Short and Long Run Causality Measures: Theory
and Inference
Jean-Marie Dufour y
Université de Montréal
Abderrahim Taamoutiz
Université de Montréal
First version: October 20, 2004
This version : May 18, 2006
Preliminary
The authors thank Kilian Lutz, Benoit Perron, Nadège-Désirée Yameogo, the participants of 45th Annual Meeting of the Société canadienne de science économique at
Charlevoix (2005), Joint Statistical Meetings of the American Statistical Association at
Minneapolis (2005), 22nd CESG Conference at Vancouver (2005), and CIREQ Time Series
Conference at Montréal (2005) for their helpful comments.
y
Canada Research Chair Holder (Econometrics). Centre interuniversitaire de recherche
en économie quantitative (CIREQ), Centre interuniversitaire de recherche en analyse des
organisations (CIRANO), and Département de sciences économiques, Université de Montréal. Mailing address: Département de sciences économiques, Université de Montréal,
C.P. 6128 succursale Centre-ville, Montréal, Québec, Canada H3C 3J7. TEL: 1 514
343 2400; FAX: 1 514 343 5831; e-mail: jean.marie.dufour@umontreal.ca. Web page:
http://www.fas.umontreal.ca/SCECO/Dufour .
z
Centre interuniversitaire de recherche en économie quantitative (CIREQ), Centre interuniversitaire de recherche en analyse des organisations (CIRANO), et Département de sciences économiques, Université de Montréal. Adresse: Département de sciences
économiques, Université de Montréal, C.P. 6128 succursale Centre-ville, Montréal, Québec,
Canada H3C 3J7. E-mail: abderrahim.taamouti@umontreal.ca.
i
ABSTRACT
The concept of causality introduced by Wiener (1956) [32] and Granger
(1969) [15] is de…ned in terms of predictability one period ahead. Recently, Dufour and Renault (1998) [9] generalized this concept by considering
causality at a given (arbitrary) horizon h, and causality up to any given horizon h, where h is a positive integer and can be in…nite (1 h 1). This
generalization is motivated by the fact that, in the presence of an auxiliary
variable Z, it is possible to have the situation where the variable Y does not
cause variable X at horizon 1, but causes it at horizon h > 1. In this case,
this is an indirect causality transmitted by the auxiliary variable Z.
Another related problem consists in measuring the importance of causality between two variables. Existing causality measures have been established
only for the horizon 1 and fail to capture indirect causal e¤ects. This paper
proposes a generalization of such measures for any horizon h. We propose
nonparametric and parametric measures for feedback and instantaneous effects at any horizon h. Parametric measures are de…ned in terms of impulse
response coe¢ cients of vector moving average representation (V M A). By
analogy with Geweke (1982) [11], we show that it is always possible to de…ne a measure of dependence at horizon h which can be decomposed into
a sum of feedback measures from X to Y; from Y to X; and instantaneous
e¤ect at horizon h. We propose a new approach to evaluate these measures
based on the simulation of a large sample from the process of interest. We
also propose a valid nonparametric con…dence intervals, using the bootstrap
technique. Finally, we suggest an empirical application which illustrates the
evolution in time of the degree of causality between macroeconomic and
…nancial variables.
Keywords: Causality; auxiliary variable; indirect e¤ect; short-run causality
measure; long-run causality measure; instantaneous causality; dependence;
predictability; estimation by simulation; valid bootstrap con…dence intervals.
ii
Contents
1 Introduction
1
2 Motivation
3
3 Framework
5
4 Nonparametric causality measures
7
5 Parametric causality measures
12
5.1 Parametric measures in the context of V ARM A(p; q) process 12
5.2 Characterization of causality measures in the context of V M A(q)
process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6 Estimation
23
7 Evaluation by simulation of causality measures
29
8 Con…dence intervals
33
9 Empirical illustration
40
10 Conclusion
42
i
1
Introduction
In this paper we propose measures of causality at di¤erent horizons which
can help to detect the well known indirect causality that appears only at
higher horizons. We also propose a valid nonparametric con…dence intervals
for these measures, using the bootstrap technique.
Our causality measures can be applied in di¤erent contexts and may help
to solve some puzzles of the economic and …nancial literature. For example
they can improve the well known debate on long-term predictability of stock
returns.
In the present paper we applied the causality measures at di¤erent
horizons to illustrate the evolution in time of the degree of causality between macroeconomic and …nancial variables. Taamouti, Garcia, and Dufour (2006), …rst, use these measures to quantify and compare the strength
of dynamic leverage and volatility feedback e¤ects in high-frequency equity
data. Second, they measure the dynamic impact of bad and good news on
volatility. Finally, using simulations they assess the ability of these measures
to detect the symmetric property of some parametric volatility models as
GARCH and nonlinear GARCH models.
Recall that the concept of causality introduced by Wiener (1956) [32] and
Granger (1969) [15] is now a basic notion for studying dynamic relationships
between time series. This concept is de…ned in terms of predictability at
horizon 1 of a variable X from its own past, the past of another variable
Y; and possibly a vector Z of auxiliary variables. Following Granger (1969)
[15], we de…ne temporal causality from X to Y one period ahead as follows:
Y causes X in the sense of Granger if observations of Y up to time (t 1)
can help to predict X(t) given the past of X and Z up to time (t 1). More
precisely, we say that Y causes X in the sense of Granger if the variance of
the forecast error of X obtained by using the past of Y is smaller than the
variance of the forecast error of X obtained without using the past of Y .
The theory of causality has generated a considerable literature. In the
context of the bivariate ARM A models, Kang (1981) [20], derived necessary and su¢ cient conditions for noncausality. Boudjellaba, Dufour, and
Roy (1992, 1994) ([3], [4]) developed necessary and su¢ cient conditions of
noncausality for multivariate ARM A models. Parallel to the literature on
the noncausality conditions, some authors developed tests for the presence of
causality between time series. The …rst test is due to Sims (1972) [30] in the
context of bivariate time series. Other tests were developed for V AR models
[ see Pierce and Haugh (1977) [27], Newbold (1982) [25], Geweke (1984) [12]
] and V ARM A models [ see Boudjellaba, Dufour, and Roy (1992, 1994) ([3],
1
[4])].
Recently, Dufour and Renault (1998) [9] generalized the concept of causality in the sense of Granger (1969) [15] by considering causality at a given
(arbitrary) horizon h and causality up to horizon h, where h is a positive
integer and can be in…nite (1 h 1) [ see also Sims (1980) [31], Hsiao
(1982) [18], and Lütkepohl (1993) [24] ]. Such generalization is motivated
by the fact that, in the presence of an auxiliary variable Z, it is possible to
have the variable Y not causing variable X at horizon 1, but causing it at
a horizon h > 1. In this case, this is an indirect causality transmitted by
the auxiliary variable Z. Dufour and Renault (1998) [9] introduced necessary and su¢ cient conditions of noncausality between vectors of variables at
any horizon h for stationary and nonstationary processes with …nite second
moments.
The analysis of Wiener-Granger distinguishes among three types of causality: two unidirectional causalities (called feedbacks) from X to Y; from Y
to X; and an instantaneous causality due to the e¤ect of the present value
of variables on the other. In practice, it is possible that these three types of
causality coexist, hence the importance of …nding means to measure their
degree and to determine the most important one. Unfortunately, existing
tests of causality fail to accomplish this task, they only inform us about
the presence or not of causality. Geweke (1982) [11] extended the causality
concept by de…ning measures of the degrees of feedback and instantaneous
e¤ects, that can be decomposed in time and frequency …eld. Gouriéroux,
Monfort and Renault (1987) [14] proposed causality measures based on the
Kullback information. Polasek (1994) [28] showed how causality measures
can be calculated using Akaike Information Criterion (AIC). Polasek (2000)
[29] introduced new causality measures in the context of univariate and multivariate ARCH models and their extensions based on a Bayesian approach.
Existing causality measures have been established only for a one period
horizon and fail to capture indirect causal e¤ects. This paper proposes a
generalization of such measures for any horizon h. We propose nonparametric and parametric measures for feedback and instantaneous e¤ects at any
horizon h. Parametric measures are de…ned in terms of impulse response
coe¢ cients of V M A representation. By analogy with Geweke (1982, 1984)
([11], [13]), we show that it is always possible to de…ne a measure of dependence at horizon h which can be decomposed into a sum of feedback
measures from X to Y; from Y to X 1 ; and instantaneous e¤ect at horizon
1
See de…nition of the feedback from Y to X in the beginning of this introduction.
Similar de…nition for the feedback from X to Y can be obtained.
2
h. We propose a new approach to evaluate these measures based on the
simulation of a large sample from the process of interest. We also propose
a valid nonparametric con…dence intervals, using the bootstrap technique.
The plan of the paper is as follows. Section 2 provides the motivation behind an extension of causality measures at horizon h > 1. Section 3 presents
the framework allowing the de…nition of causality at di¤erent horizons. In
section 4 we propose nonparametric short run and long run causality measures. In section 5 we give a parametric equivalent, in the context of linear
stationary invertible processes, of the causality measures suggested in section 4. Thereafter, we characterize our measures in the context of …nite
moving average models V M A(q) with …nite order q. In section 6 we discuss
di¤erent estimation approaches. In section 7 we suggest a new approach to
estimate these measures based on the simulation. In section 8 we establish
the asymptotic distribution of measures and the asymptotic validity of their
nonparametric bootstrap con…dence intervals. Section 9 is devoted to an
empirical application and the conclusion relating to the results is given in
section 10.
2
Motivation
The causality measures we propose in this paper are an extension of those
developed by Geweke (1982,1984) ([11], [13]) and others [see introduction].
The existing causality measures quantify the e¤ect of a vector of variables
on another at one period horizon. The contribution of such measures is
particularly limited in the presence of auxiliary variables, since it is possible
that a vector Y causes another vector X at horizon h strictly higher than
1 even if there is no causality at horizon 1. In this case we speak about
an indirect e¤ect induced by the auxiliary variables Z: Clearly causality
measures de…ned for horizon 1 are unable to quantify this indirect e¤ect.
This paper aims …rst to propose causality measures at di¤erent horizons to
quantify the degree of short and long run causality between a vectors of
random variables. Second we propose causality measures that detect and
quantify the indirect e¤ects due to the presence of an auxiliary variable:
To illustrate the importance of such causality measures we consider the
following examples.
Example 1 We suppose that the information set contains only two vari0
ables X and Y . We consider a bivariate stationary process (X; Y ) which
follows a V AR(1) model:
3
X(t + 1)
Y (t + 1)
=
0:5
0:4
0:7
0:35
X(t)
Y (t)
+
"X (t + 1)
"Y (t + 1)
.
(1)
X(t + 1) is given by the following equation:
X(t + 1) = 0:5 X(t) + 0:7 Y (t) + "X (t + 1):
(2)
Since the coe¢ cient of Y (t) in equation (2) is equal to 0:7, we can conclude
that Y causes X in the sense of Granger. However, this does not give
any information about the existence or not of causality for horizons strictly
higher than 1 nor on its degree (weak or strong). To study the causality at
horizon 2 [ see Dufour and Renault (1998) [9] ] let us consider the system
(1) at time (t + 2), we have
X(t + 2)
Y (t + 2)
0:53 0:595
0:34 0:402
0:5 0:7
+
0:4 0:35
"X (t + 2)
+
.
"Y (t + 2)
=
X(t)
Y (t)
"X (t + 1)
"Y (t + 1)
(3)
X(t + 2) is given by:
X(t + 2) =
0:53 X(t) + 0:595Y (t)
(4)
+0:5"X (t + 1) + 0:7"Y (t + 1)
+"X (t + 2):
The coe¢ cient of Y (t) in equation (4) is equal to 0:595, so we can conclude
that Y causes X at horizon 2. But, how can we measure the importance of
this “long run” causality? Existing measures cannot answer this question.
Example 2 We suppose now that the information set contains not only
the two variables of interest X and Y but also an auxiliary variable Z: We
0
consider a trivariate stationary process (X; Y; Z) which follows a V AR(1)
model:
0
1 2
X(t + 1)
@ Y (t + 1) A=4
Z(t + 1)
0:60 0
0
0:40
0
0:60
0
"X (t + 1)
+@ "Y (t + 1)
"Z (t + 1)
4
0:80
0
0:10
1
A.
30
1
X(t)
5@ Y (t) A
Z(t)
(5)
X(t + 1) is given by:
X(t + 1) = 0:6 X(t) + 0:8 Z(t) + "X (t + 1):
(6)
Since the coe¢ cient of Y (t) in equation (6) is 0, we can conclude that Y
does not cause X at horizon 1. If we consider the model (5) at time (t + 2),
we get:
0
1 2
X(t + 2)
0:60 0:00 0:80
@ Y (t + 2) A=4 0:00 0:40 0:00
Z(t + 2)
2 0:00 0:60 0:10
0:60 0:00 0:80
+4 0:00 0:40 0:00
00:00 0:60 10:10
"X (t + 2)
+@ "Y (t + 2) A.
"Z (t + 2)
32 0
1
X(t)
5 @ Y (t) A
Z(t) 1
30
"X (t + 1)
5@ "Y (t + 1) A
"Z (t + 1)
(7)
X(t + 2) is given by:
X(t + 2) =
0:36 X(t) + 0:48Y (t)
(8)
+0:56 Z(t) + 0:6"X (t + 1)
+0:8"Z (t + 1) + "X (t + 2):
The coe¢ cient of Y (t) in equation (8) is equal to 0:48, which implies that
Y causes X at horizon 2: This shows that the absence of causality at h = 1
does not exclude the possibility of a causality at horizon h > 1. This indirect
e¤ect is transmitted by the variable Z:
0:6
0:8
Y ! Z ! X;
0:48 = 0:60 0:80, where 0:60 and 0:80 are the coe¢ cients of the one period
e¤ect of Y on Z and the one period e¤ect of Z on X; respectively: So,
how can we measure the importance of this indirect e¤ ect? Again, existing
measures cannot answer this question.
3
Framework
The notion of noncausality considered here is de…ned in terms of orthogonality conditions between subspaces of a Hilbert space of random variables
with …nite second moments. We denote L2 L2 ( ; A; Q) the Hilbert space
of real random variables de…ned on a common probability space ( ; A; Q);
with covariance as inner product.
5
We consider three multivariate stochastic processes fX(t) : t 2 Zg ; fY (t) : t 2 Zg ;
and fZ(t) : t 2 Zg, with
0
X(t) = (x1 (t); :::; xm1 (t)) ;
0
Y (t) = (y1 (t); :::; ym2 (t)) ;
xi (t) 2 L2
(i = 1; :::; m2 );
2
(i = 1; :::; m3 );
yi (t) 2 L
0
Z(t) = (z1 (t); :::; zm3 (t)) ;
(i = 1; :::; m1 );
2
zi (t) 2 L
where m1
1; m2
1; m3
0; and m1 + m2 + m3 = m: We denote
Xt = fX (s); s
tg, Yt = fY (s), s
tg; and Zt = fZ(s), s
tg the
¯
¯
¯
information sets which contain all the past and present values of X, Y; and
Z; respectively: We denote It the information set which contains Xt , Yt ; and
¯ ¯
Zt . It At , for At =Xt ;Yt ; Zt ; contains all the elements of It except those of
¯
¯ ¯ ¯
At . These information sets can be used to predict the value of X at horizon
h, denoted X(t + h), for all h 1.
For any information set Bt , we denote by P (xi (t + h) j Bt ) the best
linear forecast of xi (t + h) based on the information set Bt ; u(xi (t + h) j
Bt ) = xi (t + h) P (xi (t + h) j Bt ) the corresponding prediction error,
and 2 (xi (t + h) j Bt ) the variance of this prediction error: In other words,
the best linear forecast of X(t + h) is denoted by P (X(t + h) j Bt ) =
0
(P (x1 (t + h) j Bt ); :::; P (xm1 (t + h) j Bt )) : U (X(t + h) j Bt ) = (u(x1 (t + h) j
0
Bt ) ; :::; u(xm1 (t + h) j Bt )) is the corresponding vector of prediction error,
and (X(t+h) j Bt ) its covariance matrix: Each component P (xi (t+h) j Bt )
of P (X(t + h) j Bt ); for 1
i
m1 ; is then the orthogonal projection of
xi (t + h) on the subspace Bt :
Dufour and Renault (1998) [9] provide the following de…nition of noncausality at horizon h and up to horizon h, for h 1.
De…nition 1 (Dufour and Renault (1998) [9]): For h
1; where h
positive integer,
(i) Y does not cause X at horizon h given It Yt , denoted Y 9 X j It
¯
h
if:
P (X(t + h) j It Yt ) = P (X(t + h) j It ) 8t > w;
¯
where w represents a “starting point”2 ;
(ii) Y does not cause X up to horizon h given It Yt , denoted Y 9
¯
(h)
It Yt ; if Y 9 X j It Yt for k = 1; 2; :::; h;
¯
¯
k
2
is a
Y;
¯t
X j
The “starting point”w is not speci…ed. In particular w may equal 1 or 0 depending
on whether we consider a stationary process on the integers (t 2 Z) or a process fX(t) :
t 1g on the positive integers given initial values preceding date 1.
6
(iii) Y does not cause X at any horizon given It
It
Y ; if Y 9 X j It
¯t
k
Yt ; denoted Y 9 X j
¯
(1)
Yt for all k = 1; 2; :::
¯
This de…nition corresponds to an unidirectional causality from Y to X.
It means that Y causes X at horizon h if to predict X(t + h); the past of Y
improves the forecast based on the information set It Yt . For more details,
¯
the reader can refer to Dufour and Renault (1998) [9].
An alternative de…nition can be expressed in terms of the covariance
matrix of the forecast errors.
De…nition 2 For h 1; where h is a positive integer,
(i) Y does not cause X at horizon h given It Yt , if:
¯
det (X(t + h) j It Yt ) = det (X(t + h) j It ) 8t > w;
¯
where det (X(t + h) j At ); At = It ; It Yt ; represents the determinant of
¯
the covariance matrix of the forecast error of X(t + h) given At :
(ii) Y does not cause X up to horizon h given It Yt ; if:
¯
(X(t + k) j It Yt ) = (X(t + k) j It )
¯
8 t > w and k = 1; 2; :::; h;
(iii) Y does not cause X at any horizon given It
Yt ; if:
¯
(X(t + k) j It Yt ) = (X(t + k) j It )
¯
8 t > w and k = 1; 2:::
4
Nonparametric causality measures
In this section and for the remainder of this paper we consider an information
set It which contains two vector valued random variables of interest X and
Y; and an auxiliary valued random variable Z: In other words, we suppose
that It = H[Xt [Yt [Zt , where H represents a subspace of the Hilbert space,
¯ ¯ ¯
possibly empty, containing time independent variables (e.g. the constant in
a regression model).
The causality measure we adopt in this paper is an extension of the
measure adopted by Geweke (1982, 1984) ([11], [13]). The motivations remain the same as those enumerated in Geweke (1982) [11], among which
the most important are: 1) it is nonnegative and 2) it is 0 only when there
is no causality. Speci…cally, we propose the following causality measures at
horizon h 1.
7
De…nition 3 For h 1; where h is a positive integer, a causality measure
from Y to X at horizon h, called the INTENSITY of the causality from Y
to X at horizon h; denoted C(Y !XjZ ); is given by:
h
C(Y !XjZ ) = ln
h
det (X(t + h) j It Yt )
¯
;
det (X(t + h) j It )
Similarly we can de…ne a causality measure from X to Y at horizon h,
called the INTENSITY of the causality from X to Y at horizon h; denoted
C(X!Y jZ ); by:
h
C(X!Y jZ ) = ln
h
det (Y (t + h) j It Xt )
¯
:
det (Y (t + h) j It )
Remark 1 For m1 = m2 = m3 = 1,
2
C(Y !XjZ ) = ln
h
and
(X(t + h) j It Yt )
¯
;
2 (X(t + h) j I )
t
2
C(X!Y jZ ) = ln
h
(Y (t + h) j It Xt )
¯
:
2 (Y (t + h) j I )
t
Then C(Y !XjZ ) quantify the degree of the causal e¤ect from Y to X at
h
horizon h given the past of X and Z: In terms of predictability, it measures
the quantity of the information brought in the past of Y that can improve the
forecast of X(t + h): Following Geweke (1982) [11], this measure can be also
interpreted in terms of proportion of reduction in the variance of forecast
error of X(t + h) by taking into account the past of Y . This proportion is
equal to:
2
100 [
2
(X(t + h) j It Yt )
(X(t + h) j It )
]%=100 [1-exp(-C(Y !XjZ ))]%.
¯
2 (X(t + h) j I
h
Yt )
t
¯
We can rewrite the conditional casualty measures given by de…nition 3
in terms of unconditional casualty measures [see Geweke (1984) [13]]:
C(Y !XjZ ) = C(Y Z!X )
h
h
where
8
C(Z!X );
h
det (X(t + h) j It Yt Zt )
;
¯
¯
det (X(t + h) j It )
h
det (X(t + h) j It Yt Zt )
C(Z!X ) = ln
:
¯
¯
det (X(t + h) j It Yt )
h
¯
C(Y Z!X ) = ln
C(Y Z!X ) and C(Z!X ) represent the unconditional causality measures from
0
h
0 0
h
(Y ; Z ) to X and from Z to X; respectively. Similarly we have
C(X!Y jZ ) = C(XZ!Y )
h
h
C(Z!Y );
h
where,
det (Y (t + h) j It Xt Zt )
,
¯
¯
det (Y (t + h) j It )
det (Y (t + h) j It Xt Zt )
:
C(Z!Y ) = ln
¯
¯
det (Y (t + h) j It Xt )
h
¯
CXZ!Y
= ln
h
We de…ne an instantaneous causality measure between X and Y at horizon h as follows.
De…nition 4 For h
1; where h is a positive integer, an instantaneous
causality measure between Y and X at horizon h, called the INTENSITY of
the instantaneous causality between Y and X at horizon h; denoted C(X$Y jZ );
h
is given by:
C(X$Y jZ ) = ln
h
det (X(t + h) j It ) det (Y (t + h) j It )
:
det (X(t + h); Y (t + h) j It )
where det (X(t+h); Y (t+h) j It ) represents the determinant of the covari0
ance matrix of the forecast error of the joint process X ; Y
given the information set It .
0
0
at horizon h
Remark 2 If we take m1 = m2 = m3 = 1; then
det (X(t + h); Y (t + h) j It )
= 2 (X(t + h) j It ) 2 (Y (t + h) j It )
(cov(X(t + h); Y (t + h) j It ))2 :
So the instantaneous causality measure between X and Y at horizon h can
be written as follows,
9
(cov(X(t + h); Y (t + h) j It ))2
)
2 (X(t + h) j I ) 2 (Y (t + h) j I )
t
t
1
;
2 (X(t + h); Y (t + h) j I )
t
C(X$Y jZ ) = ln (1
h
= ln
1
1
where (X(t+h); Y (t+h) j It ) represents the correlation coe¢ cient between
X(t + h) and Y (t + h) conditional on the information set It . This instantaneous causality measure is higher when this coe¢ cient becomes higher.
We can also de…ne a measure of dependence between X and Y at horizon
h. This will enable us to check if, at horizon h; the processes X and Y must
be considered together or if they should be treated independently given the
information set It Yt .
¯
De…nition 5 For h
1; where h is a positive integer, a measure of dependence between X and Y at horizon h, called the INTENSITY of the
dependence between X and Y at horizon h, denoted C (h) (X; Y j Z), is given
by:
C(h) (X,YjZ)=ln
:
det (X(t + h) j It Yt ) det (Y (t + h) j It
¯
det (X(t + h); Y (t + h) j It )
Xt )
¯
We can easily show that the measure of dependence at horizon h is equal
to the sum of feedbacks measures from X to Y , from Y to X, and of the
instantaneous causality measure at horizon h. We have
C (h) (X; Y j Z) = C(X
!Y jZ )
h
+ C(Y
!XjZ )
h
+ C(X$Y jZ ):
(9)
h
It is possible to build a recursive formulation of causality measures. This
one will depend on the predictability measure introduced by Diebold and
Kilian (1998) [5].
Diebold and Kilian (1998) [5] propose a predictability measure based on
the ratio of expected losses of short and long run forecasts:
P (L;
t ; j; k)
=1
E(L(et+j;t ))
;
E(L(et+k;t ))
where t is an available information set at time t, L is a loss function, j
and k represent respectively the short and the long-run, et+s;t = X(t + s)
P (X(t + s) j t ), s = j; k; is the forecast error at horizon (t + s):
10
This predictability measure can be constructed according to the horizons
of interest and it allows for general loss functions and for univariate or
multivariate information sets. In this paper we suppose that the loss function
L is a quadratic function,
L(et+s;t ) = e2t+s;t ; for s = j; k;
and that t is a multivariate information set. We have the following recursive
formulation.
Proposition 1 Let h1 , h2 be two di¤ erent horizons. For h2 > h1
C(Y
!XjZ )-C(Y !XjZ )
h1
= ln [1
h2
PX (It
Y ; h1 ; h2 )]
¯t
ln [1
1,
PX (It ; h1 ; h2 )];
(10)
where PX ( : ; h1 ; h2 ) represents the predictability measure for variable X,
PX (It
det (X(t + h1 ) j It
det (X(t + h2 ) j It
Y ; h1 ; h2 ) = 1
¯t
det (X(t + h1 ) j It )
:
det (X(t + h2 ) j It )
PX (It ; h1 ; h2 ) = 1
Similarly, we have:
C(X
!Y jZ )-C(X !Y jZ )
h1
h2
= ln [1
Y)
¯t ;
Y)
¯t
PY (It
X ; h1 ; h2 )]
¯t
ln [1
PY (It ; h1 ; h2 )];
(11)
where PY ( : ; h1 ; h2 ) represents the predictability measure for variable Y ,
PY (It
det (Y (t + h1 ) j It
det (Y (t + h2 ) j It
X ; h1 ; h2 ) = 1
¯t
det (Y (t + h1 ) j It )
:
det (Y (t + h2 ) j It )
PY (It ; h1 ; h2 ) = 1
Corollary 1 For h
C(Y
X)
¯t ;
X)
¯t
2;
!XjZ )
h
= C(Y
!XjZ )
+ ln[1
1
ln[1
PX (It
PX (It ; 1; h)]
Y ; 1; h)]:
¯t
Similarly, we have
C(X
!Y jZ )
h
= C(X
!Y jZ )
+ ln[1
1
ln[1
PY (It
PY (It ; 1; h)]
X ; 1; h)]:
¯t
This corollary follows immediately from proposition 1 above.
11
For h2
h1 , the function Pk ( : ; h1 ; h2 ), k = X; Y; represents the measure of short-run forecast relative to the long-run forecast, and C(k !ljZ )
h1
C(k
!ljZ ); for l 6= k and l; k = X; Y; represents the di¤erence between the
h2
degree of short run causality and that of long run causality. By Diebold
and Kilian (1998) [5], Pk ( : ; h1 ; h2 )
0 means that the series is highly
predictable at horizon h1 relative to h2 ; and Pk ( : ; h1 ; h2 ) = 0; means that
the series is nearly unpredictable at horizon h1 relative to h2 .
In particular cases, the interpretation of (10) is easy. For example, consider the case where PX (It ; h1 ; h2 ) PX (It Yt ; h1 ; h2 ), which means that
¯
the knowledge of the past of Y can help to improve the forecast of X at
horizon h1 relative to horizon h2 . Then the unidirectional e¤ect from Y
to X is stronger at horizon h1 than at horizon h2 . In other words, if h1
represents the short-run and h2 represents the long-run, then the causality
from Y to X is stronger in the short-run than in the long-run. Moreover, if
PX (It ; h1 ; h2 ) = PX (It Yt ; h1 ; h2 ), then we have a constant unidirectional
¯
e¤ect at horizon h1 relative to h2 .
5
Parametric causality measures
Now we consider a more speci…c set of linear invertible processes which
includes as special cases V AR, V M A; and V ARM A models with …nite
order. Under this set we show that it is possible to obtain a parametric
expression for short run and long run causality measures in terms of impulse
response coe¢ cients of V M A representation.
This section is divided into two subsections. In the …rst one we calculate parametric measures of short run and long run causality in the context of autoregressive moving average model. We suppose that the process
0
0
0
0
fW (s) = (X (s); Y (s); Z (s)) ; s tg (hereafter unconstrained model) is a
V ARM A(p; q) model, where p and q can be in…nite. The process fS(s) =
0
0
0
(X (s); Z (s)) ; s
tg (hereafter constrained model) can be deduced from
the unconstrained model using corollary 6.1.1 of Lütkepohl (1993) [24]. This
model follows a V ARM A(p; q) model with p mp and q (m 1)p + q.
In the second subsection we provide a characterization of those parametric
measures in the context of V M A(q) model, where q is …nite.
5.1
Parametric measures in the context of V ARM A(p; q) process
Without loss of generality, let us consider the discrete vectorial process of
0
0
0
0
zero mean fW (s) = (X (s); Y (s); Z (s)) ; s tg de…ned on L2 and charac12
terized by the following autoregressive moving average representation:
W (t) =
p
X
j W (t
j=1
=
p
X
j=1
+
2
4
2
j) +
'j u(t
j) + u(t)
j=1
(12)
30
XXj
XY j
XZj
Y Xj
YYj
Y Zj
ZXj
ZY j
ZZj
'XY j
'Y Y j
'ZY j
'XZj
'Y Zj
'ZZj
'XXj
4 'Y Xj
j=1
'ZXj
q
X
q
X
1
X(t j)
5 @ Y (t j) A
Z(t j)
30
1 0
1
uX (t j)
uX (t)
5 @ uY (t j) A +@ uY (t) A,
uZ (t j)
uZ (t)
with
E [u(t)] = 0;
Or compactly
h
i
0
E u(t)u (s) =
u
0
for s = t
:
for s 6= t
(L)W (t) = '(L)u(t);
where
2
XX (L)
XY (L)
XZ (L)
Y X (L)
Y Y (L)
Y Z (L)
ZX (L)
ZY (L)
ZZ (L)
'XX (L)
'(L)=4 'Y X (L)
'ZX (L)
'XY (L)
'Y Y (L)
'ZY (L)
Pp
'XZ (L)
'Y Z (L)
'ZZ (L)
(L)=4
2
ii (L)
= Imi
ik (L)
=
j=1
Pp
'ik (L) =
j
ikj L ;
j=1
'ii (L) = Imi +
Pq
j
iij L ,
Pq
j
j=1 'iij L ,
j
j=1 'ikj L ;
for i 6= k, and i; k = X; Y; Z:
13
3
5,
3
5,
We suppose that u(t) is orthogonal to the subspace of the Hilbert space
fW (s); s
(t 1)g and that u is a symmetric positive de…nite matrix.
Under stationarity, the V M A(1) representation of model (12) is given by:
W (t) =
(L)u(t);
(13)
where
(L) =
(L)
1
'(L) =
1
X
j
jL
=
1
X
j=0
j=0
2
4
XXj
XY j
XZj
Y Xj
YYj
Y Zj
ZXj
ZY j
ZZj
3
5Lj ;
and
0
= Im :
From the previous section, measures of dependence and feedbacks are de…ned in terms of covariance matrices of the constrained and unconstrained
forecast errors3 . So to calculate these measures we need to know the structure of the constrained model(imposing noncausality). This can be deduced
from the structure of the unconstrained model (12) using the following
proposition and corollary.
Proposition 2 (Lütkepohl (1993, pages 231-232) [24]): Linear Transformation of a V M A(q) process): Let u(t) be a K-dimensional white noise
process with nonsingular covariance matrix u and let
W (t) =
+
q
X
j u(t
j) + u(t);
j=1
be a K-dimensional invertible V M A(q) process. Furthermore let F be an
(M K) matrix of rank M: Then the M -dimensional process S(t) = F W (t)
has an invertible V M A(q) representation:
S(t) = F +
q
X
j "(t
j) + "(t);
j=1
where "(t) is M -dimensional white noise with nonsingular covariance matrix
M ) coe¢ cient matrices and q q:
" ; the j ; j = 1; :::; q; are (M
3
Here we mean the forecast errors of the constrained and unconstrained models.
14
Corollary 2 (Lütkepohl (1993, pages 232-233) [24]): Linear Transformation of a V ARM A(p; q) process): Let W (t) be a K-dimensional, stable, invertible V ARM A(p; q) process and let F be an (M K) matrix of rank M:
Then the process S(t) = F W (t) has a V ARM A(p; q) representation with
p
q
Kp;
(K
1)p + q:
Remark 3 If we suppose that W (t) follows a V AR(p) = V ARM A(p; 0)
model, then the linear transformation S(t) = F W (t) has a V ARM A(p; q)
representation with p Kp and q (K 1)p:
Suppose that we are interested in measuring the causality from Y to X
at given horizon h. So we need to apply corollary (2) to de…ne the structure
0
0
of process fS(s) = (X (s); Z (s))0 ; s tg. If we left-multiply equation (13)
by the adjoint matrix of the matrix (L), denoted (L) , then we get
(L)
(L)W (t) =
(L) '(L)u(t);
(14)
where (L) (L) = det [ (L)]. Since the determinant of (L) is a sum of
products involving one operator from each row and each column of (L),
the degree of the AR polynomial, here det [ (L)] ; is at most mp. We write
det [ (L)] = 1
1L
p
mp:
:::
p
pL ;
where
It is easy to check that the degree of the operator (L) '(L) is at most
p(m 1) + q. So equation (14) can be written as follows:
det [ (L)] W (t) =
(L) '(L)u(t):
(15)
This equation is another stationary invertible V ARM A representation of
process W (t); called the …nal equation form.
0
0
The model of process fS(s) = (X (s); Z (s))0 ; s
tg can be obtained
by taking F equal to:
F =
Im1
0
0
0
15
0
Im3
:
Pre-multiplying (15) with F results in
det [ (L)] S(t) = F (L) '(L)u(t):
(16)
The right-hand side of equation (16) is a linearly transformed …nite order
V M A process which, by proposition 2, has a V M A(q) representation with
q p(m 1) + q . We have the following constrained model,
det [ (L)] S(t) = (L)"(t) =
XX (L)
XZ (L)
ZX (L)
ZZ (L)
"(t)
(17)
with
E ["(t)] = 0;
where
h
i
0
E "(t)" (s) =
for s = t
for s 6= t;
"
0
ii (L)
= Imi +
ik (L)
=
Pq
Pq
j
j=1 ii;j L ;
j
j=1 ikj L ;
for i 6= k and i; k = X; Z;
Note that, in theory the coe¢ cients ikj , i; k = X; Z; j = 1; ::::q; and elements of the covariance matrix " can be computed from coe¢ cients ikj ,
'ikl ; i; k = X; Z; Y ; j = 1; :::; p; l = 1; :::; q; and elements of the covariance
matrix u . This is possible by solving the following system:
" (v)
=
u (v),
v = 0; 1; 2; :::;
(18)
where " (v) and u (v) are the autocovariance functions of the processes
(L)"(t) and F (L) '(L)u(t); respectively. For large numbers m, p; and q;
the system (18) can be solved by using optimization methods. In the next
two sections we will discuss other approaches to estimate the constrained
and unconstrained models.
The following example shows how one can calculate the theoretical parameters of the constrained model in terms of those of unconstrained model
in the context of a bivariate V AR(1) model.
16
Example 3 We consider the following bivariate V AR(1) model:
X(t)
Y (t)
=
XX
XY
YX
YY
X(t
Y (t
=
X(t
Y (t
1)
1)
1)
1)
uX (t)
uY (t)
+
(19)
+u(t)
In a reduced form
X(t)
Y (t)
(L)
where
1
(L)=
= u(t);
XX L
XY
Y XL
1
L
YY
;
L
with
E [u(t)] = 0; 8
h
0
E u(t)u(s)
i
<
=:
u
2
uX
=
cov(uX (t); uY (t))
2
uY
cov(uY (t); uX (t))
0
for s = t;
for s 6= t:
We suppose that all roots of det( (z)) = det(I2 z) are outside of the unit
circle. Under this assumption the model (19) has the following V M A(1)
representation:
X(t)
Y (t)
=
1
X
j=0
=
1
X
uX (t
uY (t
j
j=0
XX;j
XY;j
Y X;j
Y Y;j
j)
j)
uX (t
uY (t
j)
j)
;
where
j
=
0
= I2 :
j 1
=
j
; j = 1; 2; :::;
If we suppose that we are interested in determining the model of marginal
process X(t), then by corollary (2) and for F = [1; 0] we have,
det( (L))X(t) = [1; 0] (L) u(t);
where,
(L) =
1
YYL
Y XL
XY L
1
17
XX L
;
and
det( (L)) = 1
(
YY
+
XX )L
(
2
XX Y Y )L :
Y X XY
Thus,
X(t) ( Y Y + XX )X(t 1) ( Y X XY
= XY uY (t 1)
1)+ uX (t):
Y Y uX (t
XX Y Y )X(t
2)
The right-hand side, denoted $(t); is the sum of an M A(1) process and a
white noise process. By proposition 2, $(t) has an M A(1) representation,
$(t) = "X (t) + "X (t 1). To calculate parameters and V ar("X (t)) = 2"X
in terms of parameters of the unconstrained model, we have to solve system
(18) for v = 0 and v = 1;
8
V ar ["X (t) + "X (t 1)] = V ar [uX (t)
1) + XY uY (t
>
Y Y uX (t
>
<
>
>
:
E [("X (t) + "X (t 1))("X (t
+ XY uY (t 1))(uX (t 1)
()
8
<
:
(1 +
2
2
"X
=
)
2
"X
= (1 +
YY
1) + "X (t 2))] = E[(uX (t)
2) + XY uY (t 2))];
Y Y uX (t
Y Y uX (t
2 ) 2
uX
YY
+
2
XY
2
uY
2
XY
uY uX
and
2 .
"X
YY
2 :
uX
Here we have two equations and two unknown parameters
parameters must satisfy constraints j j< 1 and 2"X > 0:
These
The V M A(1) representation of model (17) is given by:
1
S(t) = det [ (L)]
(L)"(t) =
1
X
j "(t
j)
(20)
j=0
=
1
X
j=0
XXj
XZj
ZXj
ZZj
where
0
= I(m1 +m2 ) :
18
"X (t
"Z (t
j)
j)
;
1)] ;
1)
To quantify the degree of causality from Y to X at horizon h; we …rst need
to consider the unconstrained and constrained models of process X. The
unconstrained model is given by the following equation:
X(t) =
1
X
XXj uX (t
j=1
1
X
+
1
X
j) +
XY j uY (t
j)
j=1
XZj uZ (t
j) + uX (t);
j=1
and the constrained model is given by:
X(t) =
1
X
XXj "X (t
j) +
j=1
1
X
XZj "Z (t
j) + "X (t):
j=1
Second, we calculate the covariance matrices of the unconstrained and constrained forecast errors of X(t + h). From equation (13) the forecast error
of W (t + h) is given by:
enc [W (t + h) j It ] =
h 1
X
i u(t
+h
i);
i=0
associated to the covariance matrix
(W (t + h)
j
=
It ) =
h 1
X
i
V ar [u(t)]
0
(21)
i
i=0
h 1
X
i
u
0
i:
i=0
So, the unconstrained forecast error of X(t + h) is given by:
enc (X(t + h) j It ) =
+
h 1
X
h 1
X
XXj uX (t
+h
j) +
j=1
XZj uZ (t
h 1
X
XY j uY (t
j=1
+h
j) + uX (t + h)
j=1
associated to the unconstrained covariance matrix
(X(t + h) j It ) =
h 1
X
i=0
19
[enc
X
i
u
0
nc0
i eX ];
+h
j)
where
enc
X =
Im1
0
0
:
Similarly the forecast error of S(t + h) is given by:
ec [S(t + h) j It
Yt ] =
¯
associated to the covariance matrix
(S(t + h) j It
h 1
X
i "(t
+h
i);
i=0
Yt ) =
¯
h 1
X
i
"
0
i:
i=0
Consequently, the constrained forecast error of X(t + h) is given by:
ec (X(t+h) j It Yt ) =
¯
h 1
X
XXj
j=1
h 1
X
"X (t+h j)+
"Z (t+h j)+"X (t+h)
XZj
j=1
associated to the constrained covariance matrix
(X(t + h) j It
Yt ) =
¯
h 1
X
Im1
0
ecX
i
"
0
i
0
ecX ;
i=0
where
ecX =
:
Finally, we have the following result.
Theorem 1 Under assumptions 12, 13; and for h 1; where h is a positive
integer,
#
"
Ph 1 c
0
0
det( i=0
eX i " i ecX )
C(Y !XjZ ) = ln
;
P
0 nc0
h
det( hi=01 enc
e
)
u
i
i
X
X
where
enc
X =
Im1
0
0
;
ecX =
Im1
0
:
Example 4 If we consider a bivariate V AR(1) process [see example (3) ],
then one can calculate the causality measure at any horizon h using only the
unconstrained parameters. For example, the causality measure at horizons
1 and 2 are given by:
20
(1 +
C(Y
!X )=ln[
2
YY
)
2
uX
2
XY
+
2
uY
+
1
C(Y
ln[
!X )=
2
4
2
YY
2[(1+
4
2
uX +[(1+ Y Y
)
2
2
2
XX ) uX + XY
2
2
uX + XY
2
uY
][(1+
2
uY
2
YY
)
q
q
((1+
((1 +
YY
2
2
uX + XY
2
2
YY
)
2
uX
2
XY
+
2
uY
4
2
YY
X
2
YY
4
uX
2
2
XY
2
uY
)2 4
YY
2
2
uX ]
2
YY
4
uX ]
The last two formulas are obtained under assumptions cov(uX (t); uY (t)) = 0
and ((1 + 2Y Y ) 2uX + 2XY 2uY )2 4 2Y Y 4uX 0:
Now we will calculate the parametric measure of instantaneous causality at given horizon h. We know from section 4 that this measure is de…ned only in terms of covariance matrices of unconstrained forecast errors.
The covariance matrix of the unconstrained forecast error of joint process
0
0
X (t + h); Y (t + h)
0
is given by:
(X(t + h); Y (t + h) j It ) =
h 1
X
G
i
u
0
iG
i=0
where
G=
Im1
0
0
Im2
0
0
:
We also have,
(X(t + h) j It ) =
h 1
X
(Y (t + h) j It ) =
h 1
X
and
where
enc
Y =
0
[enc
X
i
u
[enc
Y
i
u
0
nc0
i eX ];
i=0
0
nc0
i eY ]:
i=0
Im2
Thus, we have the following result.
21
0
4
uX
2
uX
2
2
2
2 4
u + XY uY )
qX
((1+ Y Y )2 2u +
)2
2 )2
uY
:
0
;
]
]
Theorem 2 Under assumptions 12, 13; and for h
integer,
C(X
!Y jZ )=ln
h
"
Ph 1
det( i=0 [enc
X
where
nc0
i eX ]) det(
Ph 1
det( i=0 [G i u
Im1
0
G=
Ph
0
u
i
0
Im2
eX =
Im1
eY =
0
1; where h is a positive
0
0
1 nc
i=0 [eY
0
iG
0
i
u
0
nc0
i eY ])
])
#
;
;
0
0
;
Im2
0
:
Remark that the parametric measure of dependence at horizon h can be
deduced from its decomposition given by equation (9).
5.2
Characterization of causality measures in the context of
V M A(q) process
0
0
0
0
Now suppose that the process fW (s) = (X (s); Z (s); Y (s)) ; s
an invertible V M A(q) model:
W (t) =
q
X
j=1
=
q
X
j=1
j u(t
2
4
0
with
In a reduced form:
j) + u(t)
(22)
XXj
XY j
XZj
Y Xj
YYj
Y Zj
ZXj
ZY j
ZZj
uX (t)
1
30
uX (t
5 @ uY (t
uZ (t
+@ uY (t) A
uZ (t)
tg follows
E [u(t)] = 0;
h
i
0
E u(t)u (s) =
u
0
for s = t;
:
for s 6= t
W (t) = (L)u(t);
where
22
1
j)
j) A
j)
2
and
(L)=4
XX (L)
(L)
Y Y (L)
ZY (L)
XZ (L)
XY
Y X (L)
ZX (L)
ii (L)
= Imi +
ik (L)
=
Pq
Pq
ZZ (L)
j
ikj L ;
i; k = X; Z; Y:
From proposition (2) and by taking F=
Im1
0
the constrained process S(t) is an M A(q) with q
S(t) =
(L)"(t) =
5;
j
iij L ,
j=1
j=1
for i 6= k,
Y Z (L)
3
q
X
j "(t
0
0
0
Im2
, the model of
q. We have,
j)
j=0
=
q
X
j=0
XX;j
XZ;j
ZX;j
ZZ;j
"X (t
"Z (t
j)
j)
;
with
E ["(t)] = 0;
h
i
0
E "(t)" (s) =
"
0
for s = t
:
for s 6= t
Using proposition 1 we have the following result.
Theorem 3 Let h1 and h2 two di¤ erent horizons. Under assumption (22)
we have
C(Y !X=Z ) = C(Y !X=Z ); 8 h2 h1 q
h1
h2
This result follows immediately from proposition ?? and by the fact that
for all h2 h1 q; PX (It Yt ; h1 ; h2 ) = PX (It ; h1 ; h2 ):
¯
6
Estimation
From section 5 we have that short run and long run causality measures
depend on parameters of the model describing the process of interest. So
23
the estimation of these measures can be done by replacing the unknown
parameters by their estimates from a …nite sample.
More generally, there are three di¤erent approaches in estimating our
causality measures based on the di¤erent estimation approaches of the model
of the process of interest. The …rst one, called nonparametric approach,
which is the focus of this section supposes that the form of the parametric
model appropriate for the process of interest is unknown and it approximates
this one with a V AR(k) model, where k depends on the sample size [ see
Parzen (1974) [26], Bhansali (1978) [2], Lewis and Reinsel (1985) [23]]. The
second approach supposes that the process follows a …nite order V ARM A
model. The standard methods for the estimation of V ARM A models, such
as maximum likelihood and nonlinear least squares, require nonlinear optimization. This might not be feasible because the number of parameters
can increase quickly. To circumvent this problem several authors [ see Hannan and Rissanen (1982) [17], Hannan and Kavalieris (1984b) [16], Koreisha
and Pukkila (1989) [22], Dufour and Pelletier (2003) [8], and Dufour and
Jouini (2004) [7]] have developed a relatively simple approach based only
on linear regressions. This approach enables estimation of V ARM A models
using a long V AR whose order depends on the sample size. The last and
simplest approach assumes that the process follows a …nite order V AR(p)
model which can be estimated by OLS.
In practice, the precise form of the parametric model appropriate for
a process is unknown. Parzen (1974) [26], Bhansali (1978) [2], Lewis and
Reinsel (1985) [23], and others, have considered a nonparametric approach
of predicting future values using an autoregressive model …tted to a series
of T observations. This approach is based on a very mild assumption of an
in…nite order autoregressive model for the process which includes …nite-order
stationary V ARM A processes as a special case.
In this section, we use the nonparametric approach to estimate the short
and long-run causality measures. First, we discuss the estimation of the
…tted autoregressive constrained and unconstrained models. Next, we point
out some assumptions necessary for the convergence of the estimated parameters. Second, using theorem 6 of Lewis and Reinsel (1985) [23], we de…ne
approximations of covariance matrices of the constrained and unconstrained
forecast errors at horizon h. Last, we use these approximations to construct
an asymptotic estimator of our short and long-run causality measures.
In what follows we focus on the estimation of the unconstrained model.
0
0
0 0
Let us consider a stationary vector process fW (s) = (X(s) ; Y (s) ; Z(s) ) ; s
t)g. Based on Wold’s theorem this process can be written in the form of a
V M A(1) model,
24
W (t) = u(t) +
1
X
'j u(t
j);
j=1
E [u(t)] = 0;
i
0
u for s = t;
E u(t)u (s) =
0
for s 6= t
h
;
P
where u is a positive de…nite matrix. We assume that 1
j=0 k 'j k< 1 and
P
0
j
detf'(z)g 6= 0 forj z j 1, where k 'j k= tr('j 'j ) and '(z) = 1
j=0 'j z ;
with '0 = Im ; an (m m) identity matrix: Under these assumptions W (t)
is invertible and can be written as an in…nite autoregressive process,
W (t) =
1
X
j W (t
j) + u(t);
(23)
j=1
P1
P1
j
1 satis…es
where
j=1 k j k< 1 and (z) = Im
j=1 j z = '(z)
detf (z)g =
6 0 for j z j 1:
Let (k) = ( 1 ; 2 ; :::; k ) denote the …rst k autoregressive coe¢ cients
in the V AR(1) representation. Given a realization fW (1); :::; W (T )g; we
can approximate (23) by a …nite order V AR(k) model, where k depends on
sample size T . The OLS estimator of the autoregressive coe¢ cients of the
…tted V AR(k) model is given by the following equation:
^ (k) = (^ 1k ; ^ 2k ; :::; ^ kk ) = ^ 0 ^ 1 ;
k1 k
PT
0
0
0 0
1
^
where k = (T k)
t=k+1 w(t)w(t) , for w(t) = (W (t) ; :::; W (t k+1) ) ;
P
0
and ^ k1 = (T k) 1 Tt=k+1 w(t)W (t + 1) . An estimator of the covariance
matrix of the error terms is given by:
^k =
u
T
X
0
u
^k (t)^
uk (t) =(T
k);
t=k+1
Pk
j):
where u
^k (t) = W (t)
j=1 ^ jk W (t
Theorem 1 of Lewis and Reinsel (1985) [23] ensures the convergence
of ^ 1k ; ^ 2k ; :::; ^ kk under three assumptions: 1) E j ui (t)uj (t)uk (t)ul (t) j
i; j; k; l
m; 2) k is chosen as a function of T such that
4 < 1; for 1
2 =T ! 0 as k; T ! 1; and 3) k is chosen as a function of T such that k 1=2
k
P1
j=k+1 k j k! 1 as k; T ! 1. Theorem 4 of the same authors derives
the asymptotic distribution for these estimators under 3 assumptions: 1) E j
25
ui (t)uj (t)uk (t)ul (t) j
i; j; k; l m; 2) k is chosen as a function
4 < 1; 1
of T such that k 3 =T ! 0 as k; T ! 1; and 3) it exist fl(k)g a sequence of
0
(km2 1) vectors such that 0 < M1 k l(k) k2 = l(k) l(k) M2 < 1; for
k = 1; 2; ::: We also note that ^ ku converges to u , as k and T ! 1 [ see
Lütkepohl (1993, pages 308-309) [24] ].
Remark 4 The upper bound K of the order k in …tted V AR(k) model depends on the assumptions required to ensure the convergence and the asymptotic distribution of the estimator. For the convergence of the estimator we
2
need to assume that kT ! 0, as k and T ! 1; consequently we can choose
p
as an upper bound K = C T ; where C is a constant: For the asymptotic dis3
tribution of the estimator we need to assume that kT ! 0, as k and T ! 1,
p
and thus we can choose as an upper bound K = C 3 T .
Now we will calculate an asymptotic approximation of the covariance
matrix of the forecast error at horizon h. The forecast error of W (t + h)
based on the V AR(1) model is given by:
enc [W (t + h) j W (t); W (t
1); :::] =
h 1
X
'j u(t + h
j);
j=0
associated to the covariance matrix
[W (t + h) j W (t); W (t
1); :::] =
h 1
X
'j
0
u 'j :
j=0
In the same way, the covariance matrix of the forecast error of W (t + h)
based on V AR(k) model, is given by:
k [W (t
+ h)
j
W (t); W (t
= E[(W (t + h)
1); :::; W (t
k
X
k + 1)]
(h)
^ jk W (t + 1
j))
j=1
(W (t + h)
k
X
j=1
where
26
(h)
^ jk W (t + 1
0
j) ]:
(h+1)
(h+1)
^ jk
(h)
= ^ (j+1)k + ^ 1k ^ jk ; for j 2 f1; ::; g, h
(1)
^ jk
1;
= ^ jk ;
(0)
= Im :
^ jk
[see Dufour and Renault (1998) [9]]. Moreover, let us note that:
k
X
W (t + h)
(h)
^ jk W (t + 1
j=1
1
X
= (W (t + h)
(h)
j W (t
j)
+1
(24)
j))
j=1
k
X
(h)
(
^ jk W (t + 1
1
X
j)
j=1
h 1
X
=
'j u(t + h
(h)
j W (t
+1
j))
j=1
k
X
(h)
(
^ jk W (t + 1
j)
j=0
1
X
j)
j=1
(h)
j W (t
+1
j)):
j=1
where
(h+1)
j
(1)
j
(0)
j
=
(h+1)
j+1
=
j;
+
(h)
j;
1
for j 2 f1; ::; g, h
1;
= Im :
Since the errors terms u(t + h j), for 0 j (h 1), are independent of
0
0
(W (t); W (t 1); :::; ) and (^ 1k ; ^ 2k ; :::; ^ kk ), the two terms on the right-hand
side of the equation (24) are independent. Thus,
k [W (t
+ h)
j
=
W (t); W (t
1); :::; W (t
k + 1)]
[W (t + h) j W (t); W (t
1); :::; ]
k
X
(h)
+E[(
^ jk W (t + 1
j)
j=1
k
X
(
(h)
^ jk W (t + 1
j=1
1
X
(h)
j W (t
+1
j))
j=1
j)
1
X
j=1
27
(25)
(h)
j W (t
+1
0
j)) ]:
An asymptotic approximation, as k and T ! 1, of the last term in the
equation (25) is given by theorem 6 of Lewis and Reinsel (1985) [23]:
k
X
(h)
E[(
^ jk W (t + 1
1
X
j)
j=1
k
X
(h)
(
^ jk W (t + 1
+1
j))
j=1
j)
j=1
1
X
(h)
j W (t
0
+1
j)) ]
j=1
h 1
km X
'j
T
(h)
j W (t
0
u 'j :
j=0
Consequently, an asymptotic approximation of the covariance matrix of the
forecast error is equal to:
k [W (t
+ h)
j
W (t); W (t
(1 +
km
)
T
1); :::; W (t
h 1
X
'j
k + 1)]
(26)
0
u 'j :
j=0
An estimator of this quantity is obtained by replacing the parameters 'j
and u by their estimators '
^ kj and ^ ku ; respectively.
We can also obtain an asymptotic approximation of the covariance matrix of the constrained forecast error at horizon h following the same steps
as before. We denote this covariance matrix by:
k [S(t
+ h)
j
S(t); S(t
1); :::; S(t
k + 1)]
h 1
(1 +
k(m1 + m3 ) X
)
T
j
"
0
j;
j=0
where j ; for j = 1; ::; (h 1), represents the coe¢ cient of V M A representation of the process constrained S and " is the covariance matrix of
0
0 0
"(t) = ("X (t) ; "Z (t) ) : Based on these results an asymptotic approximation
of the causality measure from Y to X is given by:
a
C (Y
"
P
det[ hj=01 ecX j
P
!X=Z ) = ln
h
det[ hj=01 enc
X 'j
28
c0
j eX ]
0 nc0
u 'j eX ]
"
0
#
+ ln 1
km2
;
T + km
where,
enc
X =
Im1
0
0
;
ecX =
Im1
0
:
An estimator of this quantity will be obtained by replacing the unknown
parameters by their estimators,
C^ a (Y
7
"
P
0 #
0
det[ hj=01 ecX ^ kj ^ k" ^ kj ecX ]
+ ln 1
P
!X=Z ) = ln
0
0
h
det[ hj=01 enc
^ kj ^ ku '
^ kj enc
X'
X ]
km2
:
T + km
Evaluation by simulation of causality measures
In this section we propose a simple simulation-based technique to calculate our causality measures at any horizon h, for h 1. To illustrate this
technique we consider the same examples of section 1 limiting ourselves to
horizons 1 and 2.
Since one source of bias in the autoregressive coe¢ cients is …nite sample
size, our technique consists of simulating a large sample from the unconstrained model whose parameters are supposed to be known or estimated
from a real data set. Once the large sample (hereafter large simulation) is
simulated we use it to estimate the parameters of the constrained model.
In what follows we describe an algorithm to calculate the causality measure at given horizon h using large simulation technique:
(1) Given parameters of the unconstrained model and its initial values,
simulate a large sample of T observations, where T can be equal to 1000000,
under assumption that probability distribution of disturbance is completely
speci…ed4 .
(2) Estimate the constrained model using large simulation.
(3) Calculate the constrained and unconstrained covariance matrices of
the forecast errors at horizon h:
(4) Calculate the causality measure at horizon h using constrained and
unconstrained covariance matrices of step (3):
4
The form of the probability distribution of disturbances does not a¤ect the value of
causality measures.
29
Now let reconsider example one of section one:
X(t + 1)
Y (t + 1)
=
=
X(t)
Y (t)
0:5
0:4
+ u(t):
0:7
0:35
(27)
X(t)
Y (t)
uX (t + 1)
uY (t + 1)
+
;
where
E[u(t)] = 0;
E[u(t)u(s)0 ] =
I2 if s = t
:
0 if s 6= t
(28)
Our illustration is done in two steps. First, we calculate the theoretical
values of causality measures at horizons 1 and 2. In case of bivariate V AR(1)
model it is easy to calculate the causality measure at any horizon h using
only the unconstrained parameters [see example (4)]. This is not the case for
models with more than one lag and two variables [see section (5)]. Second,
we evaluate our causality measures using large simulation technique.
The theoretical values of causality measures at horizons 1 and 2 are
recovered as follows:
(1) We calculate variances of the forecast errors of X at horizons 1 and
2 using its own past and past of Y . Note that,
0
[(X(t + h); Y (t + h)) j Xt ; Yt ] =
¯ ¯
From this covariance matrix we have,
h 1
X
i
i0
:
i=0
V ar[X(t + 1) j Xt ; Yt ] = 1;
¯ ¯
and
V ar[X(t + 2) j Xt ; Yt ] =
¯ ¯
0
1
X
e
i
i0
0
e = 1:74;
i=0
where e = (1; 0) :
(2 ) We calculate variances of the forecast errors of X at horizons 1 and
2 using only its own past. In this case we need to determine the structure
30
of the constrained model. This one is given by the following equation [see
example 3]:
X(t + 1) = (
YY
+
XX )X(t)
+(
Y X XY
XX Y Y )X(t
1)
+"X (t + 1) + "X (t):
where Y Y + XX = 0:85 and Y X XY
XX Y Y = 0:105: The parameters
and V ar("X (t)) = 2"X are the solutions to the following system:
8
(1 + 2 ) 2"X = 1:6125
<
:
:
2 = 0:35
"X
The set of possible solutions is f( ; 2"X ) = ( 4:378; 0:08); ( 0:2285; 1:53)g.
To get an invertible solution we must choose the combination which satis…es
the condition j
j< 1; i.e. the combination ( 0:2285; 1:53): Thus, the
variance of the forecast error of X at horizon 1 using only its own past is
given by:
[X(t + 1) j Xt ] = 1:53;
¯
and the variance of the forecast error of X at horizon 2 is:
[X(t + 2) j Xt ] = 2:12:
¯
Thus, we have,
C(Y
!X )
1
= 0:425; C(Y
!X )
= 0:197:
2
In a second step we use the algorithm described in the beginning of this
section to evaluate the causality measures using large simulation technique.
Hereafter, we give the results5 that we obtain for di¤erent lag orders p of
the constrained model.
5
For illustration we take T = 600; 000:
31
Table 1: Evaluation by simulation of C(Y
p
1
2
3
4
5
10
15
20
25
30
35
C(Y
!X )
1
0:519
0:430
0:427
0:425
0:426
0:425
0:426
0:425
0:425
0:426
0:425
C(Y
!X )
1
and C(Y
!X )
2
!X )
2
0:567
0:220
0:200
0:199
0:198
0:197
0:199
0:197
0:199
0:198
0:198
The above results show that large simulation technique works well, and this
can be explained by law of large numbers.
Now consider example two of section one:
0
1 2
X(t + 1)
0:60
@ Y (t + 1) A = 4 0:00
Z(t + 1)
0:00
0:00
0:40
0:60
0:80
0:00
0:10
30
1 0
1
X(t)
"X (t + 1)
5 @ Y (t) A + @ "Y (t + 1) A ;
Z(t)
"Z (t + 1)
(29)
where
E[u(t)] = 0;
I3 if s = t
E[u(t)u(s)0 ] =
:
0 if s 6= t
In this example the analytical calculations of the causality measures at horizons 1 and 2 is not easy. The interest of this example is to illustrate the
indirect causality:
As we have shown in section one, in this example the variable Y does
not cause the variable X at horizon 1, but causes it at horizon 2. So, we
expect that causality measure from Y to X will be equal to zero at horizon
1 and di¤erent from zero at horizon 2. Using large simulation technique for
32
a sample size T = 600; 000 and for di¤erent lag orders p of the constrained
model we get the following results:
Table 2: Evaluation by simulation of C(Y
p
C(Y
1
2
3
4
5
10
15
20
25
30
35
!XjZ )
!X )
C(Y
1
0:000
0:000
0:000
0:000
0:000
0:000
0:000
0:000
0:000
0:000
0:000
and C(Y
1
!X )
(Indirect causality)
2
!XjZ )
2
0:121
0:123
0:122
0:123
0:124
0:122
0:122
0:122
0:124
0:122
0:122
Thus, the above results con…rm our expectations about the presence of an
indirect causality from Y to X.
8
Con…dence intervals
We suppose that process of interest W
t)g6 follows a V AR(p) model,
W (t) =
p
X
j W (t
0
fW (s) = (X(s); Y (s); Z(s)) ; s
j) + u(t);
(30)
j=1
or equivalently,
(I3
p
X
j
j L )W (t)
= u(t);
(31)
j=1
Pp
j
where polynomial (z) = I3
j=1 j z satis…es det[ (z)] 6= 0 for z 2 C
1
with j z j 1; and fu(t)gt=0 is a sequence of i.i.d. random variables with
following properties:
6
Here we suppose that X, Y; and Z are univariate variables. It will be easy to generalize
results of this section to the multivariate case.
33
and
E [u(t)] = 0;
h
i
0
E u(t)u (s) =
u
u
0
if s = t
;
if s 6= t
is a positive de…nite matrix.
Remark 5 If we suppose that process W follows a V AR(1) model, then
we can use Inoue and Kilian’s (2002) [19] approach to get similar results to
those developed in this section.
The parameters of model (30) can be estimated by OLS. For a realization W
fW (1); :::; W (T )g of process (30), an estimate of = ( 1 ; :::; p )
is given by the following equation:
^ = ^0 ^
1
An estimator of
u
1
:
(32)
is given by:
^u =
T
X
0
u
^(t)^
u(t) =(T
p);
t=p+1
P
0
0
0 0
where ^ = (T p) 1 Tt=p+1 w(t)w(t) , for w(t) = (W (t) ; :::; W (t p + 1) ) ;
Pp
0
^ 1 = (T p) 1 PT
^(t) = W (t)
j):
t=p+1 w(t)W (t + 1) ; and u
j=1 ^ j W (t
Now suppose we are interested in measuring causality from Y to X at
horizon h: In this case we need to know the structure of the marginal process
0
fS(s) = (X(s); Z(s)) ; s t)g: This one has a V ARM A(p; q) representation
with p 3p and q 2p,
S(t) =
p
X
j S(t
j) +
j=1
q
X
i "(t
i) + "(t);
i=1
where f"(t)g1
t=0 is a sequence of i.i.d. random variables that satis…es
E ["(t)] = 0;
h
i
0
E "(t)" (s) =
"
0
if s = t
;
if s 6= t
and " is a positive de…nite matrix.
The equation (33) can be written in the following reduced form,
(L)S(t) = (L)"(t);
34
(33)
where (L) = I2
:::
Lp and (L) = I2 + 1 L + ::: + q Lq : We
1L
Pq p j
suppose that (z) = I2 + j=1 j z satis…es det[ (z)] 6= 0 for z 2 C and
j z j 1: Under this assumption the V ARM A(p; q) process is invertible and
has a following pure V AR(1) representation,
1
X
S(t)
c
j S(t
j) = (L)
1
(L)S(t) = "(t):
(34)
j=1
Let c = ( c1 ; c2 ; :::) denote the matrix of all autoregressive coe¢ cients
in model (34) and c (k) = ( c1 ; c2 ; :::; ck ) denote its …rst k autoregressive
coe¢ cients. Suppose that we approximate (34) by a …nite order V AR(k)
model, where k depends on sample size T . In this case an estimator of the
autoregressive coe¢ cients of V AR(k) model is given by:
^ c (k) = (^ c ; ^ c ; :::; ^ c ) = ^ 0 ^ 1 :
k1 k
1k
2k
kk
An estimator of
"
is given by:
^ "k =
T
X
0
^"k (t)^"k (t) =(T
k);
t=k+1
P
0
0
0
where ^ k = (T k) 1 Tt=k+1 Sk (t)Sk (t) , for Sk (t) = (S (t); :::; S (t k +
Pk
P
0
0
c
1)) ; ^ k1 = (T k) 1 Tt=k+1 Sk (t)S(t+1) ; and ^"k (t) = S(t)
j=1 ^ jk S(t
j):
Based on the above notations, the theoretical value of the causality measure from Y to X at horizon h may de…ned as follows:
C(Y
!XjZ ) = ln (
h
G(vec( c ); vech(
H(vec( ); vech(
" ))
u ))
);
where
G(vec(
c
); vech(
" ))
=
h 1
X
0
ec
j=0
0
ec = (1; 1) ;
and
35
c(j)
1
c(j)
" 1 ec ;
H(vec( ); vech(
u ))
h 1
X
=
0
(j)
1
e
(j)
u 1 e;
j=0
0
e = (1; 1; 1) ;
c(j)
c(j 1)
c(j 1)
c(0)
c(1)
c ; for j
with 1 = 2
+ 1
2, 1 = I2 ; and 1 = c1 [see
1
Dufour and Renault (1998) [9]]. vec denotes the column stacking operator
and vech is the column stacking operator that stacks the elements on and
below the diagonal only:
From corollary 6.1.1 of Lütkepohl (1993, page 232) [24] it exists a function f (:) : R9(p+1) ! R4(k+1) which associates the constrained parameters (vec( c ); vech( " )) to the unconstrained parameters (vec( ); vech( u ))
such that (vec( c ); vech( " )) = f (vec( ); vech( u )) [see example (4)].
C(Y
!XjZ )
= ln(
h
G(f (vec( ); vech( u )))
):
H(vec( ); vech( u ))
An estimate of the quantity C(Y
^ Y
C(
!XjZ )
= ln(
h
!XjZ )
is given by:
h
G(vec( ^ c (k)); vech( ^ ";k ))
);
H(vec( ^ ); vech( ^ u ))
(35)
where G(vec( ^ c (k)); vech( ^ ";k )) and H(vec( ^ ); vech( ^ u )) are estimates of
the population quantities.
Now, let consider the following assumptions.
Assumption 1 :
[see Lewis and Reinsel (1985)]
1) E j "h (t)"i (t)"j (t)"l (t) j
h; i; j; l 2;
4 < 1; for 1
2) k is chosen as a function of T such that k 3 =T !
1;
P01as k; T !
1=2
c
3) k is chosen as a function of T such that T
j=k+1 k j k! 0 as
k; T ! 1:
4) Series used to estimate parameters of V AR(k) and series used for
prediction are generated from two independent processes which have the
same stochastic structure.
^ Y
Proposition 3 (Consistency of C(
is a consistent estimator of C(Y
!XjZ ))
h
!XjZ ):
h
36
^ Y
Under assumption 1; C(
!XjZ )
h
^ Y
To establish the asymptotic distribution of C(
!XjZ ),
let start by re-
h
calling the following result [see Lütkepohl (1990, page 118-119) [24] and
Kilian (1998a, page 221) [21]]:
vec( ^ )
vech( ^ u )
p)1=2
(T
vec( )
vech( u )
d
! N (0; );
(36)
where
1
=
0
0
2(D3 D3 )
u
0
1 D0 (
u
3
0
;
1
u )D3 (D3 D3 )
D3 is the duplication matrix, de…ned such that vech(F ) = D3 vech(F ) for
any symmetric 3 3 matrix F .
Assumption 2 :
f (:) is continuous and di¤erentiable function.
According to (36) and using delta method we have following result [see
proof in appendix].
^ Y
Proposition 4 (Asymptotic distribution of C(
!XjZ ))
tions 1 and 2 we have,
(T
^ Y
p)1=2 (C(
!XjZ )
C(Y
h
!XjZ ))
h
Under assump-
h
d
! N (0;
C );
where
@C(Y
C
=(
@C(Y
!XjZ )
h
0
@(vec( ) ; vech(
0
u)
)
) (
!XjZ )
0
h
0
@(vec( ) ; vech(
0
u)
)
);
and
1
=
0
u
0
0
2(D3 D3 )
1 D0 (
u
3
0
u )D3 (D3 D3 )
1
:
See proof in appendix.
The analytical derivative of the causality measure with respect to the
0
vector (vec( ); vech( u )) is not feasible. One way to build con…dence intervals for our causality measures is to use large simulation technique [see
37
section 7] to calculate the derivative numerically. Another way is by building
bootstrap con…dence intervals.
As mentioned by Inoue and Kilian (2002) [19], for the bounded measures
[ as in our case C(Y !XjZ ) 2 [0; 1)], the bootstrap approach is more relih
able then delta-method. The raison is because “the delta-method interval
is not range respecting and may produce con…dence intervals that are logically invalid. In contrast, the bootstrap percentile interval by construction
preserves these constraints”[see Inoue and Kilian (2002, pages 315-318) [19]
and Efron and Tibshirani (1993) [10]].
In the rest of this section we show how one can construct a valid con…dence interval for causality measures using bootstrap technique. In what
follows, we establish the validity of the percentile bootstrap con…dence intervals.
Let consider the following bootstrap approximation to the distribution
of causality measure at given horizon h.
(1) Estimate by OLS a V AR(p) process and save the residuals
u
~(t) = W (t)
p
X
^ j W (t
j); for t = p + 1; :::; T;
j=1
where ^ j , for j = 1; :::; p, are given by equation (32).
(2) Generate (T p) bootstrap residuals u
~ (t) by random sampling with
replacement from the residuals u
~(t); t = p + 1; :::; T:
(3) Generate a random draw for the vector of p initial observations
0
0
0
w(0) = (W (1); :::; W (p)) :
(4) Given ^ [see equation (32) ]; u
~ (t); and w(0); generate bootstrap
data for the dependent variable W (t) from equation:
W (t) =
p
X
^ j W (t
j) + u
~ (t); for t = p + 1; :::; T:
j=1
(5) Calculate the bootstrap OLS regression estimates
^ = (^ 1 ; ^ 2 ; :::; ^ p ) = ^
38
1^
1;
and
T
X
^ =
u
0
u
~ (t)~
u (t) =(T
p);
t=p+1
P
0
0
0
where ^ = (T p) 1 Tt=p+1 w (t)w (t) , for w (t) = (W (t); :::; W (t
P
0
0
p
1)) ; ^ 1 = (T
p) 1 Tt=p+1 w (t)W (t + 1) ; and u
~ (t) = W (t)
P+
p
W
(t
j):
^
j=1 j
(6) Estimate the constrained model of the marginal process (X; Z) using
the bootstrap sample fW (t)gTt=1 :
(7) Calculate the causality measure at horizon h; denoted C^ (j) (Y
!XjZ ),
h
using equation (35).
(8) Choose B such
times.
1
2
(B + 1) is an integer and repeat steps (2)
(7) B
Conditionally on the sample we have,
p)1=2
(T
vec( ^ )
vech( ^ u )
vec( ^ )
vech( ^ u )
d
! N (0; );
(37)
where
1
=
u
0
0
0
2(D3 D3 )
1 D0 (
u
3
0
u )D3 (D3 D3 )
;
1
D3 is the duplication matrix de…ned such that vech(F ) = D3 vech(F ) for
any symmetric 3 3 matrix F .
Proposition 5 (Asymptotic validity of the residual-based bootstrap)
Under assumptions 1 and 2 we have,
p)1=2 (C^ (Y
(T
!XjZ )
h
^ Y
C(
!XjZ ))
h
d
! N (0;
C );
where
@C(Y
C
=(
!XjZ )
@C(Y
h
@(vec( )0 ; vech(
0
u) )
) (
!XjZ )
h
@(vec( )0 ; vech(
0
u) )
0
);
and
1
=
0
u
0
0
2(D3 D3 )
1 D0 (
u
3
39
0
u )D3 (D3 D3 )
1
:
Remark 6 Proposition (5) may be used to justify the application of biascorrected bootstrap con…dence intervals, proposed by Kilian (1998,a) [21], to
improve the performance of the small sample percentile bootstrap intervals
of our causality measures.
Kilian (1998,a) [21] proposes an algorithm to remove the bias in impulse
response function prior to bootstrapping the estimate. As he mentioned, the
small sample bias in impulse response function may arise from bias in slope
coe¢ cient estimates or from the nonlinearity of this function, and this can
translate into changes in interval width and location. However, if we suppose
that ordinary least-squares (OLS) small-sample bias can be responsible for
bias in the estimated impulse response function, replacing the biased slope
coe¢ cient estimates by bias-corrected slope coe¢ cient estimates may help
to reduce the bias in the impulse response function.
Kilian (1998,a) shows that the additional modi…cations proposed in the
bias-corrected bootstrap con…dence intervals method to improve the performance of the percentile bootstrap intervals in small samples do not alter
its asymptotic validity. The reason is that the e¤ect of bias corrections is
negligible asymptotically. For more details on the bias-corrected bootstrap
con…dence intervals method, the reader can refer to Kilian (1998,a) [21].
9
Empirical illustration
In this section we apply our causality measures to quantify the degree of
relationships between macroeconomic and …nancial variables. The data set
considered is the one used by Bernanke and Mihov (1998) [1] and Dufour,
Pelletier and Renault (2003) [6]. This data set consists of monthly observations on nonborrowed reserves (N BR), federal funds rate (r), gross domestic
product de‡ator (P ), and real gross domestic product (GDP ). The monthly
data on GDP and GDP de‡ator were constructed by state space methods
from quarterly observations7 . Our sample goes from January 1965 to December 1996 for a total of 384 observations.
All variables are in logarithmic form [see Figures 1-4 in appendix]. These
variables were also transformed by taking …rst di¤erences [see Figures 5-8
in appendix], so the causality relations will have to be interpreted in term
of the growth rate of variables.
We have performed an Augmented Dickey-Fuller test (hereafter ADF test) for nonstationnarity of the four variables of interest and their …rst
7
For more details, the reader can consults Bernanke and Mihov (1998).
40
di¤erences. The values of the test statistics as well as the critical values
corresponding to 5% signi…cance level are given in tables 4 and 5 [see appendix].
The following table 3 summarizes the results of the stationarity tests for
all variables.
Table 3: Stationarity test results
N BR
r
P
GDP
Variables in logarithmic form
No
No
No
No
First di¤erence
Y es
Y es
No
Y es
As we can read from table 3, all variables in logarithmic form are nonstationary. However, their …rst di¤erences are stationary except for the GDP
de‡ator, P: We also did a nonstationary test for the second di¤erence of variable P: The test statistic values are equal to 11:04826 and 11:07160 for
an ADF -test with only an intercept and both intercept and trend, respectively. The critical values in both cases are equal to 2:8695 and 3:4235;
so the second di¤erence of variable P is stationary.
Once the data is made stationary, we use a nonparametric approach for
the estimation and the Akaike’s information criterion to specify the orders
of the V AR(k) models which describe the constrained and unconstrained
processes. To choose the upper bound K on the admissible lag orders, we
apply the results by Lewis and Reinsel (1985) [23].
Using the Akaike’s criterion for the unconstrained V AR model, which
correspond to four variables, we observe that this one is minimal at k = 16.
We use same criterion to specify the orders of the constrained V AR models,
which correspond to di¤erent combinations of three variables, and we …nd
that orders are all less than or equal to 16. To compare the determinants
of the covariance matrices of the constrained and unconstrained forecast
errors at horizon h, we take the same order k = 16 for the constrained and
unconstrained models.
We calculate di¤erent causality measures for horizons h = 1; :::; 40 [ see
Figures 9-14 in appendix]. Higher values of measures indicate greater causality. We also calculate the corresponding nominal 95% bootstrap con…dence
intervals as it is described in the previous section.
41
Based on results, we have following conclusions. From Figure 9 we remark that nonborrowed reserves (N BR) has an important e¤ect on federal
funds rate (r) at horizon 1 comparatively with other variables [see Figures
10 and 11 ]. Thus, we can conclude that nonborrowed reserves causes the
federal funds rate only in the short-term. This e¤ect is well known in the
literature and can be explain by the theory of supply and demand of the
money. We also note that nonborrowed reserves has a short-term e¤ect on
the GDP [see Figure 10] and it can causes the GDP de‡ator until horizon
4:
Figure 14 shows that the e¤ect of GDP on federal funds rate is signi…cant
for the …rst four horizons. However, the e¤ect of federal funds rate on GDP
de‡ator is signi…cant only at horizon 1 [see Figure 12].
The other more signi…cant result concern the causality from r to GDP:
Figure 13 shows that the e¤ect of the interest rate on GDP is signi…cant
until horizon 16. These results are consistent with conclusions obtained by
Dufour et al. (2005).
Table 6 represents the results of other causality directions until horizon
20. As we can read from this table there is noncausality in these directions.
The above results do not change when we consider the second rather
than …rst di¤erence of variable P .
10
Conclusion
New concepts of causality have been introduced in the literature by Dufour
and Renault (1998, Econometrica) [9]: they de…ne a causality at a given
(arbitrary) horizon h, and causality up to any given horizon h, where h is
a positive integer and can be in…nite (1
h
1). These concepts are
motivated by the fact that, in the presence of an auxiliary variable Z, it is
possible to have situation where the variable Y does not cause variable X
at horizon 1, but causes it at horizon h > 1. In this case, this is an indirect
causality transmitted by the auxiliary variable Z.
Another related problem consists in measuring the importance of causality between two variables. Existing causality measures have been established only for the horizon 1 and fail to capture indirect causal e¤ects. This
paper proposes a generalization of such measures for any horizon h. We
propose parametric and nonparametric measures for feedback and instantaneous e¤ects at any horizon h. Parametric measures are de…ned in terms
of impulse response coe¢ cients of V M A representation. By analogy with
Geweke (1982) [11], we show that it is always possible to de…ne a mea-
42
sure of dependence at horizon h which can be decomposed into a sum of
feedback measures from X to Y; from Y to X; and instantaneous e¤ect at
horizon h. We also show how these causality measures can be related to the
predictability measures developed in Diebold and Kilian (1998) [5].
We propose a new approach to estimate these measures based on simulation of a large sample from the process of interest. We also propose a
valid nonparametric con…dence intervals, using the bootstrap technique.
From an empirical application we …nd that nonborrowed reserves causes
federal funds rate only in short-term, the e¤ect of real gross domestic product on federal funds rate is signi…cant for the …rst four horizons, the e¤ect
of federal funds rate on gross domestic product de‡ator is signi…cant only
at horizon 1; and …nally federal funds rate causes the real gross domestic
product until horizon 16:
43
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46
Proof of proposition 1. From de…nition 3 we have,
C(Y
!XjZ )
h2
det (X(t + h2 ) j It Yt )
¯
det (X(t + h2 ) j It )
det (X(t + h1 ) j It Yt )
= ln
¯
det (X(t + h1 ) j It )
det (X(t + h1 ) j It ) det (X(t + h2 ) j It Yt )
+ ln
¯
det (X(t + h1 ) j It Yt ) det (X(t + h2 ) j It )
¯
= C(Y !XjZ )
= ln
h1
det
det
det
ln
det
(X(t + h1 ) j It )
(X(t + h2 ) j It )
(X(t + h1 ) j It Yt )
:
¯
(X(t + h2 ) j It Yt )
¯
Using the predictability measure of Diebold and Kilian (1998) [5] we have,
+ ln
PX (It
det (X(t + h1 ) j It
det (X(t + h2 ) j It
Yt ; h1 ; h2 ) = 1
¯
and
det (X(t + h1 ) j It )
:
det (X(t + h2 ) j It )
PX (It ; h1 ; h2 ) = 1
Hence the result of proposition 1
C(Y
!XjZ )
h1
C(Y
!XjZ )
= ln [1
PX (It
Yt ; h1 ; h2 )] ln [1
¯
h2
^ Y
Proof (Consistency of C(
Yt )
¯ ;
Yt )
¯
!XjZ )).
PX (It ; h1 ; h2 )] :
Under assumption 1 we have
h
G(vec( ^ c (k)); vech( ^ ";k ))
= (1 +
2k
T )G(vec(
= (1 + O(T
for
2
3
<
c ); vech(
))G(vec(
" ))
c ); vech(
" ));
< 1:
The last equality follows from condition 2 of assumption 1: if we consider
that k = T for > 0, then condition 2 implies that k 3 =T = T 3 1 with
47
1 and T (2T
0 < < 13 : Similarly, we have 2k
T = 2T
Thus,
ln(G(vec( ^ c (k)); vech( ^ ";k )))
= ln(G(vec(
c ); vech(
= ln(G(vec(
c ); vech(
" ))
1)
! 2 for
+ ln(1 + O(T
2
3
<
< 1:
))
(38)
for 23 <
+ O(T
);
< 1 and T ! 1:
In other ways we have ^ !
; ^u !
p
in (vec( ); vech(
" ))
u;
p
and for H(:) a continuous function
u ))
H(vec( ^ ); vech( ^ u )) ! H(vec( ); vech(
u ));
p
or
ln(H(vec( ^ ); vech( ^ u ))) ! ln(H(vec( ); vech(
p
u ))):
(39)
From ( 38) and (39) we get
^ Y
C(
!XjZ ) = ln(
h
for
2
3
<
G(vec( c ); vech(
H(vec( ); vech(
" ))
u ))
) + O(T
) + op (1);
< 1 and T ! 1:
So, under condition 2 of assumption 1, we have,
^ Y
C(
!XjZ )
h
! C(Y
p
!XjZ ):
h
Proof of Proposition 4. We have showed [see proof of consistency]
that
ln(G(vec( ^ c (k)); vech( ^ ";k )))
= ln(G(vec(
c ); vech(
" ))
+ O(T
)
(40)
= ln(G(f (vec( ); vech(
for
2
3
<
u )))
+ O(T
);
< 1:
Under assumption 2 we have,
ln(G(f (vec( ^ ); vech( ^ u )))) ! ln(G(f (vec( ); vech(
p
48
u )))):
(41)
From (40) and (41) we get
ln(G(vec( ^ c (k)); vech( ^ ";k )))
= ln(G(f (vec( ^ ); vech( ^ u )))) + O(T
for
2
3
<
(42)
) + op (1);
< 1:
Thus,
^ Y
C(
!XjZ )
= ln(
h
for
2
3
<
G(f (vec( ^ ); vech( ^ u )))
) + O(T
H(vec( ^ ); vech( ^ u ))
< 1:
^
^
~ Y
We denote ln( G(f (vec(^ );vech(^ u ))) ) by C(
H(vec( );vech(
u ))
!XjZ ):
h
^ Y
Op (1); the asymptotic distribution of C(
~ Y
of C(
) + op (1);
^
^
Since ln( G(f (vec(^ );vech(^ u ))) ) =
H(vec( );vech(
u ))
!XjZ ) will be the same as that
h
!XjZ ).
h
~ Y
Under assumption 2 and by a …rst order Taylor expansion of C(
around the true parameters we get,
~ Y
C(
!XjZ )
h
= C(Y
@C(Y
+(
!XjZ )
h
!XjZ )
h
!XjZ )
h
0
@(vec( ) ; vech(
0
u) )
)
vec( ^ )
vech( ^ u )
vec( )
vech( u )
+ op (T
1
2
);
and
~ Y
p)1=2 (C(
(T
@C(Y
' (
!XjZ )
C(Y
h
!XjZ )
h
0
@(vec( ) ; vech(
0
u) )
)
!XjZ ))
h
p)1=2 vec( ^ )
u)
(T
(T
p)1=2 vech( ^
vec( )
vech( u )
According to (36),
(T
~ Y
p)1=2 (C(
!XjZ )
h
C(Y
!XjZ ))
h
d
! N (0;
C ):
Thus,
(T
p)1=2 (C^Y
!XjZ
h
CY
49
!XjZ )
h
d
! N (0;
C );
:
where
@C(Y
C
=(
!XjZ )
@C(Y
h
0
@(vec( ) ; vech(
0
u)
)
) (
!XjZ )
0
h
0
@(vec( ) ; vech(
0
u)
)
);
and
1
=
0
u
0
0
2(D3 D3 )
1 D0 (
u
3
0
u )D3 (D3 D3 )
1
:
D3 is the duplication matrix, de…ned such that vech(F ) = D3 vech(F ) for
any symmetric 3 3 matrix F .
Proof (Asymptotic validity of the residual-based bootstrap).
Conditionally on the sample one can use a similar argument as in the proof
of the asymptotic distribution to establish the asymptotic validity of the
residual-based bootstrap. It su¢ ces to take vec( ^ ) and vech( ^ u ) as true
parameters.
50
Figure 1: NBR in logarithmic form
11.2
ln(NBR)
11
10.8
10.6
ln(NBR)
10.4
10.2
10
9.8
9.6
9.4
9.2
Jan65
Jan70
Jan75
Jan80
Jan85
Time (t)
Jan90
Jan95
Jan00
Figure 2: r in logarithmic form
3
ln(r)
2.8
2.6
2.4
ln(r)
2.2
2
1.8
1.6
1.4
1.2
1
Jan65
Jan70
Jan75
51
Jan80
Jan85
Time (t)
Jan90
Jan95
Jan00
Figure 3: P in logarithmic form
5
series 1
4.8
4.6
ln(P)
4.4
4.2
4
3.8
3.6
3.4
3.2
Jan65
Jan70
Jan75
Jan80
Jan85
Time (t)
Jan90
Jan95
Jan00
Jan95
Jan00
Figure 4: GDP in logarithmic form
8.9
ln(GDP)
8.8
8.7
8.6
ln(GDP)
8.5
8.4
8.3
8.2
8.1
8
7.9
Jan65
Jan70
Jan75
52
Jan80
Jan85
Time (t)
Jan90
Figure 5: The first dfferentiation of ln(NBR)
0.08
Growth rate of NBR
0.06
Growth rate of NBR
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
Jan65
Jan70
Jan75
Jan80
Jan85
Time (t)
Jan90
Jan95
Jan00
Figure 6: The first dfferentiation of ln(r)
0.3
Growth rate of r
0.2
Growth rate of r
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
Jan65
Jan70
Jan75
53
Jan80
Jan85
Time (t)
Jan90
Jan95
Jan00
15
x 10
-3
Figure 7: The first dfferentiation of ln(P)
Growth rate of P
Growth rate of P
10
5
0
-5
Jan65
Jan70
Jan75
Jan80
Jan85
Time (t)
Jan90
Jan95
Jan00
Figure 8: The first dfferentiation of ln(GDP)
0.025
Growth rate of GDP
0.02
0.015
Growth rate of GDP
0.01
0.005
0
-0.005
-0.01
-0.015
-0.02
-0.025
Jan65
Jan70
Jan75
54
Jan80
Jan85
Time (t)
Jan90
Jan95
Jan00
Table 4: Augmented Dickey-Fuller Test (variables in logarithmic form)
N BR
R
P
GDP
With Intercept
ADF test statistic 5% Critical Value
0:510587
2:8694
2:386082
2:8694
1:829982
2:8694
1:142940
2:8694
With Intercept and Trend
ADF test statistic 5% Critical Value
1:916428
3:4234
2:393276
3:4234
0:071649
3:4234
3:409215
3:4234
Table 5: Augmented Dickey-Fuller Test (First di¤erence)
N BR
r
P
GDP
With Intercept
ADF test statistic 5% Critical Value
5:956394
2:8694
7:782581
2:8694
2:690660
2:8694
5:922453
2:8694
55
With Intercept and Trend
ADF test statistic 5% Critical Value
5:937564
3:9864
7:817214
3:9864
3:217793
3:9864
5:966043
3:9864
Figure 9: Causality measures from Nonborrowed reserves to Federal funds rate
0.25
95% percentile bootstrap interval
OLS point estimate
Causality Measures
0.2
0.15
0.1
0.05
0
0
5
10
15
20
Horizons
25
30
35
40
Figure10: Causality measures from Nonborrowed reserves to GDP Deflator
0.16
95% percentile bootstrap interval
OLS point estimate
0.14
Causality Measures
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5
10
5615
20
Horizons
25
30
35
40
Figure 11: Causality measures from Nonborrowed reserves to Real GDP
0.16
95% percentile bootstrap interval
OLS point estimate
0.14
Causality Measures
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5
10
15
20
Horizons
25
30
35
40
Figure 12: Causality measures from Federal funds rate to GDP Deflator
0.18
95% percentile bootstrap interval
OLS point estimate
0.16
Causality Measures
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5
10
5715
20
Horizons
25
30
35
40
Figure 13: Causality measures from Federal funds rate to Real GDP
0.16
95% percentile bootstrap interval
OLS point estimate
0.14
Causality Measures
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5
10
15
20
Horizons
25
30
35
40
Figure 14: Causality measures from Real GDP to Federal funds rate
0.35
95% percentile bootstrap interval
OLS point estimate
0.3
Causality Measures
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
58
20
Horizons
25
30
35
40