Causality Tests using Linear and Nonparametric Quantile ...
Transcript of 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.
M.-Y. Chen Causality
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.
M.-Y. Chen Causality
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)
M.-Y. Chen Causality
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.
M.-Y. Chen Causality
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.
M.-Y. Chen Causality
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)
M.-Y. Chen Causality
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 .
M.-Y. Chen Causality
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)
M.-Y. Chen Causality
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.
M.-Y. Chen Causality
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′.
M.-Y. Chen Causality
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).
M.-Y. Chen Causality
The Wald test statistic is
WT (τ) = T (Rβτ − r)′Γ−1T (Rβτ − r)
d−→ χ2(q),
where q is the number of hypotheses.
M.-Y. Chen Causality
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)].
M.-Y. Chen Causality
under the null hypothesis, the Wald test statistic
WT (τ) = T (Rβτ − r)′Ω−1T (Rβτ − r)/[τ(1− τ)]
d−→ χ2(q),
M.-Y. Chen Causality
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)
M.-Y. Chen Causality
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)
M.-Y. Chen Causality
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).
M.-Y. Chen Causality
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).
M.-Y. Chen Causality
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.
M.-Y. Chen Causality
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)
M.-Y. Chen Causality
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.
M.-Y. Chen Causality
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.
M.-Y. Chen Causality
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] .
M.-Y. Chen Causality
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
) .
M.-Y. Chen Causality
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]
.
M.-Y. Chen Causality
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).
M.-Y. Chen Causality
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.
M.-Y. Chen Causality