A Consistent Nonparametric Test for Causality in Quantile



variance σ12(z) is o(1) with i.i.d data. We have

E% [H1T(s,t,Qθ)-H1T(s,t,Qθ-CT)]2

≤ Λ E% { [1t(Qθ) - Ft(Qθ)][1s(Qθ) - Fs(Qθ)]

-[1t(Qθ-CT)-Ft(Qθ-CT)][1s(Qθ-CT)-Fs(Qθ-CT)] }2

≤Λ E%{Ft(Qθ)[1-Ft(Qθ)]Fs(Qθ)[1-Fs(Qθ)]}

+E%{Ft(Qθ-CT)[1-Ft(Qθ-CT)]Fs(Qθ-CT)[1-Fs(Qθ-CT)]}

-2E{ [Ft(min(Qθ,Qθ-CT)-Ft(Qθ)Ft(Qθ-CT)]

×[Fs(min(Qθ,Qθ-CT)-Fs(Qθ)Fs(Qθ-CT)] }

E%{[Ft(Qθ)-Ft(Qθ)Ft(Qθ)][Fs(Qθ)-Fs(Qθ)Fs(Qθ)]}

E%{ [Ft(min(Qθ,Qθ-CT)-Ft(Qθ)Ft(Qθ-CT)]

×[Fs(min(Qθ,Qθ-CT)-Fs(Qθ)Fs(Qθ-CT)] }

E%{ [Ft(Qθ-CT)-Ft(Qθ-CT)Ft(Qθ-CT)]

×[Fs(Qθ-CT)-Fs(Qθ-CT)Fs(Qθ-CT)] }

E%{ [Ft(min(Qθ,Qθ-CT)-Ft(Qθ)Ft(Qθ-CT)]

×[Fs(min(Qθ,Qθ-CT)-Fs(Qθ)Fs(Qθ-CT)] }

≤Λ Ct = o (1).                                                       (A.20)

where the last equality holds by the smoothness of conditional distribution function and its

bounded first derivative due to Assumption (A.8). Thus we have

Thm,[ J,( Qθ ) - Jɪ( Qθ - C )] = Op (1)                                  (A.21)
[2]
Thm'2[J2(Qβ)- J2(Qβ -C)] = Op(1):

Noting that H2t (5, t, Qθ ) = 0 because of Fyz (Qθ (xs ) | zs ) - θ = 0, we have

J2(Qθ)-J2(Qθ-CT)

15



More intriguing information

1. How Low Business Tax Rates Attract Multinational Headquarters: Municipality-Level Evidence from Germany
2. TOWARDS THE ZERO ACCIDENT GOAL: ASSISTING THE FIRST OFFICER MONITOR AND CHALLENGE CAPTAIN ERRORS
3. Computing optimal sampling designs for two-stage studies
4. The name is absent
5. Epistemology and conceptual resources for the development of learning technologies
6. Special and Differential Treatment in the WTO Agricultural Negotiations
7. The name is absent
8. The name is absent
9. WP 92 - An overview of women's work and employment in Azerbaijan
10. The name is absent