≤ J_t ≤ J_tl , and                                               (A.11-3)

| Thm/²(J_t - J_t)| ≤ max {∣Thm/²(J_t - J_tu)| , |Thm/²(J_t - J_tl)| } (A. 11-4)

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