A Consistent Nonparametric Test for Causality in Quantile



Using (A.10) and (A.11-4), we have the following inequality;

Pr[Thml( J - JT)| > δ I supQ,(x)-Qθ(x)C}

Pr{ max { Thm2(J -J„,)| , TW'Jt - Jtl)| } > δ I supQβ(x)-Q(x)CT}

, for all δ > 0 .                                                             (A.12)

Invoking Lemma 1 and condition A2(iii), we have

Pr{ sup Qθ(x)-Qθ(x)Ct} 1 as T → ∞ .                   (A.13)

By (A.8) and (A.9), as T → ∞, we have

Pr{ max { Thm/2(Jt - Jtu) , Thm/2(Jt - Jtl) } > δ } 0, for all δ > 0.

(A.14)

Therefore, as T → ∞,

the L.H.S. of the inequality (A.12) - Pr { Thm/2(Jt - Jt )δ } 0 and

the L.H.S. of the inequality (A.12) 0 .

In summary, we have that if both Thm/2(JT-JTU) =op(1) and Thm/2(JT-JTU) =op(1),
then
Thm/2(^Jt - Jt ) = op (1).                                                         

Step 3: Asymptotic equivalence.

In the remaining proof, we focus on showing that Thm/2(JT-JTU) =op(1), with the proof
of
Thm/2(JT-JTL) =op(1) being treated similarly. The proof of Step 3 is close in lines
with the proof in Zheng (1998). Denote

HT(s,t,g)Kts{1(ytg(xt))-,}{1(ysg(xs))-,} and             (A.15)

J[g]


1

T (T -1) hm


T

t=1


T
HT(s,t,g).
st


(A.16)


Then we have JTJ[Q,] and JTUJ[Q, -CT]. We decompose HT (s,t, g) into
three parts;

HT(s,t,g)=Kts{1(ytg(xt))-F(g(xt)zt)}{1(ysg(xs))-F(g(xs)zs)}

13



More intriguing information

1. Ex post analysis of the regional impacts of major infrastructure: the Channel Tunnel 10 years on.
2. The name is absent
3. The name is absent
4. The name is absent
5. The name is absent
6. Hemmnisse für die Vernetzungen von Wissenschaft und Wirtschaft abbauen
7. 5th and 8th grade pupils’ and teachers’ perceptions of the relationships between teaching methods, classroom ethos, and positive affective attitudes towards learning mathematics in Japan
8. Business Networks and Performance: A Spatial Approach
9. Education and Development: The Issues and the Evidence
10. The name is absent
11. TOWARDS THE ZERO ACCIDENT GOAL: ASSISTING THE FIRST OFFICER MONITOR AND CHALLENGE CAPTAIN ERRORS
12. An Economic Analysis of Fresh Fruit and Vegetable Consumption: Implications for Overweight and Obesity among Higher- and Lower-Income Consumers
13. Cancer-related electronic support groups as navigation-aids: Overcoming geographic barriers
14. Non-causality in Bivariate Binary Panel Data
15. Survey of Literature on Covered and Uncovered Interest Parities
16. Tissue Tracking Imaging for Identifying the Origin of Idiopathic Ventricular Arrhythmias: A New Role of Cardiac Ultrasound in Electrophysiology
17. El Mercosur y la integración económica global
18. Housing Market in Malaga: An Application of the Hedonic Methodology
19. AN ECONOMIC EVALUATION OF COTTON AND PEANUT RESEARCH IN SOUTHEASTERN UNITED STATES
20. PROPOSED IMMIGRATION POLICY REFORM & FARM LABOR MARKET OUTCOMES