A Consistent Nonparametric Test for Causality in Quantile



used as a proper candidate for consistent testing H0 (Li, 1999, p. 104). Since

Е(^, I zt ) = Fyz {Qθ (xt ) I zt }- θ , we have

J = E {εt E(εtzt ) fz (zt )}.

(6)


The test is based on a sample analog of E{ε E(ε z) fz(z)}. Including the density

function fz (z) is to avoid the problem of trimming on the boundary of the density function,
see Powell, Stock, and Stoker (1989) for an analogue approach. The density weighted
conditional expectation
E(ε z) fz(z) can be estimated by kernel methods

E(£t I zt )fz ( zt ) =


---i---Σ Ks'
-1) h ~ ts


(7)


where m=p+q is the dimension of z , Kts= K {(zt -zs)/h} is the kernel function and

h is a bandwidth. Then we have a sample analog of J as

1 TT

T ≡--------∑ ∑ Ksεtεs

Tm         tsts

T(T- 1)h  t=1  st

1 TT

t,t 1,h∑ ∑ K [I{У, Q(x)}-θ][I{у.Qθ(χ.)}-θ]    (8)

T(T- 1)h t=1 st

The θ-th conditional quantile of yt given xt, Qθ (xt ), can also be estimated by the
nonparametric kernel method

( χt ) = Fyx ^1(θxt ),
where

(9)


(10)


L.I (У. У. )
λ

F7yx(yt I xt ) = ^≠^ντ7------

Lt.

.t

is the Nadaraya-Watson kernel estimator of Fyx (yt xt) with the kernel function of

[x4 - x I                                       _

------I and the bandwidth parameter of a . The unknown error ε can be
a J

estimated as:



More intriguing information

1. Placenta ingestion by rats enhances y- and n-opioid antinociception, but suppresses A-opioid antinociception
2. The name is absent
3. The name is absent
4. The name is absent
5. Nonparametric cointegration analysis
6. Government spending composition, technical change and wage inequality
7. A dynamic approach to the tendency of industries to cluster
8. Gender stereotyping and wage discrimination among Italian graduates
9. A Duality Approach to Testing the Economic Behaviour of Dairy-Marketing Co-operatives: The Case of Ireland
10. Putting Globalization and Concentration in the Agri-food Sector into Context