Causality Tests using Linear and Nonparametric Quantile ...

Causality Tests using Linear and

Nonparametric Quantile Regressions

Mei-Yuan Chen

Department of Finance

National Chung Hsing University

February 25, 2013

M.-Y. Chen Causality

Following Granger (1969, 1980), the random variable x is said to not

Granger cause the random variable y if

Fyt(η|(Y,X )t−1) = Fyt

(η|(Y)t−1), ∀ η ∈ R, (1)

where Fyt(·|F) is the conditional distribution of yt, and (Y,X )t−1 is the

information set generated by y and x up to time t− 1. That is, Granger

non-causality in distribution requires that the past information of x does

not alter the conditional distribution of yt. The variable x is said to

Granger causes y in distribution when (1) fails to hold.


Since estimating and testing conditional distribution are practically

cumbersome, it is more common to test a necessary condition of (1),

namely,

E[yt|(Y,X )t−1] = E(yt|(Yt−1)). (2)

It is said that x does not Granger cause y in mean if (2) holds; otherwise,

x Granger causes y in mean. Similarly, non-causality in variance is

defined in Granger, Robins, and Engle (1986) and Cheung and Ng (1996)

and non-causality in other moments. Hong, Liu, and Wang (2006)

considered non-causality in risk, a special case of (1) in which η is the

negative of the Value at Risk. Note that these notions of non-causality

are only necessary conditions of Granger non-causality in distribution.


Testing Granger Non-causality in Mean of Finite

Order

The hypothesis (2) is usually tested by regressing a linear

model

yt = α0 +

p∑

i=1

αiyt−i +

q∑

j=1

βjxt−j + et (3)

and testing the null hypothesis H0 : βj = 0, j = 1, . . . , q.

Rejecting this null hypothesis suggests that x Granger causes

y. Yet, failing to reject the null is compatible with

non-causality in mean but says nothing about causality in

other moments or other distribution characteristics.M.-Y. Chen Causality

Testing Granger Non-causality in Mean of Infinite

OrderConsider the regression model

yt = α0 +

p∑

i=1

αiyt−i +

∞∑

j=1

βjxt−j + et (4)

and testing the null hypothesis H0 : βj = 0, j = 1, . . . ,∞ for Granger

non-causality in mean. However, the hypothesis is not testable since the

order of lags of xt is infinite. By imposing a geometric lag structure for

the coefficients βj , Firoozi and Lien (2011) suggests the a procedure for

testing Granger non-causality in mean of infinite order. Specifically, the

coefficients of βj is assumed to follow

βj = πλj , ∀π, λ,with |λ| < 1. (5)


Given the restriction of (5), (4) reduces to

yt = α0 +

p∑

i=1

αiyt−i +

∞∑

j=1

(πλj)xt−j + et

= α0 +

p∑

i=1

αiyt−i + π

∞∑

j=0

(λL)jxt − πxt + et

= α0 +

p∑

i=1

αiyt−i +πλL

1− λLxt + et,

and then

(1− λL)yt = α0(1− λL) +

p∑

i=1

αi(1− λL)yt−i + (πλL)xt + (1 − λL)et.

Finally,

yt = (1 − λL)α0 + λyt−1 +

p∑

i=2

(αi − λαi−1)yt−i

−λαpyt−(p+1) + πλxt−1 + ηt.


Therefore, the null hypothesis of non Granger causality

H0 : βj = 0, j = 1, . . . ,∞ reduces to H0 : πλ = 0. And the

null can be tested by checking the significance from zero of

coefficient of xt−1 in the regression of yt on

1, yt−1, . . . , yt−(p+1) and xt−1.


Given that a distribution is completely determined by its

quantiles, Granger noncausality in distribution can also be

expressed in terms of conditional quantiles. Let Qyt(τ |F)

denote the τ -th quantile of F (·|F), then (1) is equivalent to

Qyt(τ |(Y ,X )t−1) = Qyt(τ |Yt−1), ∀ τ ∈ (0, 1). (6)


It is said that x does not Granger cause y in all quantiles if (6)

holds. It may also be defined that Granger non-causality in a

quantile range [a, b] ⊂ (0, 1) as

Qyt(τ |(Y ,X )t−1) = Qyt(τ |Yt−1), ∀ τ ∈ [a, b]. (7)

Note that Lee and Yang (2006) considered only non-causality

in a particular quantile, i.e., the equality in (6) holds for a

given τ .


Corresponding the regression mean model (3), consider the

linear quantile regression model

yt = α0(τ) +

p∑

i=1

αi(τ) yt−i +

q∑

j=1

βj(τ) xt−j + et(τ). (8)

Denote

xt = [1, yt−1, . . . , yt−p, xt−1, . . . , xt−q]

βτ = [α0(τ), α1(τ), αp(τ), β1(τ), . . . , βq(τ)]

The linear quantile regrression (8) can be represented as

yt = x′tβτ + et(τ)


Note that the dimension of xt is k = 1 + p+ q. Denote βτ as

the quantile regression estimator for βτ and the null

hypothesis of non-Granger causality in distribution in (6) as

H0 : Rβτ = r, ∀ τ ∈ (0, 1)

with R = [0q×1, 0q×p, Iq] is a q × k matrix and r = 0q×1 is a

q × 1 vector.


Given the asymptotic normality derived by Fitzenberger

(1997), we have

D−1/2T

√T (βτ − βτ )

d−→ N (0, Ik)

where DT = L−1T JTL

−1T , LT = E(1/T )

∑Tt=1 ft(F

−1(τ))xtx′t,

JT = Var[

(1/√T )∑T

t=1 xtφτ(ut)]

, and φτ (·) = τ − I(· < 0).

And then

R√T (βτ − βτ )

d−→ N (0,RDTR′),

or equivalently

Γ−1/2T R

√T (βτ − βτ )

d−→ N (0, Iq),

where ΓT = RDTR′.


Suppose DT → DT and then

ΓT = RDTR′ = R

τ(1− τ)

f 2(F−1(τ))

(

∑Tt=1 xtx

′t

T

)−1

R′

is consistent for ΓT . Therefore,

Γ−1/2T

√TR(βτ − βτ )

d−→ N (0, Iq).


The Wald test statistic is

WT (τ) = T (Rβτ − r)′Γ−1T (Rβτ − r)

d−→ χ2(q),

where q is the number of hypotheses.


More generally, if ΓT is estimated by

ΓT = R

(

∑Tt=1 ft(F

−1(τ))xtx′t

T

)−1

JT

(

∑Tt=1 ft(F

−1(τ))xtx′t

T

)−1

R′

= R

(

∑Tt=1 ft(F

−1(τ))xtx′t

T

)−1 [

τ(1 − τ)

∑Tt=1 xtx

′t

T

]−1

(

∑Tt=1 ft(F

−1(τ))xtx′t

T

)−1

R′

= R

(

∑Tt=1 ft(F

−1(τ))xtx′t

T

)−1 [∑T

t=1 xtx′t

T

]−1

(

∑Tt=1 ft(F

−1(τ))xtx′t

T

)−1

R′/[τ(1− τ)]

= ΩT /[τ(1 − τ)].


under the null hypothesis, the Wald test statistic

WT (τ) = T (Rβτ − r)′Ω−1T (Rβτ − r)/[τ(1− τ)]

d−→ χ2(q),


Let W q(τ) denote a q-vector of independent Brownian

motions and hence, for τ ∈ (0, 1),

Bq(τ) = W q(τ)− τW q(1)

represents a q-vector of independent Brownian Bridges. As a

result, for any fixed τ ∈ (0, 1),

Bq(τ)d−→ N (0, τ(1− τ)Iq). (9)


The normalized Euclidean norm of Bq(τ),

Qq(τ) = ‖Bq(τ)‖/√

τ(1− τ)

is referred to as a standardized tied-down Bessel process of

order q. Q2q(τ)

d→ χ2q is then followed. In addition, we have,

under the null hypothesis,

WT (τ) ⇒‖ Bq(τ)√

τ(1 − τ)‖2, τ ∈ (0, 1)

and then

supτ∈(0,1)

WT (τ) ⇒ supτ∈(0,1)

‖ Bq(τ)√

τ(1 − τ)‖2, τ ∈ (0, 1)


For testing the non-Granger causality in quantile τ ∈ [a, b], the

test statistic is considered as

supτ∈[a,b]

WT (τ) ⇒ supτ∈[a,b]

‖ Bq(τ)√

τ(1 − τ)‖2, τ ∈ [a, b].

By the result provided in DeLong (1981),

P

supτ∈[a,b]

‖ Bq(τ)√

τ(1− τ)‖2< c

= P

sups∈[1,s2/s1]

‖ W q(s)√s

‖2< c

,

for s1 = a/(1− a), s2 = b/(1− b), and W q a vector of q

independent Brownian motions. The critical values for some q

and s2/s1 have been tabulated in DeLong (1981) and Andrews

(1993).


Nonparametric Test of Jeong and Hardle (2008)Jeong and Hardle (2008) extends Zheng (1998)’s approach to the case of

dependent data and then to the test of Granger causality in quantile.

Suppose the causality of a bivariate case, or only yt, wt are observable,

is considered. Denote ut−1 = yt−1, . . . , yt−p, wt−1, . . . , wt−q and

wt = wt−1, . . . , wt−q. Granger causality in mean of Granger (1988) is

defined as

(i) wt does not cause yt in mean with respect to ut−1 if

E(yt|ut−1) = E(yt|ut−1\wt−1);

(ii) wt is a prima facie cause in mean of yt with respect to but−1 if

E(yt|ut−1) 6= E(yt|ut−1\wt−1).


Motivated by the definition of Granger-causality in mean,

Jeong and Hardle (2007) define the Granger-causality in

quantile θ as

(i) wt does not cause yt in mean with respect to ut−1 if

Qθ(yt|ut−1) = Qθ(yt|ut−1\wt−1);

(ii) wt is a prima facie cause in mean of yt with respect to

but−1 if

Qθ(yt|ut−1) 6= Qθ(yt|ut−1\wt−1),

where Qθ(yt|·) ≡ infyt|F (yt|·) ≥ θ is the θth (0 < θ < 1)

conditional quantile of yt.


Denote Fyt|ut(·) as the conditional distribution function of yt

given ut and is assumed to be absolutely continuous in yt for

almost all ut. Then

Fyt|utQθ(yt|ut) = θ

and from the definition of the Granger-causality in quantile θ

defined above, the hypotheses to be tested are

H0 : PFyt|utQθ(yt|ut\wt) = θ = 1 (10)

Ha ; PFyt|utQθ(yt|ut\wt) = θ < 1 (11)


Based on the idea of Zheng (1998) which reduces the problem

of testing a quantile restriction to a problem of testing a

particular type of mean restriction, the null hypothesis of (10)

is true if and only if E[Iyt ≤ Qθ(yt|ut\wt)|ut] = θ or

Iyt ≤ Qθ(yt|ut\wt) = θ + ǫt such that E(ǫt|ut) = 0.

Jeong and Hardle (2007) consider the following distance

measure to construct the test,

J ≡ E[Fyt|ut(Qθ(yt|ut\wt))− θ]2f

ut(ut),

where fut(ut) is the marginal density function of ut. It is

obvious that J ≥ 0 and the equality holds if and only if H0 is

true, with strick inequality holding under Ha.


Since E(ǫt|ut) = Fyt|ut[Qθ(yt|ut\wt)]− θ,

J ≡ E[Fyt|ut(Qθ(yt|ut\wt))− θ]2f

ut(ut)

= E[E(ǫt|ut)]2f

ut(ut)

= Eǫt[E(ǫt|ut)]fut(ut)

Thus the test suggested by Jeong and Hardle (2007) is based

on a sample analog of EǫtE(ǫt|ut)fut(ut). Including the

density function fut(ut) is to avoid the problem of trimming

on the boundary of the density function, see Powell, Stock,

and Stoker (1989) for an analogue approach.


As the density weighted conditional expectation E(ǫt|ut)fut(ut) can be

estimated by kernel method

E(ǫt|ut)fut(ut) =

1

(T − 1)hm

T∑

s6=t

Ktsǫt,

where m = p+ q, Kts = K(ut − us)/h is the kernel function and h is

a bandwidth, Jeong and Hardle (2007) consider a sample analog of J as

Jt ≡ 1

T (T − 1)hm

T∑

t=1

T∑

s6=t

Ktsǫtǫs

=1

T (T − 1)hm

T∑

t=1

T∑

s6=t

Kts [Iyt ≤ Qθ(yt|ut\wt) − θ]

[Iys ≤ Qθ(ys|ut\ws) − θ] .


Besides, the θ-th conditional quantile of yt given ut\wt,

Qθ(yt|ut\wt), can also be estimated by the nonparametric

kernel method

Qθ(yt|ut\wt) = F−1(θ|ut\wt),

where

Fyt|ut\wt(yt|ut\wt) =

∑Ts 6=tK

(

(ut\wt)−(us\ws)h

)

I(ys ≤ yt)

∑Ts 6=tK

(

(ut\wt)−(us\ws)h

) .


Therefore, ǫt can be estimated as

ǫt ≡ Iyt ≤ Qθ(yt|ut\wt) − θ.

Finally, the test statistic suggested by Jeong and Hardle (2007) is

Jt ≡ 1

T (T − 1)hm

T∑

t=1

T∑

s6=t

Ktsǫtǫs

=1

T (T − 1)hm

T∑

t=1

T∑

s6=t

Kts

[

Iyt ≤ Qθ(yt|ut\wt) − θ]

[

Iys ≤ Qθ(ys|ut\ws) − θ]

.


The limiting distribution of JT is given in the following theorem in Jeong

and Hardle (2007),

(i) Under the null hypothesis, Thm/2JT →d N(0, σ2), where

σ2 = 2Eσ4ǫt|ut

fut(ut)

∫

K2(u)du and

σ2ǫt|ut

= E(ǫ2t |ut) = θ(1− θ).

(ii) under the null hypothesis, σ ≡ 2θ2(1− θ)2 1T (T−1)hm

∑

s6=t K2ts is a

consistent estimator of σ2.

(iii) Under the local alternative

Ha : Fyt|ut\wtQθ(yt|ut\wt) + T−1/2h−m/4l(ut)|ut) = θ,

Thm/2JT →d N(µ, σ2), where

µ = Ef2yt|ut

[Qθ(yt|ut)]l2(ut)fut

(ut).


Note that strictly stationarity for yt,wt and some suitable

conditions for ǫt, kernel functions K and bandwidth

parameters h are required for above theorem.


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