Full text of DOI:10.1111/1468-0262.00168

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.
Econometrica, Vol. 68, No. 6 November, 2000 , 1465᎐1490
NONPARAMETRIC TEST FOR CAUSALITY WITH
LONG-RANGE DEPENDENCE1
2
BY JAVIER HIDALGO
This paper introduces a nonparametric Granger-causality test for covariance stationary
linear processes under, possibly, the presence of long-range dependence. We show that
the test is consistent and has power against contiguous alternatives converging to the
parametric rate Ty1r2. Since the test is based on estimates of the parameters of the
representation of a VAR model as a, possibly, two-sided infinite distributed lag model, we
Ž
.
first show that a modification of Hannan’s 1963, 1967 estimator is root-T consistent and
asymptotically normal for the coefficients of such a representation. When the data are
long-range dependent, this method of estimation becomes more attractive than least
squares, since the latter can be neither root-T consistent nor asymptotically normal as is
the case with short-range dependent data.
KEYWORDS: Causality, long-range dependence, spectral analysis, distributed lag model,
consistent test.
1. INTRODUCTION
Ž
.
Since Granger 1969 , when analyzing relationships between economic vari-
ables, the concept of causality has been a basic subject routinely invoked in the
econometricreconomic literature. Typically, tests for causality are commonly
Ž
Ž ..
performed in the context of unrestricted vector autoregressive VAR P mod-
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.
els with P a finite known positive number. See among others, Granger 1969 or
Geweke 1982 when the data are short-range dependent, or for variables
showing stochastic-trend behavior, see Sims, Stock, and Watson 1990 or Toda
and Phillips 1993 . Some extensions are in Hosoya 1991 who analyses causality
for stationary short-range dependent processes that do not necessarily have a
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.
Ž
.
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.
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.
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.
VAR representation or Lutkepohl and Poskitt 1996 , and references therein,
¨
who allow for an infinite order VAR model. However, the above models do not
cover the so-called long-range dependent processes that have attracted immense
attention in recent years in the econometric literature. The object of this paper
is thus to introduce and analyze a causality test that allows for long-range
dependence. In addition, the test does not rely on any specific finite parameteri-
zation of the model and covers processes that do not possess a finite order
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.
vector autoregressive moving average VARMA representation such as Bloom-
field’s 1973 exponential model.
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.
1Previous versions of this article have been circulated under the title ‘‘Estimation of Distributed
Lag Models with Long-range Dependence.’’
2
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.
This article is based on research funded by the Economic and Social Research Council ESRC
reference number: R000238212. I thank Andrew Harvey, Peter Robinson, a co-editor, and two
referees for helpful comments which led to a much improved version of the article. Of course, any
remaining errors are the sole responsibility of the author.
1465

1466
JAVIER HIDALGO
To fix ideas, let a bivariate observable covariance stationary vector wt s
Ž
.
yt , xt Ј satisfy
ϱ
Ž
.
Ž .
ts1, 2 . . . ,T ,
A L w s
A wtyj s⑀t
Ý
t
j
j
s0
where ⑀t is a bivariate martingale difference sequence and A0 is the identity
matrix. The interest is in testing the null hypothesis H0: yt £xt, that is yt does
not cause xt, against the alternative hypothesis H1: yt «xt, that is yt causes xt.
Ž .
The main attributes of the test, described below, are: i it is nonparametric,
Ž .
that is, we do not impose any specific parametric model for the data, ii it is
y1r2
Ž .
consistent, iii it has power against T
Ž .
contiguous alternatives, and iv its
limit distribution can be obtained, say, from the distribution of the supremum of
the standard Brownian motion. Thus, the paper extends previous work in two
main directions. First, we allow for a general covariance stationary linear
process and second, since it is nonparametric, we avoid the danger of possible
misspecification.
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.
Now we briefly discuss the main ideas of the test. Following Sims 1972 or
Ž
.
Hosoya 1977 , a test for H0 is equivalent to testing whether the coefficients
Ž .
c j are simultaneously equal to zero for all j-0 in
ϱ
Ž
.
Ž .
c j xtyj qut ,
1.1
yt s
Ý
syϱ
j
w
<
x
where, by construction, E ut xs, yϱ-s-ϱ s0, and xt and ut are, possibly,
long-range dependent processes. Alternatively, the null hypothesis H0 is equiva-
lent to
0
Ž
.
Ž
.
w
x
c jy1 cos ␲ j␭ s0 ᭙␭g 0, 1 ,
Ý
syϱ
j
or
0
␮
yi␲ j␭
Ž
.
Ž
.
w
x
S* ␮ sRe
c jy1 e
d␭ s0 ᭙␮g 0, 1 ,
H Ý
ž
/
ž
/
0
jsyϱ
Ž .
where Re a denotes the real part of a complex number a. Therefore the
hypothesis testing can be described as
Ž
.
Ž
.
w
x
Ž
.
w
x
1.2
H0 : S* ␮ s0 ᭙␮g 0, 1 against H1: S* ␮ /0 in ⌬; 0, 1
where ⌬ has Lebesgue measure greater than zero.
Ž . Ž .
Given estimates of c j , say c j , and using Riemann’s discrete approximation
ˆ
Ž
.
of integrals by sums, S* ␮ can be estimated by
w
x
M␮
0
1
yi j␭
p
Ž
.
Ž
.
Ž .
c jy1 e
ˆ
1.3
ST ␮ sRe
Ý
Ý
ž
/
ž M
/
p
s1
jsyϱ

TEST FOR CAUSALITY
1467
Ž .
where ␭p s␲ prM, ps1, . . . , M, and MsM T is a number that increases
slowly with T, that is My1 qMTy1 ª0. The test can thus be based on whether
Ž
. w x
is significantly different than zero for all ␮g 0, 1 by the
or not ST
␮
Ž
.
implementation of a functional of ST ␮ , say a Kolmogorov-Smirnov test.
The remainder of the paper is organized as follows. In the next section, we
Ž .
describe and motivate the estimation technique of the coefficients c j , whose
statistical properties are given in Section 4. Section 3 describes the statistical
framework. Also, since our statistics and proofs are based on the spectral density
matrix estimator of wt and the cross-spectrum between xt and ut, we give some
useful statistical properties of them. In Section 5, we study the properties of
Ž
.
1.3 and we provide the test for Granger-causality and its implementation. In
Section 6, we discuss the choice of M and how the results of Sections 4 and 5
are extended to general p-dimensional data. Finally, Section 7 gives a summary
of the paper. The proofs of the results are confined to Mathematical Appendix
A; they apply some technical Lemmas given in Mathematical Appendix B.
2. ESTIMATION PROCEDURE AND ITS MOTIVATION
Ž .
In this section we describe the estimation technique of the coefficients c j in
1.1 and discuss why it is more desirable than least squares LSE estimates in
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.
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.
the presence of long-range dependence. To accomplish this, we adopt the
frequency domain approach whose merits have been proven to be numerous and
applied in several contexts. For example, it was used when testing for causality
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.
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.
in the pioneer work by Granger 1969 , or in Geweke 1986 to analyze the
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.
neutrality of money. Recently, Hosoya 1995 has shown the usefulness of
spectral analysis in contrast to time domain methods when analyzing the
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.
causality among economic variables; see also Geweke 1982 . Another example
is in the efficient estimation of the parameters in a regression model; see the
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.
cornerstone work by Hannan 1963 , extended to more general models useful in
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.
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.
econometrics by Hannan and Terrell 1973 and recently by Robinson 1991
who also allowed for data dependent smoothing in the spectral estimation. One
rationale of the approach lies in that only minimal conditions of the process are
needed, like stationarity, and no explicit assumptions on its dynamic specifica-
tion are thus required.
Ž
.
In the frequency domain, the lag structure given in 1.1 is described by the
ϱ
yi j␭
Ž .
Ž .
Ž .
, so c j is interpreted as
frequency response function C ␭ sÝjsyϱc j e
Ž .
y1Ž . Ž .
the jth Fourier coefficient of C ␭ sfx x ␭ fy x ␭ , that is,
2
␲
y1
i j␭
Ž
.
Ž .
Ž
.
Ž .
C ␭ e d␭,
2.1
c j s 2␲
H
0
Ž .
Ž .
␭
where fy x
␭
and fx x
are the indicated elements of the spectral density
Ž .
matrix, fw w ␭ , of wt defined from the relationship
Ž
.Ž
.
E
w1 yEw1 wjq1 yEw Ј
Ž
.
1
␲
yi j␭
Ž .
fw w ␭ e
Ž .
js0,"1,"2, . . . .
s
d␭
H
y
␲

1468
..
Due to this interpretation, Hannan 1963, 1967 see also Brillinger 1981
JAVIER HIDALGO
Ž
. Ž
Ž
Ž .
Ž
.
Ž
.
proposed to estimate c j by the sample discrete analogue of 2.1 ,
My1
2 Ý
1
i j␭
p
ˆ
Cp e
Ž
.
Ž .
2.2
c j s
,
˜
2 M
ps0
y1
ˆ
ˆ
ˆ
ˆ
ˆ
where Cp sfx x, p fy x, p, and fy x, p and fx x, p are estimates of fy x, p and fx x, p
Ž .
Ž
.
respectively, where for a generic function g ␭ , gp denotes g ␭ .
The motivation of the estimator in 2.2 , coined by Sims 1974 as HI
p
Ž
.
Ž
.
Ž
.
Hannan’s inefficient estimator, is threefold. First is the ability to estimate the
Ž .
Ž
.
coefficients c j irrespective of the number of lags specified in 1.1 , which will
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.
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.
be relevant when analyzing the properties of ST ␮ defined in 1.3 . Second,
since there is no gain by exploiting the information on the covariance structure
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.
of the errors ut , as Sims 1974 showed, the HI estimator becomes as efficient as
Ž
.
Ž .
the generalized least squares GLS estimator. This motivates the LSE of c j
Ž
.
given in Robinson 1979 , although under stronger assumptions than those we
want to impose in this paper.
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.
Finally, the third motivation, which makes the estimates in 2.2 more appeal-
ing when the data are long-range dependent, is as follows. Assume, for exposi-
tional simplicity, that model 1.1 is
Ž
.
q
Ž
.
Ž .
c j xtyj qut
Ž .
ts1, . . . , T ,
2.3
yt s
Ý
syr
j
where both q and r are finite and known a priori. When the data are
short-range dependent, it is known that, under suitable conditions, the LSE is
root-T consistent and asymptotically normal. However, under long-range depen-
Ž
.
dence, as Robinson 1994 observed, when the joint long-range dependence in
the regressor xt and error ut is sufficiently strong, that is the product of the
spectral density functions of xt and ut is not integrable, the LSE is no longer
root-T consistent and more importantly, it loses the central limit theorem
property.
Ž
.
Motivated by this observation, Robinson and Hidalgo 1997 showed that a
Ž
class of frequency-domain weighted LSE, including GLS with parametric error
spectral density function as a special case, is root-T consistent, asymptotically
normal and Gauss-Markov efficient in model 2.3 . More generally, their results
are also valid when c j is known up to a set of parameters ␪, that is
c j sc j; ␪ for all j, in 1.1 . The intuition why the estimator in Robinson and
Hidalgo 1997 is root-T consistent and asymptotically normal is because the
.
Ž
.
Ž .
Ž
Ž .
Ž
.
.
Ž
.
weighted function possesses a zero sufficiently strong to compensate for the
singularity of the spectral density function induced by the joint long-range
y1
Ž .
Ž
dependence of xt and ut. So, since f
␭ possesses a zero at ␭s0 see C8
x x
y1
ˆ
.
Ž
.
below , we can expect that fx x, p becomes asymptotically a weighted function
satisfying the conditions of Robinson and Hidalgo 1997 . In fact, it is shown in
Theorem 1 below that the modification of the HI estimator given in 4.1
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.
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.

TEST FOR CAUSALITY
1469
achieves root-T consistency and asymptotic normality, so that the HI estimator
is indeed a desirable estimator.
It is worth mentioning that Hidalgo and Robinson 1999 have recently
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.
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.
provided asymptotic justification of Hannan’s 1963 GLS estimate when the
Ž .
spectral density fuu ␭ is unknown and both xt and ut are, possibly, long-range
Ž
.
dependent. However, the last approach in the general framework of model 1.1
seems difficult to implement in empirical studies and based on the previous
comments, it is expected that there is no gain in efficiency. It should be noted
Ž .
Ž .
␭ do not coincide, applying
that when the singularities of fuu
␭
and fx x
Ž
.
Robinson and Hidalgo’s 1997 results, under suitable conditions, the LSE will
be root-T consistent and asymptotically normal.
3. STATISTICAL FRAMEWORK AND ASYMPTOTIC PROPERTIES OF THE
SPECTRAL DENSITY MATRIX ESTIMATOR
Ž
.
Ž
.
As the estimator in 2.2 and the test for H0 and their proofs employ the
Ž .
Ž .
Ž .
estimate of the spectral density fx x ␭ and the cross-spectra fy x ␭ and fu x ␭ , it
seems appropriate to examine the properties of the spectral density matrix
estimate of a, for example, ¨-dimensional covariance stationary linear process
Ž .
zt. Let f ␭ be the spectral density matrix of zt defined from the relation
Ž
.
Ž .
Ž
.Ž
.
3.1
␥ j sE z1 yEz1 zjq1 yEz Ј
Ž
.
1
␲
yi j␭
Ž .
f ␭ e
Ž .
js0,"1,"2, . . . .
s
d␭
H
y
␲
Ž
.
Let ztg denote the gth element of zt , the autocovariance spectrum function of
ztg by ␥g g
Ž .Ž Ž .. Ž .
␭ and the cross-covariance cross-spectrum of ztg and zth by
j
fg g
Ž .Ž Ž ..
j
␥
fgh ␭ .
We wish to estimate f ␭ on the basis of T observations ZT s zt, ts1, . . . , T .
Writing
gh
Ž .
ꢀ
4
T
T
1
z eit␭
z eyit␭ Ј
,
Ž .
I ␭ s
Ý
Ý
t
t
2␲ T ž
/ ž
/
t
s1
ts1
we estimate f by
m
1
ˆ
Ž
.
Ž .
f ␭ s
3.2
I ␭ q␭ ,
Ý
sym
ž
/
j
2mq1
j
w
x
Ž
.
where ms Tr4M with M as defined in Section 1 and␭j s 2␲ j rT, js0,"
w
x
1, . . . , " Tr2 .
ˆ
Ž .
To examine the properties of f ␭ , let us introduce the following conditions
Ž
.
w
.
CONDITION C1: For gs1, . . . , ¨, there exist Cg g 0, ϱ , dg g 0, 1r2 and ␣g
Ž
x
0, 2 , such that
y2 dg
␣
Ž .
Ž
Ž
..
fg g ␭ sCg ␭
Ž .
1qO ␭
as ␭ª0q
w
x
and fg g ␭ )0 for all ␭g 0, ␲ .

1470
JAVIER HIDALGO
Ž .
CONDITION C2: For gs1, . . . , ¨, fg g ␭ is twice continuously differentiable in any
open set outside the origin, and
Ѩ j
yjy2 dg
Ž .
Ž
.
fg g ␭ sO ␭
as ␭ª0q for js1, 2.
j
Ѩ␭
Let us define the coherence between ztg and zth as
Ž .
Ž .
1r2 Ž . 1r2 Ž .
␭ fhh ␭ .
Rgh ␭ sfgh ␭ r f
Ž
.
g g
<
Ž .<
␭ is twice continuously differentiable
CONDITION C3: For g-hs2, . . . , ¨, Rgh
Ž
x
in any open set outside the origin and for some ␤g 1, 2 ,
␤
<
Ž .
Ž .<
Ž
.
Rgh ␭ yRgh 0 sO ␭
as ␭ª0q .
ꢀ
4
CONDITION C4: zt is a co¨ariance stationary linear process defined as
ϱ
ϱ
5
␨
j
52
-ϱ,
z s
␨ ⑀
,
Ý
Ý
t
j
tyj
j
s0
js0
Ž
.
5
5
D stands for the
where ␨ is the identity matrix, ␨ are ¨ =¨ matrices, and
j
norm of0the matrix D.
ꢀ
4
X
CONDITION C5: ⑀t is a stochastic process with finite fourth moments, where
X
Ž
<
.
Ž
<
.
Ž
.
Ž
<
.
s␮
ty1
E ⑀t
such that
generated by ⑀s, sFt, and the joint fourth cumulant of ⑀t j , ji s1, . . . , ¨ and
F
s0, E ⑀t ⑀t
F
sE ⑀t ⑀t s⌶ a.s., E ⑀ ⑀
⑀
F
ty1
ty1
t j1 t j2 t j3
3, j1, j2, j3
<
<
␮
-ϱ for all j1, j2 , j3 where F is the ␴-algebra of e¨ents
3, j1, j2, j3
t
i
i
is1, . . . , 4 satisfies
␬
,
t1 st2 st3 st4 ,
otherwise,
j1 , j2 , j3 , j4
Ž
.
s
cum ⑀t , ⑀t , ⑀t , ⑀t
1 j1
2 j2
3 j3
4 j4
½
0,
<
<
with␬smax j s1, . . . ,
␬
-ϱ.
¨
, is1, . . . , 4
j1, j2, j3, j4
i
<Ž
ϱ
. Ž .<
Ž< Ž .<
.
CONDITION C6: ѨrѨ␭ ␩ ␭ sO ␩ ␭ r␭ as ␭ª0q, where
␨ ei j␭
,
j
Ž .
␩ ␭ s
Ý
j
s0
which is twice continuously differentiable in any open set outside the origin, and
y1r2
Ž . Ž .
Ž .
f
␭ ␩ ␭ , gs1, . . . , ¨, is a nonzero finite ¨ector where ␩ ␭ denotes the gth
g g
g
Žg .
row of ␩ ␭ .
CONDITION C7: M 2rTq1rMª0.
ˆ
Ž .
Prior to stating some results on the spectral matrix estimator, f ␭ , some
discussion about our conditions is in order. Conditions C1᎐C3 deal with the
Ž .
behavior of f ␭ . For frequencies ␭ª0q, they are the same used elsewhere

TEST FOR CAUSALITY
1471
Ž
.
by, say, Robinson 1995a, b and thus his comments apply here, while for
frequencies ␭ in any open set outside the origin, they are standard. Conditions
C4 and C5 are restrictive in the linearity they impose, but not otherwise.
Although Condition C5 can be weakened to cover more general processes for
Ž
.
the innovation ⑀t as in Hosoya and Taniguchi 1982 , we keep it in its present
form to avoid extra notational complications of the rather lengthy proofs of
some of the results.
Examples of processes satisfying C1᎐C5 are as follows. Let ␰ be a ¨-dimen-
t
sional unobservable covariance stationary linear process that possesses a contin-
uous and bounded away from zero spectral density matrix and consider the filter
ϱ
Ž
.
Ž .
G j ␰
3.3
z s
.
tyj
Ý
t
j
s0
ϱ
i j␭
Ž .
Ž .
Ž .
Let Gg ␭ denote the gth row of the matrix G ␭ sÝjs0G j e
such that
dg
Ž .
Gg ␭ ␭ tends to a nonzero finite vector as ␭ª0q, for gs1, . . . , ¨. For
instance, let ␰ be a stationary invertible vector autoregressive moving average
t
Ž
.
VARMA process with iid innovations and let each ztg be formed by separate
fractional integration of the corresponding ␰ element, so that
t
yd1
yd¨
i␭
i␭
Ž .
Ž
.
Ž
.
G ␭ sdiag 1ye
, . . . , 1ye
.
Ž
.
Then C1᎐C5 hold. In the particular case of zt being scalar, its spectral density
function is
2
2
i␭
i␭
Ž
⌰ e
.
.
␴
y2 d
i␭
Ž
.
Ž .
f ␭ s
<
<
3.4
1ye
, y␲-␭F␲ ,
Ž
2␲
⌽ e
Ž .
Ž .
where 0Fd-1r2, and where ⌰ ␭ and ⌽ ␭ are the MA and AR polynomials
Ž
.
respectively, having no zeroes in or on the unit circle. 3.4 is the familiar
fractional autoregressive moving average ARFIMA model; see for instance
Granger and Joyeux 1980 or Hosking 1981 . Another model that exhibits
Ž
Ž
.
Ž
.
.
Ž
.
long-range dependence is the fractional Gaussian noise fgn process introduced
Ž
.
Ž
by Mandelbrot and Van Ness 1968 , whose spectral density function see Sinai
Ž
..
1976 is
y2y2 d
2
ϱ
Ž
.
4␴z ⌫ 2d
␭
Ž
.
Ž .
f ␭ s
Ž
.
2 Ž
.
3.5
cos ␲ d sin ␭r2
jq
Ý
3q2 d
2␲
Ž
.
2␲
j
syϱ
2
Ž
Ž
..2
Ž .
where ␴ sE zt yE zt -ϱ and ⌫ и denotes the gamma function. From
Ž
.
z Ž
.
3.4 and 3.5 , we observe that the ARFIMA and fgn models do not represent
the same process, although their spectral density functions behave as K␭
y2 d
as
␭ª0q, for a generic finite positive constant K. For a review of these models,
Ž
.
see Beran’s 1994 monograph.
Ž
.
Condition C6, when ␭ª0q, was assumed elsewhere in Robinson 1995b , so
his comments apply here also, while for frequencies ␭ in any open set outside

1472
JAVIER HIDALGO
the origin, the condition is standard in spectral density matrix estimation. The
second part of the assumption is not strong; see for instance the comments
dg
y1r2
Ž
.
made after 3.3 , once ␭ is identified as fg g
up to constants. Finally,
Condition C7 gives the upper bound on the rate of M to infinity. In particular,
M cannot increase faster than T1r2y␦ for any 0-␦-1r2.
ˆ
ˆ
Ž .
Ž .
Ž
.
Ž
.
Let fgh ␭ denote the g, h th element of f ␭ in 3.2 .
PROPOSITION 1: Assuming C1᎐C7, for g, hs1, . . . , ¨,
Ž .
M log T
U
Ž
.
<
<
E ␾
sO
fgh, p
if 1FpFM,
gh, p
ž
/
Tp
Ž .
Ž .
with the con¨ention that O a so a s0 if as0 and where
m
1
U
U Ž .
ˆ
␾
gh, p sfgh, p yfgh, p and fgh ␭ s
fgh ␭ q␭ .
Ý
sym
ž
/
j
2mq1
j
PROOF: The proof is a straightforward application of Proposition 1 of Hidalgo
Ž
.
w
x
w x
and Yajima 1998 when p satisfies 1Fp- M␦ , where a denotes the integer
Ž
.
Ž .
part of a, which applies Theorem 2 of Robinson 1995a , and by Proposition 1 c
of Hidalgo and Robinson 1999 when M␦ FpFM for ␦)0.
Ž
.
w
x
Q.E.D.
PROPOSITION 2: Assuming C1᎐C7, for a, b, g, hs1, . . . , ¨,
1r2 1r2
My1T Cov ␾gh, p , ␾
sO fg g , p fhh, p fa1ar,2p b1br,2p
.<
ab, p
f
.
<
Ž
Ž
.
PROPOSITION 3: Assuming C1᎐C7, for g, hs1, . . . , ¨,
sOp PM 1r2rT1r2
.
y1r2 y1r2
Ž
.
sup
f
f
␾
Ž
.
g g , p hh, p gh, p
p
s1, . . . , P
4. THE ASYMPTOTIC DISTRIBUTION OF THE HI ESTIMATOR
Ž
.
The HI estimator in 2.2 , similar to the technical problems encountered in
many other nonrsemi-parametric estimators, is quite difficult to analyze as it
ˆ
Ž .
Ž .
stands since fx x 0 tries to estimate fx x 0 , which may be infinity. One way to
Ž
.
proceed is by trimming the term corresponding to ps0 from 2.2 . However, as
it will become clear when examining A.7 in Mathematical Appendix A, the
trimming introduces a bias problem that otherwise would not exist if the
Ž
.
Ž
.
frequency ␭0 s0 was included. Thus, we modify 2.2 to
My1Ј
2 Ý
1
y1
4.1
fx x, p fy x, p ei j␭
,
p
ˆ
ˆ
Ž
.
Ž .
c j s
ˆ
2 M
ps1
2 My1Ј
ap ei j␭ means Ý2psMy1 1ap ei j␭ qa1. Intuitively, we have replaced the
p
p
where Ý
ps1
y1
ˆ
ˆ
.
estimator of C0 by that of C1, that is fx x, 1 fy x, 1

TEST FOR CAUSALITY
1473
Ž
.
The asymptotic properties of the estimator given in 2.2 were first established
Ž
.
by Hannan 1967 for a finite, possibly of unknown order, distributed lag model
and Brillinger 1981 for the infinite distributed lag model, when both fx x ␭ and
fuu ␭ are bounded and bounded away from zero. Thus, the aim of this section,
Ž
.
Ž .
Ž .
due to the possible adverse properties of the LSE, is to show that the estimator
Ž
.
defined in 4.1 is root-T consistent and asymptotic normal under the presence
of, possibly, long-range dependence. Finally, it is worth noting that due to the
Ž .
Ž .
Ž
.
Ž . < <
cyclical behavior of c j , that is, c j sc jq2 M , c j , j sMq1, . . . , cannot
ˆ
ˆ
ˆ
< Ž .<
be estimated, although their contribution is negligible since Ý< j<) M c j s
y2q␶ r2
y1r2
Ž
.
Ž
.
by C9 and C10 respectively given below.
O M
so T
Assume the following:
ꢀ
4
ꢀ
4
ꢀ
4
CONDITION C8: wt s yt , xt Ј and ut are co¨ariance stationary linear pro-
cesses defined as
ϱ
ϱ
ϱ
ϱ
52
<
w s
⌽ ⑀
,
⌽
-ϱ and u s
⌽ u⑀
,
2 -ϱ,
u
<
⌽
j
5
Ý
Ý
Ý
Ý
t
j
tyj
j
t
j
u, tyj
j
s0
js0
js0
js0
Ž .
where ⑀t and ⑀u, t satisfy C5, fuu ␭ and fw w
Ž .
Ž .
satisfy C1᎐C3 and fuu ␭ and
␭
ϱ
are bounded away from zero. Also, ⌽ ␭ sÝjs0⌽j ei j␭ and ⌽ ␭ s
5
Ž .5
Ž .
uŽ .
fw w
␭
Ýϱjs0⌽j ei j␭ satisfy C6 and xt and ut are mutually independent.
u
ꢀ
4
ꢀ
4
Ž< <.
Ž< <y3q␶ r2
.
for some 0-␶-1.
CONDITION C9: c j sO j
CONDITION C10: M 2Ty1 qM y4Tª0 with ␶ as in C9.
␶
Ž .
Condition C8 deals with the requirement of some smoothness of fuu ␭ and
Ž .
fw w ␭ . The requirement of independence between xt and ut , as in Robinson
Ž
.
and Hidalgo 1997 , is necessary for the proof of asymptotic normality. We
believe that it might be possible to relax it to some extent, but that will
enormously complicate the already technical proof given in Robinson and
Hidalgo 1997 . This certainly remains an open question. Condition C9 implies
that the first derivative of C ␭ is Liptchitz continuous with Liptchitz parameter
in the interval 0, 1y␶r2 . Condition C10 strengthens the admissible values of
M in C7. Specifically, the rate of increase of M to infinity cannot be slower than
Ž
.
Ž .
Ž
.
q1rŽ4y␶ .
T ␦
for arbitrarily small ␦)0.
THEOREM 1: Assuming C8᎐C10, for any finite collection j1, . . . , jq:
d
Ž . 1r2Ž Ž .
i T
Ž .
Ž
.
Ž
..
Ž
ꢀ
4r,
.
where
s1, . . . , q
c j yc j , . . . , c j yc jq Јª N 0,⍀s ⍀j j
ˆ
ˆ
1
1
q
l
r
l
␲
y1
y1
iŽ jryj .␭
l
Ž
.
Ž . Ž .
␭ fuu ␭ e
x x
⍀
s 2␲
f
d␭,
H
jr j
l
y
␲
Ž .
Ž
.
l
which corresponds to the asymptotic co¨ariance between c j and c j .
ˆ
ˆ
r

1474
JAVIER HIDALGO
ii Let fuu, p sfy y, p yfx x, p fy x, p 2 . A consistent estimator of ⍀j j , r, ls1, . . . , q,
y1
ˆ
ˆ
ˆ
ˆ
Ž .
<
<
r
l
is
My1Ј
2 Ý
1
y1
.␭
p
r
l
ˆ
⍀
ˆ
s
fx x, p fuu, p eiŽ j yj
.
jr j
l
2 M
ps1
Ž
.
Theorem 1 indicates that the results, obtained by Hannan 1967 and Brillinger
Ž
.
1981 for short-range dependent data, hold the same under long-range depen-
dence. It is important to observe the following; because fx x ␭ ;K␭
y2 dx
Ž .
and
y2 du
Ž .
fuu ␭ ;K␭
as ␭ª0q by C8, where 0Fdx , du -1r2, the asymptotic
Ž .
ˆ
covariance structure of c j , when taken as a process indexed by j, behaves as
that of a long-range dependent process with dsdu ydx. In particular, when
y1
Ž .
fx x ␭ sKfuu
Ž .
␭
Ž
ˆ
x
Ž .
ˆ
Ž
.
for all ␭g y␲, ␲ , c j is, asymptotically, an iid 0, K
y1 y1
Ž .
Ž
.
Ž . Ž .
␭ fuu ␭ .
Gaussian process. Generally, c j has a spectral density 2␲
f
x x
5. A NONPARAMETRIC CAUSALITY TEST
Ž
.
If in 2.3 r was a known finite constant, a Wald test for H0 could easily be
implemented from the results of Theorem 1. However, when there is no known
Ž .
Ž
.
parameterization of the model, e.g. of c j in 1.1 , in terms of a finite set of
parameters, ␪, the results given in Theorem 1 cannot be implemented in a
straightforward manner. In this framework, one way of testing for Granger
Ž .
causality might be by fitting an AR P model to wt where P increases slowly
Ž
.
with T as in Lutkepohl and Poskitt 1996 . However, their results depend heavily
¨
on the assumption that the data are short-range dependent. Specifically, their
Assumptions 1 and 2, and the assumption that the sample covariance of the data
converges to the population one at the rate Ty1r2, are not necessarily the case
under long-range dependence. Moreover, the asymptotic distribution of the
sample covariance may not be a normal random variable; see Taqqu 1975 or
Hannan 1976 . Thus, we adopt the approach given in 1.3 . Given the estimates
Ž .
Ž
.
Ž
.
.
Ž
.
Ž
of c j in 4.1 , consider
w
x
M␮
0
1
yi j␭
p
Ž
.
Ž
.
Ž
.
5.1
ST ␮ sRe
c jy1 e
.
ˆ
Ý
Ý
ž
/
ž M
/
p
s1
jsyMq1
Ž
.
Ž
.
ST ␮ forms the basis for the hypothesis testing in 1.2 as follows. In Corollary
1 below, we show that under H0, T1r2ST ␮ «
B ␮ , a Gaussian process
w eakl y
˜
Ž
.
Ž
.
with covariance structure
1
Ž
.
␲ min ␮1, ␮
2
y1
Ž
.
Ž
.
Ž . Ž .
␭ fuu ␭ d␭,
5.2
K ␮1 , ␮
s
f
H
2
x x
4␲
0
where ‘‘«w eakl y’’ denotes weakly convergence in D 0, 1 equipped with the
w
x
Skorohod metric.
˜
Ž .
Ž
.
Since the function K ␮, ␮ is nondecreasing and nonnegative, B ␮ admits
..
Ž
Ž
Ž .
the representation B K ␮, ␮ in distribution, where B ␮ is the standard

TEST FOR CAUSALITY
1475
w
x
Brownian motion in 0, 1 . This representation, Corollary 1 below and the
continuous mapping theorem yield
sup T1r2ST
Ž
.
␮
w
x
␮g 0, 1
<
Ž
.<
1r2 Ž
.
<
Ž
.<
«
sup
B ␮ sK
1, 1 sup B ␮
in law.
w
Ž
.x
w
␮g 0, 1
x
␮g 0, K 1, 1
Let us introduce the following condition.
4
4
<
<
<
<
CONDITION C8Ј: C8 is satisfied and in addition xt and ut are uniformly
integrable.
4q␦
<
<
Observe that a sufficient condition for the second part of C8Ј is that E xt
qE ut 4q␦ -ϱ for some ␦)0.
<
<
COROLLARY 1: Assuming C8Ј, C9, and C10, under H0,
weakly
« B ␮ ,
T1r2ST
␮
˜
Ž .
Ž
.
w
x
Ž
.
2
in D 0, 1 with the Skorohod metric and co¨ariance structure K ␮1, ␮ gi¨en in
Ž
.
5.2 .
ˆ
Ž
.
Ž
.
Let K ␮, ␮ be the consistent estimate of K ␮, ␮ ,
w
x
M␮
1
y1
ˆ
ˆ ˆ
fx x, p fuu, p .
Ž
.
K ␮, ␮ s
Ý
4M
p
s1
Then, Corollary 1 and the comments made above are useful for testing H0. For
example, the Kolmogorov-Smirnov test based on T1r2ST ␮ would reject the
Ž
.
y1r2Ž . <
null if sup K
1, 1 T1r2ST ␮ , ␮g 0, 1 exceeds an appropriate critical
Ž
.<
w
x4
ˆ
ꢀ
value obtained from the boundary crossing probabilities of a Brownian motion,
which are readily available on the unit interval. More generally see Koul and
Ž
Ž
..
Stute 1999 as
weakly
« B ␮
y1r2 Ž
.
.
y1 Ž .
Ž .
ˆ
ˆ
Ž
K
1, 1 T1r2ST K ␮, ␮
t
Ž
.
ž
/
..y1Ž .
ꢀ
w
ˆ
x
.
4
ˆ
ˆ
Ž
Ž
Ž
where K ␮, ␮
t sinf ␮g 0, 1 , K ␮, ␮ Gt , the limiting distribution of
y1r2Ž ..y1Ž ..
any continuous functional of K t can be ob-
1, 1 T1r2ST K ␮, ␮
.
ŽŽ
ˆ
Ž
Ž
.
w
x
tained from the distribution of the corresponding functional of B ␮ on 0, 1 .
As in other problems involving testing, one key area of interest is to know the
properties of the tests under contiguous or Pitman alternatives. To this end, let
us introduce
0
y1r2
Ž
.
Ž
.
Ž .
h ␲␭ ,
Ha:
c jy1 cos ␲ j␭ sT
Ý
syϱ
j

1476
JAVIER HIDALGO
Ž .
w
x
< Ž .<
where h ␽ is a continuous function in 0, ␲ such that 0- h ␽ in a set
w
x
⌬; 0,␲ with positive Lebesgue measure.
COROLLARY 2: Assuming C8Ј, C9, and C10, under Ha,
␮
weakly
˜
Ž
Ž
.
Ž
.
.
Ž
.
5.3
T1r2ST
␮
« B ␮ q h ␲␭ d␭.
H
0
Corollary 2 thus indicates that the test has power against contiguous alterna-
tives that converge to the null at the rate Ty1r2. It therefore has no zero
asymptotic relative efficiency compared to rival parametric tests based on a
Ž
.
correct parameterization of 1.1 . From Corollary 2, we can easily show that the
test is consistent. For that purpose, consider
0
Ž . Ž . Ž .
c jy1 cos ␲ j␭ sh ␲␭ .
H1:
Ý
syϱ
j
From Corollary 1 we can then immediately prove that under H1
␮
weakly
1r2
T1r2ST ␮ yT
h ␲␭ d␭ « B ␮ ,
˜
Ž .
Ž
.
Ž
.
H
0
and thus the consistency of the test.
To finish this section, it is worth noting that as the Kolmogorov-Smirnov test
has somewhat poor finite sample properties, we may alternatively attempt to
bootstrap the statistic T1r2ST ␮ , in a similar way to Hidalgo and Kreiss 1999 ,
Ž
.
Ž
.
using a combination of the Wild and Moving Block Bootstraps. Specifically, let
y1
lsl T be a number such that l ql Ty1 ª0. Consider LsTylq1 groups,
Ž .
such that the qth group is formed from the observations
Ž
.
Ž
.
Ј.
qqy1
yq , xq Ј, . . . , ylqqy1 , x l
Then the bootstrap analogue of T1r2ST ␮ is
Ž
.
w
x
M␮
L
0
1
STU ␮ sL
Re
l 1r2
c
Ž
q
y1r2
Ž
.
.
jy1
ˆ
Ž
Ý
Ý
Ý
ž
ž M
q
s1
ps1
jsyMq1
yc jy1 eyi j␭
␽
p
Ž
.
ˆ
.
q
/
/
Ž
.
Ž
.
Ž
.
where ␽ are iid 0, 1 random variables and c jy1 is identical to 4.1 but
ˆq
using onqly the observations from the qth group.
6. IMPLEMENTATION OF THE TEST AND ITS EXTENSION TO
MULTIVARIATE DATA
Ž .
In empirical studies to implement the test and estimate the parameters c j ,
Ž
w
x.
we face the problem of how to choose M or m recall that Ms Tr4m . Based

TEST FOR CAUSALITY
1477
Ž .
on the approach adopted to estimate the parameters c j , the choice of M can
be regarded as that of the number of leadsrlags in the model
w
x
MsÝTr4m
Ž
.
Ž .
c j xtyj qu
Ž .
ts1, . . . , T .
6.1
yt s
˜
t
w
x
jsyMsy Tr4m
Ž
.
One standard criterion for choosing M, and thus m, in 6.1 is the minimiza-
tion of the Akaike’s 1974 AIC criterion
Ž
.
log ␴ 2 q2 MrT
ˆM
Ž
.
6.2
where, using Kolmogorov’s formula,
My1
2 Ý
1
␴ 2 sexp
log f
␭
,
ˆ
Ž
.
j
ˆM
ž
/
uu
ž 4␲ M
/
js1
that is, the estimator of the one-step-prediction error based on the residuals
w
x
MsÝTr4m
Ž .
c j x
ˆ
yt y
.
tyj
w
x
jsyMsy Tr4m
The motivation of the AIC criterion is twofold. First, it gives an approximately
unbiased measure for the prediction error and secondly, it provides an asymp-
Ž
totic unbiased ‘‘estimate’’ of the Kullback-Leibler information using its Whittle’s
approximation ,
.
Ž .
f ␭
␲
T
Ž
.
Ž
.
Ž .
log g ␭ q
KL f, g s2T log 2␲ q
d␭,
H
ž
/
Ž .
g ␭
2␲
y
␲
where f and g are the true and candidate spectral density functions for a given
data set.
Ž
.
An alternative criterion to the AIC given in 6.2 , which can be used, is based
on the choice of ms Tr4M that minimizes the cross-validation criterion
w
x
w
x
Tr2 m
Ž
.
␭
j
Iuu, m
ˆˆ
m
Ž
.
Ž
.
j
6.3
Qm s
log fuu, Ž j. ␭ q
,
Ý
m
½
. 5
ˆ
Ž
fuu, Ž j.
␭
j
js1
m
ˆ
Ž
.
Ž
.
Ž
.
where Iuu, m
ˆ
␭
ˆ l
is the periodogram of the residuals in 6.1 and fuu, Ž j. ␭ s
j
ˆˆ
2
m
m
ˆ
Ž
.
Ž
.
Ž
.
fy y, Ž j. ␭ y C ␭ fx x, Ž j.
sity function of ut given in part ii of Theorem 1: that is
␭
is the one-leave-out estimate of the spectral den-
j
j
j
Ž .
m
1
m
ˆ
ˆ
Ž
Ž
.
.
j
fuu, Ž j. ␭ s
Iy y ␭ q␭ y C ␭ 2 Ix x ␭ q␭
.
Ý
sym, l/0
ž
/
ž
/
j
l
j
l
j
ž
/
2mq1
l
Ž
.
The motivation of the above criteria Qm in 6.3 is that, as shown by, for
ˆ
.
Ž
.
Ž
example, Hurvich and Beltrao 1990 , the expected value of KL fuu, fuu , where
˜
ˆ
fuu is a nonparametric estimate of fuu, is asymptotically equivalent to the

1478
JAVIER HIDALGO
Ž
Ž
..
cross-validation criterion apart from the constant 2T log 2␲ for the estimate
ˆ
Ž
.
Ž
.
fuu. Moreover, it is worth mentioning that the two criteria 6.2 and 6.3 are
asymptotically equivalent; see for instance Stone 1977 or Kavalieris 1989 .
Now, let us describe how the results of Sections 4 and 5 are extended to a
psp1 qp2 dimensional covariance stationary vector wt s yt, xt Ј. Assume that
wt admits the AR ϱ representation
Ž
.
Ž
.
X
X
Ž
.
Ž .
ϱ
Ž
.
Ž
.
Ž .
ts1, 2, . . . , T ,
6.4
A L w s
A wtyj s⑀t
Ý
t
j
j
s0
where ⑀t is a p-dimensional martingale difference sequence and A0 is the
identity matrix. As was argued in Section 1, the null hypothesis, H0, that yt £xt ,
Ž .
is equivalent to testing whether the p1 =p2 matrices c j are zero for j-0 in
the infinite distributed lag representation of 6.4 :
Ž
.
ϱ
Ž .
c j xtyj qut ,
yt s
Ý
syϱ
j
w
<
x
where, by construction, E ut xs,yϱ-s-ϱ s0.
Ž .
In this case, the HI estimator of c j is defined as
My1Ј ˆ
ˆ
2 Ý
1
y1
fy x, p fx x, p ei j␭
,
p
Ž .
c j s
ˆ
2 M
p
s1
whereas the test for H0 is given by
w
x
M␮
0
1
yi j␭
p
Ž
.
Ž
.
ST ␮ sRe
vec
c jy1 e
.
ˆ
Ý
Ý
jsyMq1
ž
/
ž M
/
p
s1
We then achieve the following results.
THEOREM 2: Assuming C9, C10, and an ob¨ious extension of C8 to multi¨ariate
data, for any finite collection j1, . . . , jg ,
d
Ž . 1r2Ž Ž Ž .
Ž ..
Ž Ž
.
Ž
...
Ž
ꢀ
4r,
l
s1, . . . , q
.
i T
where
vec c j yc j , . . . , vec c j yc jq Јª N 0, ⍀s ⍀j j
ˆ
ˆ
1
1
q
r
l
␲
y1
iŽ jryj .␭
Ž
.
Ž
y1 Ž
Ž ..
fx x y␭ mfuu ␭ e
.
l
⍀
s 2␲
d␭,
H
jr j
l
y
␲
Ž Ž ..
which corresponds to the asymptotic co¨ariance matrix between vec c j
and
ˆ
r
Ž Ž ..
vec c j
.
ˆ
l
y1
ˆ
ˆ
ˆ
ˆ
ˆ
Ž .
ii Let fuu, p sfy y, p yfy x, p fx x, p fx y, p. A consistent estimator of ⍀j j , r, ls
r
l
1, . . . , q, is
My1Ј
2 Ý
1
y1
.␭
p
r
l
ˆ
⍀
ˆ
ˆ
s
f
x x, yp mfuu, p eiŽ j yj
.
ž
/
jr j
l
2 M
ps1

TEST FOR CAUSALITY
1479
PROOF: The proof of this theorem or any other result in this section follows
by routine extension to those of Sections 4 and 5 and whose details can be seen
Ž
.
in Hidalgo 1998 .
COLLARY 3: Assuming C9, C10 and an ob¨ious extension of C8Ј to multi¨ariate
data, under H0,
weakly
« vec B ␮
T1r2ST
␮
˜
Ž .
Ž
.
Ž
.
in Dp =p 0, 1 with the Skorohod metric, and where vec B ␮ is a p1 =p2-
Gaussian process with co¨ariance structure
1
2
˜
Ž Ž ..
w
x
1
Ž
.Ž y1 Ž
.
Ž ..
␲ min ␮1, ␮
2
Ž
.
K ␮1 , ␮
s
fx x y␭ mfuu ␭ d␭.
H
2
4␲
0
From Corollary 3, the test for H0 can be implemented in a similar fashion to
that employed in Section 5. Finally, to examine the properties of the tests under
contiguous alternatives, introduce
0
y1r2
Ž
.
Ž
.
Ž .
h ␲␭ ,
Ha:
c jy1 cos ␲ j␭ sT
Ý
syϱ
j
Ž .
w
x
5 Ž .5
where h ␽ is a continuous function in 0, ␲ such that 0- h ␽ in a set
w
x
⌬; 0, ␲ with positive Lebesgue measure.
COROLLARY 4: Assuming the same conditions of Corollary 3, under Ha,
␮
weakly
T1r2ST
␮
« vec B ␮ q h ␲␭ d␭ .
˜
Ž
Ž
.
.
Ž
.
H
ž
/
0
7. CONCLUSIONS
In this paper we have first shown that the asymptotic properties of Hannan’s
Ž
.
1967 HI estimator hold the same under long-range dependence, making the
estimation procedure a very desirable one in view of the possible adverse
Ž
.
properties of the LSE, as shown in Robinson 1994 . Secondly, we have proposed
a nonparametric Granger-causality test when the data possibly exhibit long-range
dependence. By means of spectral analysis, we have shown the asymptotic
properties of the test and examined its implementation. More importantly, we
have shown that the test has power against contiguous alternatives converging to
the null at the rate Ty1r2. That is, although the test is nonparametric, it has an
asymptotic relative efficiency greater than zero when compared to parametric
tests based on a correct specification of the model.
Economics Dept., London School of Economics, Houghton St., London WC2A
2AE, U.K.; f.j.hidalgo@lse.ac.uk
Manuscript recei¨ed October, 1996; final re¨ision recei¨ed No¨ember, 1999.

1480
JAVIER HIDALGO
MATHEMATICAL APPENDIX A
Henceforth, Ýj and Ýj, k denote Ýmjsym and Ýmj, ksym, respectively, and K denotes a finite
positive constant.
Ž
.
is
PROOF OF PROPOSITION 2: The cov ␾gh, p, ␾
ab, p
T
1
ꢀ
Ž
.
Ž
.
Ž
.
Ž
.
␥
tys ␥hb ryu q␥g b tyu ␥ha rys
Ý Ý
g a
2
2mq1 4␲ 2T 2
Ž
.
Ž
.
A.1
j
, k t, r , s, us1
iŽtyr.␭ m pqjyiŽ syu.␭
2
2 m pqk
Ž
.4
e
qcum ztg , zrh , zsa , zub
.
We deal with the contribution of the first and third term in braces, the second being similarly
Ž
.
Ž
.
handled to the first. The contribution of the first term of A.1 is by 3.1
1
␲
Ž
.
Ž
.
Ž
.
d␪
2 m pqj
H ␪ H ␭kyj y␪ fg a ␪1 q␭
H
, k
Ý
1
1
1
2
½
2mq1 4␲ 2T 2
Ž
.
y
␲
j
Ž
.
A.2
␲
Ž
.
Ž
.
Ž
.
d␪
2 m pqk
=
H
H y␪ H ␪2 q␭
fhb ␪2 y␭
,
2
jyk
2
5
y
␲
T
yit␻
Ž
.
Ž
.
where H ␻ sÝ
e
. We study the first factor inside the braces of A.2 , the second being
ts1
␲
Ž
.
Ž
.
Ž
.
identical. Adding and subtracting fg a
␭
Hy␲ H ␪ H ␭kyj y␪ d␪1, that factor is
2 m pqj
1
1
␲
Ž
.
Ž
.Ž
Ž
.
Ž
..
d␪
2 m pqj
H ␪ H ␭kyj y␪
fg a ␪1 q␭
yfg a
␭
H
1
1
2 m pqj
1
y
␲
␲
Ž
.
Ž . Ž
H ␪ H ␭kyj y␪ d␪1 ,
1
␲
.
qfg a
␭
H
2 m pqj
1
y
y1
Ž
Ž
faa
.
Ž
..
Ž
.
Ž
.
whose first term is O ⌬g a, j 2mpqj
T log 2mpqk by Theorem 2 c, d of Robinson 1995a ,
1r2
Ž
.
1r2Ž
. Ž
, whereas the second term is 0 if j/k or 2␲ Tfg a ␭
2 m pqj
.
where ⌬g a, j sf
if jsk.
␭
␭
g g
2 m pqj
2 m pqj
Ž
.
Thus, as Tªϱ, the modulus of A.2 is bounded by
log2 2mpqk
2mpqk
Ž
.
K
1
⌬
⌬
2 Ý
g a, j hb, k
2mpqj
Ž
.
Ž
Ž
2mq1
j
, k
Ž
.
K
log 2mpqj
2mpqj
Ž
.
A.3
q
q
⌬
⌬
2 Ý
g a, j hb, j
.
.
2mq1
1
j
<
Ž
.< <
Ž
.<
.
2 m pqj
fg a
␭
fhb
␭
2 Ý
2 m pqj
2mq1
j
1r2 1r2 1r2
The first term of A.3 is O mp y2 log2 mp Fp where Fp sfaa, p g g, p bb, p
f
f
fh1hr, 2p , because, by
Ž
.
ŽŽ
.
Ž
.
.
Ž
.
C1᎐C3, ⌬g a, j and ⌬hb, j satisfy the conditions of Lemma 1 see Appendix B . Similarly, by Lemma 1,
the second term of A.3 is O my2 py1log mp Fp , whereas the third term is
Ž
.
Ž
Ž
.
.
<
< <
<
Fp Rg a, p Rhb, p
Ž
Ž ..
1qo 1
,
2mq1
<
< <
<
<
< <
<
since ␭
and fg a, p fhb, p sFp Rg a, p Rhb, p satisfies the conditions of Lemma 1. Thus, the
2 m ps␭
Žp
.
first term of A.1 is
<
< <
<
Ž
.
Ž
.
Fp Rg a, p Rhb, p
log mp
log2 mp
Ž
Ž ..
1qo 1 qO Fp
q
.
2
m2 p
½
5
ž
/
2mq1
Ž
.
mp

TEST FOR CAUSALITY
1481
Ž
.
Next, proceeding as above, the second term of A.1 is
log2 mp
Ž
.
O
Fp
.
2
ž
/
Ž
.
mp
1r2
Ž
.
Ž .
Ž .
Thus, we are left with the third term of A.1 . Writing ␩ ␭ s␩ ␭ ⌶
ward calculations, we deduce that it is bounded, in absolute value, by
and after straightfor-
˜
K
Ž
.
Ž
.
Ž
.
˜
2 m pqk
Ž
␭
bs
4
.
2 m pqk
␩
␭
␩
˜
y␭
␩
˜
y␭
␩
˜
3
⌸
H
2 Ý Ý
g s
2 m pqj
as
2 m pqj
h s
1
2
3
T 2 2mq1
Ž
.
s
j
, k
l
Ž
.
A.4
Ž
.
Ž
.
Ž
. Ž .
H ␭2 m pqk q␰ d␭d␮d␰
2 m pqk
=H ␭2 m pqj y␭ H ␭y␮y␭
H ␮y␰y␭
2 m pqj
Ž .
Ž .
˜
where ␩ ␭ is the indicated element of ␩ ␭ , the first sum is over s s1, . . . , ¨ and ls1, . . . , 4, and
˜
ab
l
y2
w
x
Ž
.
y2 Ž
.
2mq1
⌸s y␲, ␲ . By Cauchy-Schwarz inequality, a typical term in A.4 is bounded by KT
times
2
Ž
.
Ž
y␭
2
.
Ž
.
Ž
.
2 m pqj
␩
␭
␩
H ␭2 m pqj y␭ H ␭y␮y␭
d␭d␮
˜
˜
H
Ý
g s
2 m pqj
as
2 m pqj
1
2
½
⌸
j
Ž
.
Ž
.
Ž
˜
.
2 m pqk
A.5
=
␩
˜
y␭
␩
␭
H
Ý
h s
2 m pqk
bs
3
4
2
⌸
k
1r2
2
Ž
.
Ž
.
=H ␮y␰y␭
H ␭2 m pqk q␰
d␮d␰
.
2 m pqk
5
Ž
.
The first integral in A.5 is
Ž
.
Ž
.
Ž
. Ž
y␭ H ␭y␮y␭
2 m pqj
1
.
2 m pqj
1
␩
␭
␩
y␭
H ␭
˜
˜
H
Ý
g s
2 m pqj
as
2 m pqj
1
1
2
1
ž
2
⌸
j1
Ž
.
Ž
.
Ž
.
Ž
.
d␭d␮.
2 m pqj
2
=
␩
y␭
␩
␭
H ␭y␭
H ␮y␭q␭
˜
˜
Ý
g s
2 m pqj
as
2 m pqj
2 m pqj
1
2
2
2
2
/
j2
Because
and
Ž
.
Ž
. Ž
d␮s2␲ H ␭
j yj
2 2
.
H ␭y␮y␭
H ␮y␭q␭
H
2 m pqj
2 m pqj
1
1
⌸
Ž
.
Ž
.
Ž
.
,
j yj
2
H ␭
y␭ H ␭y␭
d␭s2␲ H ␭
H
2 m pqj
2 m pqj
1
2
1
⌸
if follows, by C6, that
1r2
2
A.5 sO T 2
⌬
⌬
a g , k K ␭
⌬
⌬
sO T mFp
Ž
.
Ž
.
Ž .
bh, k K ␭
jyk
Ž
.
Ý
Ý
j, k
g a, j
jyk
hb, j
ž
ž
/
/
j
, k
Ž .
where K ␭ is the Fejer kernel, because
`
y2
Ž
.
Ž
.
jyk
Ž
.
Ž
Ž ..
2mq1
⌬
⌬
g a, k K ␭
s Trm faa, p fg g , p 1qo 1 .
Ý
g a, j
j
, k
Ž
.
Ž
.
<
Ž
.<
Ž
.
So A.4 sO Fp MrT , which concludes that cov ␾gh, p, ␾
sO Fp MrT .
Q.E.D.
ab, p

1482
JAVIER HIDALGO
y1r2
PROOF OF PROPOSITION 3: Write ␦gh, p sf
fyhh1,rp2. By elementary inequalities and because
g g, p
<
<
<
<
a ,
l
sup
a
FÝ
l
l
l
P
<
<
<
Ž
.<
<
Ž
.<
.
gh, p
sup
s1, . . . , P
␦
␾
F
␦
E ␾
q
sup
ps1, . . . , P
␦
␾
gh, p yE ␾
Ý
gh, p gh, p
gh, p
gh, p
gh, p
p
ps1
2
1r2
, the second term on the right is Op PM 1r2rT1r2 by Proposition 2
<
<
Ž
.
ŽŽ
.
.
Since sup
a
F Ý a
l
l
l
l
whereas, by Proposition 1, the first term on the right is bounded by
P
M log T
1
U
<
<
.
K
␦
fgh, p
Ý
gh, p
T
p
p
s1
1r2
Ž . 1r2Ž . <
Ž .<
␭
But f
␭ fhh
␭
Rgh
satisfies the conditions of Lemma 1, so by straightforward arguments,
Q.E.D.
g g
we conclude the proof of the Proposition.
Ž .
PROOF OF THEOREM 1: We establish only i , omitting for the sake of brevity the much easier
Ž .
proof of ii . By the Wold device, it suffices to show that for any finite q)0,
q
T1r2
⌿
c j yc j
Ž Ž
.
Ž
..
ˆ
Ý
l
s1
l
l
l
q
␲
d
y1
y1
iŽ j yj .␭
l l
1 2
Ž
.
Ž
.
Ž .
ª N 0, 2␲
⌿
f
␭ fuu ␭ e
d␭⌿
l2
H
Ý
l1
x x
ž
/
y
␲
l1
,
l
2s
1
where ⌿ are constants such that Ýq
⌿
s1. To that end, introduce ␣p sfy x, p yEfy x, p, Ap s
2
ˆ
ˆ
l
l
s1
l
ˆ
ˆ
ˆ
ˆ
Efy x, p, ␤p sf x x, p yEfx x, p, and Bp sEfx x, p. A typical element on the left of the last displayed
Ž .
Ž .
expression, say, c j yc j is
ˆ
My1Ј
2 Ý
1
y1
i j␭
p
Ž
Ž
Ž
.
.
.
Ž
.
A.6
A.7
A.8
Bp ␣p yByp 2Ap ␤p
e
2 M
p
s1
My1Ј
2 Ý
1
p
q
y
Byp 1Ap ei j␭ yc j
Ž .
2 M
p
s1
My1Ј
2 Ý
Ž
.
1
Ž
.
y1 Ž
.
By2␣p ␤p y ␤p qBp
␣p qAp Bp
␤
ei j␭
.
y2
2
p
p
p
2 M
p
s1
We show below that
My1
2 Ý
1
y1
p
ˆ
Ž
.
Ž
.
Ž
.
Ž
.
Ž .
A.9
A.6 q A.7 q A.8 s
f
x x, p
fu x, p ei j␭ qL j
2 M
p
s1
whereL j sOp TM y1r2 log MqMy2qrr2 qTy3r2M 2 qTy1
Ž .
M
uniformly in j. Thus, by C10,
ŽŽ
.
.
q
q
T1r2
My1
2 Ý
Ý
l
s1
y1
␭
T1r2
⌿
f
⌿ ei j
l
l
p
ˆ
fu x, p
Ž Ž ..
c j yc j s
l
.
Ž
Ž .
qop 1 .
ˆ
Ý
l
l
x x, p
ž
/
2 M
p
s1
l
s
1
Ž
.
But, by similar arguments to those of Theorem 1 of Robinson and Hidalgo 1997 , the first term on
the right of the last displayed equation converges in distribution to
q
␲
y1
y1
iŽ j yj .␭
l l
1 2
Ž
.
Ž
.
Ž .
N
0, 2␲
⌿
f
␭ fuu ␭ e
d␭⌿
.
l2
H
Ý
, 2s1
l
l1
x x
ž
/
y
␲
l1
Ž
.
So, to complete the proof we need to show our claim in A.9 .

TEST FOR CAUSALITY
1483
Ž
.
We begin examining A.8 . This term is
My1Ј
2 My1Ј
2 Ý
Ý
1
1
U
Byp 2␣p ␤p ei j␭
q
Cp Byp 2␤p2ei j␭
p
p
y
2 M
2 M
p
s1
ps1
Ž
.
A.10
My1Ј
2 Ý
Ž
.
1
Ž
.
y1 Ž
.
U
p
q
␤p qBp
␣p qAp yCp By2␤p2ei j␭
,
p
2 M
p
s1
Uy1
U
y x, p
where CUp sfx x, p
f
with f¨U , p s 2mq1 y1Ýj f¨
Ž
.
Ž
␭p q␭ .
¨
2
.
¨
j
The first term of A.10 is Op MrT by Cauchy1-Schwarz and Markov inequalities because by
1
2
Ž
f
.
Ž
.
Proposition 2,
f
␣
and
.
f
␤
1r2Ty1r2 with zth sxt and ztg syt, and
Ž .
are Op M
p
y1r2 y1r2
y1
x x, p y y, p
p
x x, p
U
U
Ž
Ž<
<.
Ž
.
U
ztg szth sxt there, Bp yfx x, p so fx x, p sO fx x, p by Proposition 1 and Lemma 1 respectively,
and Cp -K by C9. By similar arguments and since Cp -K by Lemma 2 see Appendix B , the
second term is also Op MrT . The third term of A.10 is Op
standard linearization arguments and Propositions 1 to 3,
<
<
<
<
Ž
.
y3r2M 2 uniformly in j, because by
Ž .
T
Ž
.
Ž
.
fx x, p fy x, p yCUp sOp MT
y1
y1r2
ˆ
ˆ
Ž
.
.
sup
s1, . . . , M
p
2 My1Ј
ap ei j␭ sÝ2psMy1 1ap ei j␭ qa1, we conclude that
p
p
Thus, since Ý
ps1
M 2
M
T
Ž
.
A.8 sOp
q
.
3r2
žT
/
Next the bias term A.7 : Because ␭2 M s2␲ and Ýps0
2 My1ei j␭ sÝ
e
,
2 M
i j␭
p
p
Ž
.
ps1
2
M
M
1
yi ␭
l
p
p
Ž .
Ž
.
c j s
c l
e
ei j␭
Ý
Ý
ž
/
2 M
p
s1
l
sy
M
My1
M
M
2 Ý
Ý
1
1
yi
l
␭
p
e
ei j␭
q
Ý
c l ,
p
Ž
.
Ž
.
s
c l
ž
/
2 M
2 M
p
s1
l
sy
M
l
sy
M
Ž
.
so that A.7 is by C9
ϱ
1
B1y1A1 y
c l
Ž
.
Ž
.
A.11
Ý
½
5
2 M
l
sy
ϱ
My1
ϱ
2 Ý
Ý
syϱ
1
1
yi ␭
l
p
Byp 1Ap y
e
ei j␭ qO
.
p
Ž
c l
.
q
2y␶ r2
ž M /
ž
/
2 M
p
s1
l
Next,
Uy1
U
y x, p
Uy1
U
y x, p
Byp 1Ap sfx x, p
f
f
q Byp 1Ap yfx x, p
f
Ž
.
M 2 log2T
Uy 1
U
y x, p
U
U
U
Uy1 Ž
.
Uy1 Ž
.
sfx x, p
qfx x, p Ap yfy x, p yCp fx x, p Bp yfx x, p qO
T 2 p2
ž
/

1484
JAVIER HIDALGO
<
Ž .<
by Proposition 1, standard linearization arguments, and because C ␭ -K by C9. Thus, replacing
yi
l
␭
y1
Ýϱsy ϱc l e
by its definition f
fy x, p, A.11 is
p
Ž
.
Ž
.
l
x x, p
ϱ
1
1
Uy 1
U
y x, 1
U
U
U
Ž
.
Ž
Uy1 Ž
.
Uy1 Ž
..
fx x, 1
f
y
c l
q
fx x, 1 A1 yfy x, 1 yC1 fx x, 1 B1 yfx x, 1
Ý
syϱ
½
5
2 M
2 M
l
2 Ý
My1
1
Uy 1
U
y x, p
y1
i j␭
p
Ž
.
q
fx x, p
f
yf
fy x, p
e
x x, p
2 M
p
s1
Ž
.
A.12
My1
2 Ý
1
U
U
U
Ž
Uy 1 Ž
.
Uy1 Ž
..
i j␭
p
q
fx x, p Ap yfy x, p yCp fx x, p Bp yfx x, p
e
2 M
p
s1
1
M log2T
T 2
qO
q
.
2y␶ r2
ž M
/
Ž
.
By Lemma 2 the third term of A.12 is bounded, in absolute value, by
My1
2 Ý
K
M 2
1
1
1
˜
Cp qO
sO
2y␶ r2
2y␶ r2
ž M / ž M /
Ž . Ž .
-K by C9. The fourth term of A.12 is O T logT by Proposi-
U
p
p
s1
ϱ
l
yi
l
␭
y1
p
˜
<
Ž
.
<
since Cp sÝ sy ϱ l c l e
<
<
Ž
.
tion 1 and since by Lemma 2, Cp -K, so is the second term of A.12 by the same argument.
The first term of A.12 is
Ž
.
ϱ
ϱ
1
1
Uy 1 Ž
.
U Ž
fy x
.
␭ y
1
Ž
.
Ž
.Ž
.
Ž
.
,
yi
l
␭
yi
l
␭
y2
1
1
fx x
␭
c l
e
q
c l
e
y1 sO M
Ý
Ý
1
2 M ž
/
2 M
l
sy
ϱ
l
sy
ϱ
yi
l
␭
y1
1
<
<
<
Ž .<
because
e
y1 FK l M as ␭1 s␲rM and Ýϱ
the left of the last displayed expression is O M
l c l -K by C9, so the second term on
, whereas the first term is O M
l
sy ϱ
y2
y2
Ž
.
Ž
.
by Lemma 2.
Thus, we conclude that
1
log T
Ž
.
A.7 sO
q
.
2y␶ r2
ž M
/
T
To finish the proof we need to examine A.6 . Replacing Bp and Ap Byp 1 by fx x, p and CUp ,
U
Ž
.
respectively,
My1
2 Ý
1
M1r2 log T
U
i j␭
p
Ž
.
Uy 1 Ž
.
A.6 s
fx x, p ␣p yCp ␤p e
qOp
1r2
ž T
/
2 M
T
p
s1
y1r2 y1r2
y1
1r2Ty1r2
,
Ž
after standard calculations, because by Proposition 2 f
f
␣
and f
␤ are Op
p
M
y y, p x x, p
p
x x, p
U
U
Ž<
<
Ž . Ž ..
by Proposition 1 Bp yfx x, p sO fx x, p M log T r Tp , for example, and by Cauchy inequality.
U
y1
y1
Ž
.
Ž
.
Ž .
Next, since fx x, p fx x, p y1 sO p
by Lemma 1, as fx x ␭ satisfies the conditions there,
My1
2 My1
2 Ý
Ý
1
1
y1
i j␭
y1
f
q
f
fu x, p ei j␭
ˆ
x x, p
p
p
ˆ
Ž
.
Ž
.
␣p yCp ␤p yfu x, p e
A.6 s
x x, p
2 M
2 M
p
s1
ps1
Ž
.
A.13
2 Ý
My1
1
log M
U
y1
i j␭
p
Ž
.
q
f
Cp yCp ␤p e qOp
x x, p
1r2M1r2
žT /
y1q␶ r2Ty1r2 by Lemma 2 and
.
M
2 M
p
s1
Ž
.
Ž
by Proposition 2. The third term on the right of A.13 is Op
because supps1, . . . , M fx x, p
y1
y1r2
<
<
Ž
.
by Proposition 3.
␤
sOp
T
M
p
Ž
.
To complete the proof, we are left with the first term on the right of A.13 . Because by C8
ˆ
Efu x, p s0,
ˆ ˆ ˆ ˆ ˆ ˆ ˆ
␣p yCp ␤p yfu x, p s fy x, p yCp fx x, p yfu x, p yE fy x, p yCp fx x, p yfu x, p ,
Ž
.
A.14
Ž
.
Ž
.

TEST FOR CAUSALITY
1485
ˆ
ˆ
Ž
.
whose first term on the right, by definition of fy x, p and fx x, p and 1.1 , is
ϱ
T
T
1
eyi t␭
x eis␭
s
2
m pqj
2 m pqj
c
x
2mq1 2␲ T Ý Ý l Ý
Ý
ty
l
Ž
.
ž
/ž
/
j
t
s1
ss1
l
syϱ
Ž
.
A.15
ϱ
T
T
1
l
␭
p
y
c eyi
x eyit␭
x eis␭
.
2 m pqj
2 m pqj
Ý
Ý Ý
Ý
l
t
s
Ž
.
ž
/
ž
/ž
/
2mq1 2␲ T
j
ts1
ss1
l
sy
ϱ
Observe that
T
T
1
x
eyi t␭
x eis␭
Ý
Ý
ss1
ty
l
s
2␲ T ž
/ž
/
t
s1
yi
l
␭
ꢀ
4
ꢀ
4
Ž .
is the cross-periodogram of xt and xty , whose cross-spectrum is fx x ␭ e
.
l
Ž
.
Ž
.
Thus, using A.15 and by C9, the right side of A.14 is
1
l
␭
c eyi
p Ž
.
Ý
l
2mq1
<
<
l
F
M
yi
l
␭
y2q␶ r2
j
Ž
.Ž
y1 Ix x
Ž
.
Ž
..
Ž
qO M
p
.
.
=
e
␭
yEIx x
␭
Ý
Ž
.
2 m pqj
2 m pqj
j
yi
l
␭
j
<
<
y1Ž
.
jrm and by routine extension of Proposition 2,
Because 1ye
FK l M
1
j
1r2Ty1r2 fx x, p
;
.
Ž
Ž
.
Ž
..
Ž
Ix x
␭
yEIx x
␭
sOp
M
Ý
2 m pqj
2 m pqj
žm /
2mq1
j
then
My1
2 Ý
1
1
log M
T1r2M1r2
y1
fu x, p ei j␭ qO
q
.
p
ˆ
Ž
.
A.6 s
f
x x, p
2y␶ r2
ž M
/
2 M
p
s1
Ž .
This concludes the proof of part i and the theorem.
1r2
Ž .
Ž .
PROOF OF COROLLARY 1: Because under H0 c j s0 for all j-0, by Theorem 1 T c j s
ˆ
T1r2L j qaj where
Ž .
T1r2
My1
2 Ý
y1
p
aj s
f
fu x, p ei j␭
ˆ
,
x x, p
2 M
p
s1
Ž .
suppressing reference to T in aj or L j . Thus, with the change of subindex yj by j,
w
x
M␮
My1
1
T1r2ST ␮ sRe
ayjy1 qT1r2L yjy1
..
ei j␭
p
Ž
.
Ž
Ž
Ý
Ý
M
ž
/
j
s0
ps1
M
y1
Ž
.
sin ␲ j␮
s ay1 qT1r2L y1 ␮q
ayjy1 qT1r2L yjy1
Ž
Ž
..
Ž
Ž
..
Ž
Ž ..
1qo 1
Ý
␲ j
j
s1
i j␭
y1
since Re My1 Ýw M␮x
e
ª
p
Ž
.
Ž
␲ j
.
Ž . w x
sin ␲ j␮ uniformly in ␮g 0,1 . Now,
ps1
M
ªϱ
M
y1
Ž
.
sin ␲ j␮
Ž
.
Ž .
1
sup
␮
T1r2L yjy1
sop
Ý
j
js1
1r2
y1r2
log M and ÝMjsy11jy1 sO log M .
Ž
Ž
.
Ž
.
Ž
.
since by Theorem 1 T L yjy1 sOp
M

1486
JAVIER HIDALGO
Ž
.
So the behavior of 5.1 is governed by
k
y1
My1
Ž
.
Ž
.
sin ␲ j␮
sin ␲ j␮
ay1 ␮q
a
q
ayjy1
,
Ý
Ý
yjy1
␲ j
␲ j
j
s1
jsk
where k is a fixed but large constant. The proof is thus completed if
k
y1
Ž
.
sin ␲ j␮
Ž
.
Ž .
A.16
Gk ␮ say1 ␮q
a
yjy1
Ý
␲ j
j
s1
converges to a Gaussian process indexed by ␮,
1
␲
␮
y1
EGk2 ␮ y
f
␭ fuu ␭ d␭ -⑀
Ž
.
Ž
.
Ž .
Ž .
A.17
H
x x
4␲
0
for any arbitrary ⑀)0 and k large enough, and
M
y1
Ž
.
sin ␲ j␮
Ž
.
A.18
ayjy1
Ý
␲ j
j
sk
is small uniformly in M and ␮.
We begin with assertion A.18 . For 0-s1 -s2 -M,
Ž
.
2
s2
s2
s2ys1 s2y¨
␲
j␮
Ž
.
sin ␲ j␮
e
2
ayjy1 ayjy1y
¨
Ž
.
A.19
a
F
a
F
.
Ý
Ý
Ý Ý
jss1
yjy1
yjy1
2
Ž .
j jq¨
␲ j
␲ j
␲
j
ss1
jss1
¨
s
1
Ž
.
Let ␻
denote the j, l th element of ⍀. Then,
l
< jy <
2
sÝ2y¨
sÝ2y¨
ayj y1 ayj y1y¨ ayj y1 ayj
y1y
2
ayjy1 ayjy1y
¨
¨
1
1
2
E
s
F
E
ž
/
q¨ <
2
Ž
.
Ž
.
Ž
.
j jq¨
j1 j1 q¨ j2 j2 q¨
j
ss1
j1, j2ss1
sÝ2y¨
2
␻¨2 q␻
< q␻
␻
y¨ < < j yj
1
< j yj
< j yj
1
2
1
2
Ž
Ž ..
1qo 1 ,
Ž
.
Ž
.
j1 j1 q¨ j2 j2 q¨
j1, j2ss1
d
4
Ž
.
Ž
.
because ayj js1, . . . , k ª N 0, ⍀ by Theorem 1, and by the uniform integrability of ayj by C8Ј,
<
<
ꢀ
4
E ayj ayj ayj ayj converges to the corresponding expectation of the limit distribution of ayj .
1
2
3
4
d
The latter follows because if Xt ª X and Xt is uniformly integrable, then E Xt ªE X 4. Now
4
4
<
<
<
<
<
<
l
l
q1
Ž
.
choosing s1 s2 and s2 s2
, by Cauchy inequality, the expectation of the left side of A.19 is
bounded by
1r2
l
2
␻¨2 q␻
< q␻
␻
y¨ < < j yj
1
2
lq1 y¨
2 Ý
< j yj
< j yj
q¨ <
2
1
2
1
2
K
Ý
Ž
.
Ž
.
j1 j1 q¨ j2 j2 q¨
l
¨
s
1
j1, j2s2
1r2
l
2
2
lq1 y¨
1
FK
Ý Ý
2
j12 j1 q¨
Ž
.
l
¨
s1
j1s2
1r2
l
lq1 y¨
2
␻¨2 q␻
< q␻
␻
y¨ < < j yj
1
2
2 Ý
< j yj
< j yj
q¨ <
2
1
2
1
2
qK
,
Ý
s1
Ž
.
Ž
.
j1 j1 q¨ j2 j2 q¨
l
¨
j1s2 , j1-j2
by triangle inequality. But ␻¨ ;K¨y1q␣, where ␣sdx ydu -1 recall the comments after Theorem
Ž
.
1 , so the right side of the last displayed inequality, after standard calculations, is bounded by
1r2
lq1 y¨
l
l
2
2
lq1 y¨
2
2 Ý
1
1
y1q␣
y
l
l
Ž1y␣ .
FK 2 r2 q2y
Ž
.
.
K
qK
¨
Ý Ý
Ý
s1
2
j12 j1 q¨
Ž
.
j1 j1 q¨
Ž
.
l
l
¨
s
1
¨
j1s2
j1s2

TEST FOR CAUSALITY
1487
Let
lq1
2
Ž
.
sin ␲ j␮
T s max
a
.
Ý
s2
l
yjy1
␲ j
0
F␮F1
l
j
Then,
2
y
l
l
Ž1y␣ .
E T FK 2 r2 q2y
,
Ž
.
Ž
.
l
y
l
l
Ž1y␣ .
y
l
l
Ž1y␣ .r2
and thus, with probability greater than 1yK 2 r2 q2y
, T FK 2 r4 q2y
Ž
.
Ž
.
.
l
n
Ž
.
Now choose ks2 , so that A.18 is, in absolute value, bounded by
w
Ž
.x
M q1
log2
T FK 2ynŽ1y␣ .r2 q2ynr4
Ž
.
,
Ý
l
sn
l
Ž
.
and choosing n large enough, we conclude the proof of A.18 .
Ž
.
Ž
.
Assertion A.16 . From Theorem 1, the finite dimensional distributions of Gk
␮
converge to a
Ž
.
normal random variable. To finish, we need to prove the tightness condition; by Billingsley’s 1968
Theorem 12.3, it suffices to check
..4
E Gk ␮ yGk ␮q¨ FK ¨
<
<
2
Ž
.
Ž
Ž
.
Ž
A.20
.
w
x
But that follows because for all j and u, ¨ g 0, ␲ ,
2
Ž
.
Ž Ž
..
sin ju
sin j uq¨
y
ž
/
j
j
Ž
.
Ž
.
Ž Ž
.. Ž Ž
.
.
sin ju cos juq␰ j¨
sin j uq¨ cos j uq¨ q␰ ju
s
y
¨
ž
/
j
j
Ž
.
where ␰g 0,1 , which implies that
4
2
Ž
.
Ž
Ž
..
< <
¨
sin 2␲ j␮
sin 2␲ j ␮q¨
E
ayjy1
y
FK
,
j2
ž
/
/
ž
j
j
Ž
.
Ž
.
and thus A.20 . That concludes the proof of A.16 .
Finally A.17 follows since by A.16 , with the convention that sin ax rxsa for xs0,
Ž
.
Ž
.
Ž
.
k
y1
Ž
.
Ž
.
sin ␲ j1 ␮ sin ␲ j2 ␮
Ž
2 Ž ..
E Gk
␮
s
␻
Ý
< j yj
<
2
1
␲ j1
␲ j2
j1 , j2s0
1
␲
␮
y1
Ž
.
Ž .
ª
f
␭ fuu ␭ d␭, and
H
x x
4␲
k
ªϱ
0
1
Ž .
␮1 , ␮
2
␲ min
y1
Ž
Ž
.
Ž
2 ..
Ž
.
Ž .
E Gk
␮
Gk
␮
ª
f
␭ fuu ␭ d␭.
Q.E.D.
H
1
x x
4␲
ªϱ
k
0
Ž
.
PROOF OF COROLLARY 2: First, the left side of 5.3 is
w
x
M␮
T1r2
M
0
yi j␭
p
Ž Ž
.
Ž
..
Re
c jy1 yc jy1
e
ˆ
Ý
Ý
ž
/
ž
p
s1
jsyMq1
w
x
M␮
0
1
yi j␭
T1r2
c jy1 e
.
p
Ž
.
q
Ý
Ý
jsyMq1
M
ž
/
/
p
s1

1488
JAVIER HIDALGO
˜
Ž
.
But, from Corollary 1, the first term converges weakly to B ␮ , whereas the second term, under Ha,
is My1 ÝwpMs␮1xh ␭ qo 1 by C9 and C10. From here the conclusion of the Corollary is immediate.
Ž
.
Ž .
p
Q.E.D.
MATHEMATICAL APPENDIX B
Ž .
Let g ␭ be a function that satisfies the following assumptions.
y␣
␩
Ž .
A1. g ␭ sK␭
Ž
.Ž .. Ž x
as ␭ª0q, where␣F2 and ␩g 0, 2 .
1qO
␭
Ž .
A2. g ␭ is twice continuously differentiable in any open set outside the origin.
j
j
yj
Ž Ž ..
Ž
.
A3. Ѩ log g ␭ rѨ␭ sO ␭
as ␭ª0qfor js1,2.
LEMMA 1: Assuming A1᎐A3,
1
y1
Ž
.
Ž
Ž
.
Ž
2 m p ..
Ž
Ž
..
B.1
g ␭
yg ␭
sO p g ␭
.
2mq1 Ý
jq2 m p
2 m p
j
PROOF: By A2, A3, and the mean value theorem
2␲ j
Ž
.
Ž
.
Ž
.
,
␰ jq2 m p
g ␭
sg ␭
q
gЈ ␭
jq2 m p
2 m p
ž /
T
Ž .
Ž
.
where ␰s␰ j g 0,1 , and by A1᎐A3
1q␣
␣
T
mp
gy1
␭
gЈ ␭
.
Ž
.
Ž
.
sO
␰ jq2 m p
2 m p
ž /
ž
/
ž
/
␰ jq2mp
T
Ž
.
So the left side of B.1 is bounded by
1q␣
␣
Ž
. Ý
Kg ␭
j
T
mp
T
2 m p
y1
Ž
Ž 2 m p ..
sO p g ␭
2mq1
T
␰ jq2mp
.<
j
since pG1 and Ky1 - mpr ␰ jq2mp -K.
Q.E.D.
<
Ž
LEMMA 2: Assuming C8 and C9,
4␲
1
Uy 1
U
y x, p
y1
˜
Ž
.
B.2
fx x, p
f
yf
fy x, p
s
Cp qO
.
x x, p
2y␶ r2
ž M /
Mp
Ž
.
PROOF: The left side of B.2 is
y1
1
1
Ž
Ž
.
Ž
..
Ž
.
Ž
.
.
C ␭p q␭ yC ␭ fx x ␭p q␭
f
␭p q␭
j
2mq1 Ý
2mq1 Ý
j
p
j
x x
ž
/
j
j
1
y1
Ž
.
Ž
Ž
.
Ž
..
␯
B.3
s
C ␭p q␭ yC ␭
2mq1 Ý
j
p
p
j
1
y1
Ž
Ž
.
Ž
..Ž y1 Ž
fx x
.
Ž
.
q
C ␭p q␭ yC ␭
␭
fx x ␭p q␭ y1 ␯
,
p
2mq1 Ý
j
p
p
j
j
y1
2mq1 y1Ýj fx x ␭p q␭ . But, by Taylor expansion and C8,
Ž
.Ž
.
Ž
.
where ␯p sf
␭
x x
p
j
2 M j
M j
y1
Ž
.
<
Ž
.
Ž
.
j
<
B.4
f
␭
fx x ␭p q␭ y1 s
1qO
,
x x
p
ž /
p ž
/
T
T
p

TEST FOR CAUSALITY
1489
Ž
.
Ž
.
p
whereas, by C9, C ␭p q␭ yC ␭ is
j
1y␶r2
1y␶r2
ϱ
j
j
j
j
yi ␭
l
p
˜
Ž
.
Ž
.
Ž
.
Ž
.
B.5
2␲
l c l
e
qO
s 2␲
Cp qO
,
Ý
ž T /
ž /
T ž
/
T
ž
/
T
l
sy
ϱ
ϱ
l
yi ␭
l
p
˜
Ž
.
Ž
.
Ž
.
p
where Cp sÝ sy ϱ l c l e
. Thus, adding and subtracting C ␭p q␭ yC ␭ into the first
j
term on the right of B.3 and using B.4 , B.5 and Ýmjsym js0, the right side of B.3 is
Ž
.
Ž
.
Ž
.
Ž
.
y1
y2q␶ r2
˜
Ž
.
Ž
.
.
4␲ Mp
Cp qO M
Q.E.D.
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Ž
.
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Ž
.
Ž
BERAN, J. 1994 : Statistics for Long-memory Processes. London: Chapman and Hall.
.
BILLINGSLEY, P. 1968 : Con¨ergence of Probability Measures. New York: John Wiley.
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Ž
.
Biometrika, 60, 217᎐226.
Ž
.
BRILLINGER, D. R. 1981 : Time Series, Data Analysis and Theory. San Francisco: Holden-Day.
Ž
.
GEWEKE, J. 1982 : ‘‘Measurement for Linear Dependence and Feedback between Multiple Time
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Ž
.
ᎏᎏᎏ 1986 : ‘‘The Superneutrality of Money in the USA. An Interpretation of its Evidence,’’
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Ž
.
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Ž
.
GRANGER, C. W. J., AND R. JOYEUX 1980 : ‘‘An Introduction to Long Memory Time Series and
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Ž
.
HANNAN, E. J. 1963 : ‘‘Regression for Time Series,’’ in Time Series Analysis, ed. by M. Rosenblatt.
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Ž
.
.
ᎏᎏᎏ 1967 : ‘‘The Estimation of a Lagged Regression Relation,’’ Biometrika, 54, 409᎐418.
Ž
ᎏᎏᎏ 1976 : ‘‘The Asymptotic Distribution of Serial Covariances,’’ Annals of Statistics, 4, 396᎐399.
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.
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.
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Ž
.
HIDALGO, J., AND J.-P. KREISS 1999 : ‘‘Bootstrap Specification Tests for Covariance Stationary
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Ž
.
HIDALGO, J., AND P. M. ROBINSON 1999 : ‘‘Adapting to Unknown Error Autocorrelation in
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Ž
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