*Corresponding author.
E-mail address:[email protected] (F. Yao).
Inference on one-way e
!
ect and evidence in
Japanese macroeconomic data
Feng Yao!,
*, Yuzo Hosoya"
!Faculty of Economics, Kagawa University, Takamatsu 760-8523, Japan
"Faculty of Economics, Tohoku University, Sendai 980-8576, Japan
Received 1 January 1999; received in revised form 1 August 1999; accepted 1 October 1999
Abstract
This paper provides an approach to testing a variety of causal characteristics expressed in terms of the measures of one-way e!ect for cointegrated vector time series. For this purpose, we propose Wald tests and their computational algorithm by means of the measures of one-way e!ect, incorporating Johansen's algorithm for the maximum likeli-hood estimates and the likelilikeli-hood ratio tests. Using the Wald test statistics, the paper also provides a method of con"dence-set construction for the causal measures. The approach proposed in the paper includes testing Granger's non-causality as an instance of its multiple applications. As an illustration, the paper presents a characterization of the causal structure of the recent Japanese macroeconomy on the basis of the proposed method and the derived evidence. ( 2000 Elsevier Science S.A. All rights reserved.
JEL classixcation: C12; C13; C14; C32; E49; E51; E52; O11; O47; O53
Keywords: Cointegration; VARmodel; Granger's non-causality; Measures of one-way e!ect; Wald statistic; Testing causality; Maximum-likelihood estimation; Likelihood ratio statistic; Asymptotic theory; Japanese economy
1. Introduction
Elicitating causal relationships among a set of several time series has been one of the basic subjects in the literature of economic theory as well as of
econometrics. To solve the problems of determining the direction of causality between a pair of time series and also of statistically testing the absence of feedback, Granger (1963, 1969) introduced a celebrated de"nition of causality. His concept of causality is a statistically testable criterion de"ned in terms of predictability based on the assumption that the cause chronologically precedes the e!ect and the future does not cause the past. Among the earlier representa-tive studies testing absence of feedback relation are the Granger test of zero restriction of speci"c coe$cients of an stationary autoregressive representation and the Sims test of the zero restriction of some coe$cients in moving-average representation of stationary bivariate processes (see Sims, 1972; Hosoya, 1977 for the equivalence of their concepts of non-causality which is extended even to non-stationary processes; see also Florens and Mouchart, 1982).
As regards testing Granger's non-causality in levels of a non-stationary vector autoregressive (VAR) system, Sims et al. (1990) dealt with trivariateVARsystems, to conclude that the Wald test statistic has a limitings2distribution if the time series are cointegrated and otherwise that it has a nonstandard limiting distribu-tion, whereas Toda and Phillips (1993) extended their results. LuKtkepohl and Reimers (1992), using the Wald test for Granger's non-causality in bivariate cointegrated "nite order AR process, investigated the short- and long-term interest rates in the U.S., and Mosconi and Giannini (1992) provided the likelihood ratio test of non-causality between a pair of cointegrated time series based on the Johansen model and inference. So far, the interest of the econo-metric literature seems mostly concerned with the Granger non-causality test. For the purpose of quantitative characterization of the feedback relationship between two multivariate time series, Geweke (1982) introduced an early version of the measure of causality from one time series to another in the time domain as
well as in the frequency domain. Developing Geweke's frequency-domain
ap-proach, Hosoya (1991) introduced three causal measures summarizing the interdependency between a pair of nondeterministic stationary processes. They consist of the measures of association, one-way e!ect and reciprocity, each of which is de"ned as overall as well as frequency-wise measure. The measure of association is equal to the sum of the others. Granger and Lin (1995) gave an extended measure of one-way e!ect for an non-stationary bivariate cointegrated process, giving a characterization of the measure near the zero frequency in terms of the model parameters. Hosoya (1997a) also extended all his measures to a speci"c class of non-stationary reproducible processes.
testing the nullity of the overall measure of one-way e!ect (OMO), but also the
strength of the one-way e!ect. Moreover by means of the integral of the
frequency-wise measure of one-way e!ect (FMO) on speci"c frequency bands, the long- and short-run causal relationships can also be tested (an earlier idea was given by Geweke, 1982). In view of these points, the approach proposed in this paper extends the conventional causal test theories.
In order to be speci"c, the causal test procedure in this paper incorporates Johansen's likelihood-based method (see Johansen, 1988, 1991, 1995). Namely, we determine the cointegration rank and cointegration space by his likelihood ratio test, and then exploiting the property that the ML estimator of cointegrat-ing vector has a convergence rate faster than the estimators of the other
parameters, we construct Wald test statistics which are asymptotically s2
-distributed. In order to illustrate the performance of the Wald test theory, we apply it to the study of Japanese macroeconomic time-series over the span of the recent twenty years. Our empirical analysis characterizes the recent Japanese
macroeconomy by means of the estimatedOMOandFMO.
This paper is organized as follows: Section 2 presents heuristic exposition of
theOMOand theFMOintroduced in Hosoya (1991), focusing on
interpreta-tional aspects of those measures in particular. Then those measures are extended to non-stationary processes. Based on an ECM, Section 3 constructs Wald test statistics for testing theOMOand other causal measures, and exhibits relevant computational procedures. Section 4.1, identi"es empirically pertinent ECMs for the causal analysis, where preliminary data analysis of Japanese macroeco-nomic time-series is carefully conducted. In Section 4.2, we deal with empirical
relations among GDP, M2#CD, Call Rates, Exports and Imports in Japanese
economy in the recent twenty years. The estimated cointegration rank, estimates of theFMOand theOMOand causal test statistics are listed in"gures. For the cases where causality is statistically signi"cant, the con"dence intervals of the trueOMOare also listed in the corresponding "gures. In the last part of this section, a bivariate case is discussed in detail and the set of the con"dence sets is visualized by a three-dimensional graph of the con"dence surface. Section 5 concludes the paper.
We use the following notations and symbols in the sequel. The set of all integers and the set of positive integers are denoted byZandZ`, respectively.
For a set of random variables MZ
i,i3AN with "nite second moment,
HMZ
i,i3ANimplies the closure in mean square of the linear hull ofMZi,i3ANin the Hilbert space of random variables with"nite second moment. For ap-vector process X(t) with "nite covariance matrix and for S a set of integers,
HMX(t),t3SNimpliesHMX
i(t),t3S,i"1,2,pN. DenoteIpthe identity matrix of orderp, whereas 1
pis thep-vector with all the elements of 1.AHindicates the conjugate transpose if Ais a complex matrix and the simple transpose ifAis
a real matrix. The vec operator transforms a m]n matrix B into a m)n]1
v(C) denotes then(n#1)/2 vector that is obtained fromvec Cby eliminating all supradiagonal elements of an]nmatrixC. In this way, for symmetricC,v(C) contains only the generically distinct elements ofC. For a random vectorXor for a pair of random vectors X and >, Cov(X) and Cov(X,>) indicate the variance}covariance matrix of X and of vec(X,>), respectively. The trace of a square matrixCis denoted bytr Cand the determinant is denoted bydet C. The Kronecker product of any matrixAandBis denoted byA?B, whereas the
sum of two vector subspaces H
1 and H2 is denoted by H1=H2. The lag
operator denoted by¸and the di!erence operator is denoted byD"1!¸.
2. Causal measures for cointegratedVARmodel
The section introduces the measuresOMOandFMOfor non-deterministic
stationary time-series and then extends them to non-stationary time-series in cointegrated relations (see for mathematical details Hosoya, 1991, 1997a). At the end of this section, we discuss long- and short-run relationships expressed by those one-way e!ect measures.
The construction of the causal measures, in particular the measures of one-way e!ect, is closely related to the prediction theory of stationary processes. Suppose that M;(t),<(t),t3ZN is a zero mean jointly covariance stationary process where the;(t) and<(t) arep
1]1 and p2]1 real vectors, respectively (p"p
1#p2). Suppose also that the process M;(t),<(t)N is non-deterministic and has thep]pspectral density matrix
f(j)"
C
f11(j) f12(j)f
21(j) f22(j)D
, !n(j)n,
wheref
11(j) is thep1]p1 spectral density ofM;(t)N, and thatf(j) satis"es
P
~nnlogdet f(j) dj'!R. (2.1)Under condition (2.1),f(j) has a factorization such that
f(j)"1
2nK(e~*j)K(e~*j)H, (2.2)
whereK(e~*j) is the boundary value lim
of the processM;(t),<(t)Nby its own past; then, we have
(see Rozanov, 1967, pp. 71}7, for example). The relationship (2.2) is the fre-quency domain version of the time-domain Wold decomposition representation
A
;(t) the real-matrix coe$cients in the expansion of the analytic functionK(z); namely K(z)"+=j/0KI(j)zj.
The one-way e!ect component of<(t) is the component which causesM;(t)N one-sidedly but su!ers no feedback from it in the Granger sense. We can
ex-tract such component and the residual is denoted by <
0,~1(t) from <(t) as
the linear regression residual obtained by regressing <(t) on
M;(t#1!j),<(t!j), j3Z`N. It turns out that M<
0,~1(t)N is a white-noise
process with covariance matrix R
22!R21R~111R12 and that M<(t)N does not causeM<
0,~1(t)Nin the Granger sense (see Hosoya, 1991). Although we have no way of measuring directly how the addition of the series M<(t)Nimproves the one-step ahead prediction ofM;(t)N, the series M<
0,~1(t)Nenables us to do so. A seriesM<(t)Ncauses anotherM;(t)Nin Granger's sense ifM<(t!j), j3Z`Nis
informative in predicting ;(t), whence it would be natural to measure the
strength of causality by the extent the one-step ahead prediction error of;(t) is reduced by adding the informationHM<(t!j), j3Z`N. Since M<(t)Ndoes not
1]p2matrices andN(t) is a dependent stationary process which is orthogonal to the processM<
0,~1(t)N. It follows from (2.4) thatf11(j) the spectral density matrix ofM;(t)Nis decomposed as
is equal to the one ofN(t) onHMN(t!j),j3Z`Nso that the determinant of that covariance matrix is provided in view of (2.3) by
(2n)p1exp
G
1 2nP
n
~n
logdet g(2)(j) dj
H
,whereas the corresponding quantity of the residual by regressing M;(t)N on
HM;(t!j), j3Z`Nis equal to
measuring the frequency-wise as well as overall improvement of prediction by the use of the informationHM<
0,~1(t!j), j3Z`N, respectively, equivalently as measuring the strength of causality ofM<
0,~1(t)NtoM;(t)N.
For the purpose of obtaining the explicit analytic expression of the prediction improvement, it is convenient to translate the above rather time-domain-oriented construction in terms of frequency-domain representations. In contrast to the Wold decomposition of M;(t),<(t)N which is a decomposition into an
where ;I (j) and <I (j) are (frequency-wise orthogonal) random measures such that
CovMd;I (j), d<I(j)N"f(j),
namely, the processes M;(t)N and M<(t)Nare interpreted as weighted sums of harmonic oscillations with orthogonal random weight for the respective fre-quency. On the other hand, the prediction error formula (2.3) implies that the one-step ahead prediction error of;(t) measured in terms of the determinant of the prediction error covariance matrix is the geometric mean of the
det CovMd;I (j)Nover the frequency domain !n(j)n. In other words, the
variability of d;I (j) expresses the frequency-wise contribution to the one-step ahead prediction error of;(t). In the case of the joint one-step ahead prediction of M;(t),<(t)N, a similar argument applies and the variability expressed by
det CovMd;I (j),d<I (j)Nindicates the contribution of thej-frequency oscillation to the joint prediction error of;(t) and<(t).
the other seriesM<(s),s)t!1Nwhen it is added for the prediction of;(t) and which portion of the variability in the pairMd;I (j), d<I (j)N, which is correlated in general, is attributable to the seriesM<(t)N. Although these questions are intrac-table if we try to work directly with the pair M;(t),<(t)N, the pairing
M;(t),<
0,~1(t)Nenables us to solve these questions. In view of its construction of <
where the spectral density matrix of the processM;(t),<
0,~1(t)Nis denoted by
1 matrix which consists of the "rst p1 columns of
f(j),fI
22(j)"1/2nMR22!R21R~111R12N(see Hosoya, 1991, pp. 432}3, and also see Whittle, 1963 for the spectral regression (2.6)). Relation (2.6) is nothing but the frequency-domain version of (2.4). Since the one-step ahead prediction error of;(t) on the basis ofHM;(t!j),<
0,~1(t!j),j3Z`Nis the same as that ofN(t) on the basis of its own past, it follows that
detR@
11 denotes the covariance matrix of the one-step ahead prediction error ofS(t); whereas as for the prediction of;(t) by its own past values, we have the
It turns out thatM<(t)Ndoes not causeM;(t)Nin the Granger sense if and only if
M
V?U"0 (see Hosoya, 1991, p. 432). Consequently, in conformity with Gran-ger's causality concept, we might call M
V?U the overall measure of one-way
e!ect (OMO) from<to;andM
V?U(j) the frequency-wise measure of one-way e!ect (FMO). It is obvious thatM
V?U(j) in (2.10) can also be expressed by
M
V?U(j)"log[det f11(j)/detMf11(j)!fI12(j)fI~122(j)fI21(j)N]. (2.11)
Then theOMOfrom<to; can be expressed by
M
As a next step, in order to extend this causal analysis of non-deterministic stationary time-series to non-stationary processes, consider the process
MX(t),>(t)Nwhich is generated by andHM<(t),t)0Nrespectively. The process given by (2.13) has the characteristic
that the one-step ahead prediction and the residual of X(t) based on
It should be noted, however, that the relationship (2.13) is not very well de"ned. Suppose thatB(¸) is another block diagonal matrix given by
B(¸)"
C
B11(¸) 022,0"Ip2. The left multiplication of B(¸) to each member of the equation (2.13) produces a di!erent representation of the process MX(t),>(t)N. Unless
B(¸)"I
p, the resulting generating processMB11(¸);(t),B22(¸)<(t)Nmight pos-sibly possess a spectral structure di!erent from that ofM;(t),<(t)N. In order to retain invariance of the one-way e!ect structure under such a multiplication, a certain restriction on the generating mechanism (2.13) is required. Let
f
11(j)"1/2nK(1)(e~*j)K(1)(e~*j)H and f22(j)"1/2nK(2)(e~*j)K(2)(e~ij)H be ca-nonical factorizations, respectively.
Assumption 2.1. The process (2.13) satis"es either (i) the zeros ofdet A
11(z) anddet A22(z) are all on or outside of the unit disc; or
(ii) There are no common zeroes betweendet A
11(z) anddetK(1)(z) and between
det A
22(z) anddetK(2)(z).
Remark 2.1. Assumption (i) is convenient to deal with such unit-root type processes as cointegration processes, where non-stationary is generated by a unit-root common trend. Since, then,B(¸) is limited to such lag polynomials for which the zeroes ofdet B
11(z) anddet B22(z) are on or outside the unit disc,
which follows from a more basic relationship in calculus that for realrsuch that
DrD)1,
P
n~n
logD1!2rcosj#r2Ddj"0.
If some zeros ofdet A(z) is allowed to be inside the unit circle so that the process has a greater-than-unity root, this invariance property does not hold any longer. To deal with such a circumstance, assumption (ii) would be useful in order to identify the generating process.
The preceding consideration leads us to the following extended de"nitions of
the Granger non-causality and of the measuresOMOandFMO. Suppose the
Dexnition 2.1. M>(t)Nis said not to causeMX(t)Nif and only if the prediction error or outside of the unit disc. Denote byC(¸) the adjoint matrix ofA(¸) so that
C(¸)A(¸),D(¸),
where D(¸) is the diagonal matrix having d(¸),det A(¸) as the common
diagonal element,d(¸)"+bj/0d
j¸jis a lag polynomial with scalar coe$cients
such that d
0"1 and the zeros of+bj/0djzj are either on or outside the unit circle. Left-multiplyingC(¸) to the members of Eq. (2.14), we have
M;(t),<(t)Nis a stationary MA process and that the processMX(t),>(t)Nsatis"es Assumption 2.1 (i). Therefore, in view of De"nition 2.2 above, all the measures of one-way e!ect for the possibly non-stationary processesMX(t),>(t)Nare deter-mined by the corresponding measures of the stationary processesM;(t),<(t)N.
Moreover, since the zeros ofdet C(z) are either on or outside the unit circle, the covariance matrix of the one-step ahead prediction error of=(t) is equal to R and if the spectral density matrix of M=(t)N is denoted by f(j), it has a canonical factorization
f(j)"1
2nK(e~*j)K(e~*j)H, (2.16)
where K(e~*j)"C(e~*j)R1@2 for the Cholesky factor R1@2 of R such that R"R1@2R1@2. Then the causal measures can be calculated in view of (2.11) and
(2.12) by means of the spectral densityf(j) de"ned by (2.16) and its factorK(e~*j). A variety of causal measures can be derived on the basis of theOMOand the
FMObetweenMX(t)NandM>(t)Nfor the purposes of the long-run or short-run characterization of causal relation. In caseM
Y?XO0, for example, the contri-bution of a long-run e!ect in the overall one-way e!ect is given by
D
for a certain low-frequency band [!e, e], or one might be rather interested in
the contribution of the relative e!ect for a given period band
[t
where period implies the time-length of a cycle and we used the relationt"2n/j
between periodtand frequencyj(j'0).
The long-run e!ect may be measured in another way, by the mean FMO
which is given by
whereeis a small positive number, or its limit aseP0. In order to summarize the one-way e!ect in a period band [t1,t2],
Remark 2.3. A notable peak ofM
V?U(j) atj"p/2, for example for quarterly data, would indicate that there is a signi"cant one-way e!ect from M<(t)N to
M;(t)Nin one-year period cycle. But this does not imply thatM<(t)NcausesM;(t)N with one-year lag. The time-lag relationship inM<(t)NcausingM;(t)Nshould be observed in Sims'distributed-lag representation (2.4) which connectsM<
0,~1(t)N
andM;
~1,0(t)Nin the time domain.
Remark 2.4. The existence of the Nyquist frequency (see for example Yao, 1985) should not be ignored. The discernible highest frequency isj"n, which corres-ponds to two periods (t"2n/j"2); namely, half a year for quarterly data. The economic implication is that, we cannot discern the one-way e!ect shorter than half a year for quarterly data.
3. Testing causality in cointegratedVARprocesses
This section considers the Wald tests for testing hypotheses on the measures of one-way e!ect based on the ECM given by (3.1), providing the computational procedure and also applying the test statistics to construction of con"dence-sets of those measures.
Let MZ(t)N"MX(t)H,>(t)HNH be generated by a cointegrated p-vector AR model which is represented in the error-correction form
*Z(t)"abHZ(t!1)#a+~1 j/1
C(j)*Z(t!j)#k#UP(t)#e(t), (3.1)
whereaandb arep]rmatrices (r)p), andk is a constantp-vector. Also in (3.1), P(t) is a column (s
$!1)-vector of centred seasonal dummy variables,
wheres
$ is the seasonal period so that for quarterly data,s$"4; suppose also
that Me(t)N is a Gaussian white-noise process with mean 0 and with positive-de"nite non-degenerate variance}covariance matrix R. Let h be a (r)p)]1
vector consisting of the elements of b such h"vecbH. Denoting
nt"p)(r#p)(a!1))#p)(p#1)/2, lettbe thent]1 vector which consists of
the elements of a and C(j) (j"1,2,a!1) and the elements in the lower triangular part of R; namely t"vec(vec(a, C)H,v(R)), where C"
MC(1),2,C(a!1)Nandv(R) denotes the (p)(p#1)/2)]1 vector.
The spectral density matrixfand its canonical factorKderived for the joint processMZ(t)Nby relation (2.15) are given, respectively, by
f(jDh, t)"1
2nK(e~*jDh,t)K(e~*jDh, t)H (3.2)
and
whereC(e~*jDh,t) is the adjoint matrix of the complex-valued polynomial matrix
It is important to note here that the Granger causality is de"ned only between
non-deterministic time-series and there is no one-way e!ect between such
deterministic components as the dummy variables and the intercept which appear in model (3.1); a deterministic component can be predicted exactly by its past values and there is no improvement in prediction if information of another series is added (see Hosoya, 1977 for a formal proof for non-causality between deterministic processes).
Note that in these instances,G(h, t) is di!erentiable functions with respect to (h,t).
Johansen (1988,1991) showed that, if (h, t) is the true value and (hK, tK) is the ML estimate,¹(hK!h) tends to have a mixed multivariate normal distribution
andJ¹(tK!t) tends to have a multivariate normal distribution as¹PR,
whence G(hK, tK) is a J¹ consistent estimate of G(h,t). By the stochastic Note that the"rst-order asymptotic distribution ofG(hK, tK) is completely deter-mined bytK and the non-standard limiting distribution ofhK is not involved, the sampling error ofhK being negligible in comparison with that oftK. Consequently, the test forG(h,t) and the con"dence-set construction can be conducted based on the Wald statistic
W,¹MG(hK, tK)!G(h,t)N2/H(hK, tK), (3.5)
As regards evaluation ofD
tGathK,tK, the numerical di!erentiation is practical
in view of the complexity of the exact analytic expression. Speci"cally, the gradient ofG(h, t)
for su$ciently small positivehwhereh
i is thent]1 vector with theith element
h and all the other elements zero; namely,
h
i"(0,The numerical computation of2,h, 0,2,0)H, i"1, 2,2W,(nht,t.) in (3.4) can be conducted as follows. We
set t(1)"vecMa,CN,t(2)"vec(k, U) and t(3)"l(R), and also we set
t(12)"vec(t(1), t(2)). Then the log-likelihood function of the parameter
t(12)andt(3) based on observationsZ(1),2,Z(¹) can be given as
Moore}Penrose inverse of matrixD(see Magnus and Neudecker, 1988, p. 49).
Denote bytK(12)andtK(3)the ML estimators oft(12)andt(3), respectively, then
the asymptotic variance}covariance matrix of J¹MtK(12)!t(12)N and
(see Magnus and Neudecker, 1988, p. 321). The asymptotic covariance of
J¹(tK(1)!t(1)), which is denoted byW
t(1)t(1) is then constructed fromR?Q~1 by eliminating the rows and columns corresponding toJ¹(tK(2)!t(2)). In fact we can write the symmetric (p)(r#p)(a!1))#p)s$) dimensional matrix
mensional squared matrix. The covariance matrix W
t(1)t(1) is constructed by eliminating all the last s
$ columns and the last s$ rows of the submatrices
pijQ~1, i,j"1,2,p.
t(1)t(1)denotes the variance}covariance matrix ofJ¹(tK(1)!t(1)) evaluated at (hK, tK), then
W(h, t)"
A
WKt(1)t(1) 0
0 2D`(RK?RK)D`H
B
#o1(1). (3.10)Therefore we can use the"rst right-hand side member of (3.10) as a consistent estimate ofW(h,t).
By (3.4) and (3.10), we then get a variance estimate HK "H(hK, tK). Denote
G
01 the given scalar, for the purpose of testing the null hypothesis
G(h, t)"G
01, we evaluate the test statistic=de"ned by (3.5). In order to test no-causality in Granger's sense, we set the null hypothesis asG
test statistic is given by
="¹MG(hK,tK)N2/H(hK, tK). (3.11)
If=*s2
a(1), fors2a(1) the upperaquantile of thes2distribution with one degree
of freedom, we may reject the null hypothesis of non-causality from>toX. On the other hand, in view of (3.5), the (1!a) con"dence interval of the causal measureG(h, t) is provided by
(G(hK,tK)!H
a,G(hK,tK)#Ha), (3.12)
whereHa"J(1/¹)H(hK, tK )s2a(1).
Based on model (3.1), various other measures can be constructed. For in-stance, in casep*3, we might divideZ(t) into three vectorsZ1(t),Z2(t) andZ3(t) such that
Z(t)"(Z1(t)H,Z2(t)H,Z3(t)H)H
and investigate the pair of the vectorOMO,
G(h,t)"
A
M(Z1,Z2)?Z3(h, t)Or one might be interested in the pair of a long- and a short-run mean e!ect
G(h,t)"
A
21j0:@j@:j0MY?X(jDh, t) djthen the preceding arguments are extended to this vector-valued G(h, t) in a straightforward way as follows. LetDtGin (3.4) now represent the Jacobian matrix
is asymptoticallys2-distributed withmdegrees of freedom if (h, t) is the true
value. Denote G
be conducted by settingG
0m"0. The test statistic is given by
=(m),¹MG(hK,tK)NHH(hK, tK)~1MG(hK, tK)N. (3.14)
If =(m)*s2
a(m) for the aupper quantile of s2 distribution withm degrees of
freedom,G
0m"0 is rejected at asigni"cance level; the asymptotically (1!a) con"dence set ofGcan be given by the interior and the surface of an ellipsoid; namely
[G:¹MG(hK,tK)!GNHH(hK,tK)~1MG(hK,tK)!G)N)s2
a(m)]. (3.15)
It consists of thoseGwhich are not rejected at the asigni"cance level by the Wald test statistic. In the case of m"2, for example, set G(hK, tK)"(GK
1, GK 2),
G"(G1, G2) andH(hK, tK)~1"MhK(
ij)N(hK(ij)"hK(ji)); then the con"dence sets with asymptotic con"dence coe$cient (1!a) for various aare given by the set of nested ellipses determined by
MG:F
2(G))s2a(2)/¹N, 0(a(1, (3.16)
where
F
2(G)"(G1!GK 1)2hK(11)#(G2!GK2)2hK(22)#2(G1!GK1)(G2!GK2)hK(12).
Remark 3.1. Note that our algorithm for evaluating the Wald statistic and the con"dence set does not depend upon the kind of measures of one-way e!ect so that it applies also toDM
Y?X(e) orDMY?X(t1, t2), given in Section 2.
4. Empirical analysis
In order to examine the performance of our Wald test in practice, the test theory of the causal measures developed in the foregoing sections is applied in this section to macroeconomic data analysis of Japan. The data used are the
quarterly observations of GDP(Y), M2#CD (M), Call Rates (R), Exports (Ex)
4.1. Preliminary analysis
In the following study, we use model (3.1) with the common lag-lengtha"5. The lag-length of an autoregressive process delimits the range of possible con"guration of theFMO. In order to avoid the di!erence of lag-length playing a part in"gurative di!erence ofFMO, we did not use information criteria which are rather suited for identi"cation of individual models, but used a common lag-length. As is seen below, the uncorrelation and the Gaussianity hypotheses seem mostly supported for the residuals derived by"tting the lag-lengtha"5. The"tted model we used isp-dimensional AR(5) in ECM form represented by
*Z(t)"PZ(t!1)#+4 k/1
C(k)*Z(t!k)#k#UP(t)#e(t), (4.1)
where e(t)'s (t"1,2,¹) are Gaussian white noise with mean 0 and vari-ance}covariance matrix R, and we choose P(t) the 3]1 vector of centered seasonal dummies so as not to produce seasonal trend e!ects in the level ofZ(t). The"rst"ve observations ofZ(t) are kept for initial values.
The hypothesis of independentrcointegration vectors is
H(r):P"abH, (4.2)
wherea, barep]rmatrices (r)p) such that rank(P)"r. Ifr"0, (4.1) reduces to a full-rank unit root process. Ifr"p, thenPis full rank and the processZ(t) is stationary. Denote, the same as in Johansen (1995) Theorem 6.1, the
decreas-ing sequence of eigenvalues 1'jK1'2'jK
p'0 and the matrix constituted by the corresponding eigenvectors <K "(<K
1,2,<K p). Under model (4.1), the likelihood ratio test statistic for the hypothesisH(r) againstH(p) is given by the
&trace'statistic
q(r)"!¹ +p i/r`1
ln(1!jK
i). (4.3)
The quantile tables used for the cointegration rank test are given by
Osterwald-Lenum (1992) based upon Monte Carlo simulations. The ML estimator ofbis
bK"(<K
1,2,<K r), which correspond to thernon-zero eigenvalues. The estimates of the other parameters are obtained by OLS by settingP"abKHin the model (4.1).
In the case where there is no or little prior information aboutr, we might estimateras follows: Denote byq(iD1!a) the (1!a) quantile ofq(i) and byq((i) be the observation of q(i). If q((0)(q(0D1!a), we choose r("0. For
r"1,2,p!1, letr( be the"rstrsuch that
and if there is no such r, then set r("p. This procedure de"nes an estimator
r( which takes on the values 0, 1,2,pand which converges in probability to the true value, ifa tends to 0 with appropriate speed as the sample size tends to in"nity (see Hosoya, 1989, pp. 442}3). A variety of aspects of the identi"cation problem are discussed by Johansen (1995), but we choose in our analysis the least restrictive model speci"cation. For details see Johansen (1988,1991,1995), and Johansen and Juselius (1990).
Remark 4.1. The numerical computations of the paper were conducted by FORTRAN programs (see Yao and Hosoya, 1995; Yao, 1996). By applying those programs to the "ve macroeconomic series, we investigated bivariate, trivariate as well as four-variate models. Since the size of 20-yr quarterly data
cannot be regarded as large, to be conservative, we use ¹!n
t instead of the
sample size¹in (3.8), (3.9) and (3.11).
The estimated eigenvalues and the corresponding eigenvectors of the
bivari-ate and trivaribivari-ate as well as four-varibivari-ate in ECM are given in Tables 1}3,
respectively. The variables of the models are indicated in the tables. The observed trace statistics are also listed there. We estimaterin this paper based not only on theq(r) statistic but also on the consideration of other aspects of data and the corresponding model. Consider for example the process of
deter-mining the cointegration rank rfor the case of four-variate model where the
necessary quantiles are listed in Table 3. It shows that
q((0)"48.73'43.95"q(0D0.9) andq((1)"25.38(26.79"q(1D0.9). According to the above procedure we select r("1, which is listed in Figs. 1(c1) and (c2).
Consider for another example the determination the cointegration rank rfor
bivariate modelZ"(>,R)Hwhere the necessary quantiles are listed in Table 1. Even though the observed test statistics indicate two cointegrated relations, considering the obvious non-stationary nature of the nominal GDP, we chose
r("1. The parametersaandC(k), which will be used in the following causality analysis, are then estimated by the OLS method and denoted by
a(,CK(k), (k"1, 2, 3, 4), respectively.
A criterion for the lag length selection is that the resulting residuals are uncorrelated to a reasonable degree. This is checked by Portmanteau tests. In this paper, we use the following modi"ed form given by Hosking (1980), which seems to have better performance for small sample size:
The eigenvalues and the eigenvectors and the trace statistics for bivariate models!
!Note: Y: GDP, M: M2#CD, R: call rates, Ex: exports, Im: imports.ris the cointegration rank; The trace statisticqis de"ned by (4.3). Table 2
The eigenvalues and the eigenvectors and the trace statistics for trivariate models
Table 3
The eigenvalues and the eigenvectors and the trace statistics for four-variate models!
The eigenvalues Trace statistics
(0.268 0.163 0.124 0.028) q-statistic
The eigenvectors p!r q( 90% 95%
M 0.735 0.083 !0.382 0.864 1 2.13 2.69 3.76
R 0.015 !0.021 0.047 0.006 2 12.07 13.33 15.41
Ex !0.677 !0.584 0.781 !0.096 3 25.38 26.79 29.68
Im 0.041 0.808 !0.492 !0.495 4 48.73 43.95 47.21
!Note: The 90% and 95% quantiles are from Table 1 in Osterwald-Lenum (1992). These quantities are also used for Tables 1, 2.
Table 4
TheHg-statistics for testing residual autocorrelation!
f Hg-statistic p-value f Hg-statistic p-value
>&M 52 55.8605 0.3319 R&Im 52 51.3962 0.4976 >&R 52 56.9648 0.2956 Ex&Im 52 68.1974 0.0653 >&Ex 52 62.9925 0.1413 >&Ex&Im 117 141.7412 0.0596 >&Im 52 58.3853 0.2524 >&M&R 117 129.4265 0.2037 M&R 52 42.1847 0.8325 M&R&Im 117 116.0028 0.5087
M&Ex 52 61.1332 0.1808 M&R&Ex 117 133.3932 0.1427
R&Ex 52 59.6746 0.2168 M&R&Ex&Im 208 229.2855 0.1492
!Note:f: the degrees of freedom.Hg-statistic is de"ned by (4.4).
Under the null hypothesis of uncorrelation, the distribution of this test statistic is approximated for large¹ and fors'aby s2 distribution with degrees of freedomf"p2(s!a) whereais the lag length of the model. For our cases of bivariate, trivariate and four-variate models, we chose s"18. The observed statistics are listed in Table 4. The results support that all the residuals in the models are reasonably uncorrelated.
The Gaussian assumption of the disturbance term is checked by applying the omnibus test for multivariate normality given by Doornik and Hansen (1994) to the residuals of the estimated models. LetRH
r be thep]¹matrix of the residuals with sample covariance matrixF"(f
ij). Create a matrixDwith the reciprocals of the standard deviations on the diagonal,D"diag(f~1@211 ,2,f~1@2
pp ), and then
form the correlation matrixC"DFD. De"ne the transformed matrix ofR
r by
R
c"H¸~1@2HHDRHr, (4.5) where¸is the diagonal matrix with the eigenvalues ofCon the diagonal. The
Table 5
Testing normality of residuals!
f Ep-statistic p-value f Ep-statistic p-value
>&M 4 0.0940 0.9989 R&Im 4 0.1067 0.9986 >&R 4 1.1102 0.8927 Ex&Im 4 0.8692 0.9289 >&Ex 4 2.2849 0.6835 >&Ex&Im 6 0.5600 0.9970 >&Im 4 0.6071 0.9623 >&M&R 6 5.0317 0.5398
M&R 4 5.6457 0.2272 M&R&Im 6 10.3059 0.1123
M&Ex 4 2.7026 0.6088 M&R&Ex 6 8.9454 0.1767
R&Ex 4 7.6296 0.1061 M&R&Ex&Im 8 12.5157 0.1296
!Note:Ep-statistic is de"ned by (4.6).
skewness Jb
1i and kurtosis b2i of each vector of the transformed RHc,
i"1,2,p. Under the null hypothesis of multivariate normal distribution of the residuals, the test statistic is asymptotically distributed as
E
2iis transformed from a gamma distribution tos2, and
then transformed into standard normal z
2i using the Wilson}Hilferty cubed root transformation.
The observed test statisticsE
pfor all the models used in this paper are listed in Table 5. These test statistics seem to indicate that there is no signi"cant departure from Gaussianity. The results in Tables 4 and 5 ensure us that we may proceed to the analysis of the one-way e!ect measurement on the basis of the proposed ECMs.
4.2. Empirical measurement of one-way ewect
The following analyses of the causal relationships of Japanese macroeco-nomic time series are conducted on model (4.1). The empirical examples would characterize the recent Japanese macroeconomy in view of the one-way causal-ity.
and the spectral density estimatefK(j)"1/2nKK (e~*j)KK(e~*j)H. As for the numer-ical evaluation of D
tGin (3.6), we choose h"0.0001; after having conducted
evaluation of the Jacobian matrix for numerous choice including smallerh, we found that the results were su$ciently stable forh"0.0001.
Fig. 1 lists 16 plots of the estimated FMO. There, Figs. (a1)}(a8) show
bivariate cases, and Figs. (b1)}(b6) show trivariate cases, while Figs. (c1) and (c2) are for four-variate. The estimates of cointegrating rank (r) and theOMO(M) as well as the Wald test statistic = de"ned by (3.11) are also presented in the
"gures. The 95% con"dence intervals of theOMO, in case the null hypothesis of non-causality is rejected, are also listed in the corresponding"gures. TheOMO
estimates are obtained by numerical integration of the estimated FMOs by
dividing [0,n] into 200 equal intervals. For each of the models we calculate
FMOfor frequency points ji"in/200,i"1, 2,2,200. As for the number of division of [0, n], we checked many cases of interval division up to 1200, and we found that the 200 equal-division of the interval (0,n] is"ne enough.
Although similar computations were conducted on possible combinations and pairing of the"ve variables, only a few are exhibited in the paper to save the space. In view of Fig. 1, notable"ndings are as follows:
f The estimatedOMOfrom M2#CD to nominal GDP is about three times of
that in the reverse direction, but both of the two measures are not signi"cant at 0.05 critical level [see plot (a1)]. Since the p-value of the Wald test for testing the one-way e!ect from money supply to GDP is 0.08, even though the e!ect is small, but signi"cant at 10% signi"cance level. The plot (a1) also shows that there is no one-way e!ect in the frequency band [0.4n, n], or in a period band shorter than 1 yr and a quarter. The estimate of theFMOfrom money toGDPhas a peak in the interval of [0.25n, 0.4n], suggesting that it possibly takes about 1 yr and a quarter for the e!ect to appear. The second peak near the origin indicates the existence of long-run e!ect.
f In general, Call Rate has conspicuous one-way e!ects to the other variables. In contrast, the e!ects in the reverse direction are small and not signi"cant. (see (a3)}(a6) in Fig. 1). The presence of one-way e!ect from interest rates would seem rather conformable to the conventional understanding of macro-economic activities. Also this role of interest rates seems to be consistent with what Sims (1980) found in the macroeconomic data of U.S.
f The plot (a7) shows that the one-way e!ect from Exports to GDP is signi"cant. The corresponding one-way e!ect in the frequency domain shows that the e!ect
is not long run. The estimate of theOMOfrom Imports to GDP is 2.61 and
="2.79 with ap-value 0.095 (see plot (a8)). It is only signi"cant at 0.1 critical
level. The one-way e!ect from Exports and Imports to GDP is 3.82 and
f As far as the bivariate cointegration model between exports and imports is concerned, theOMOfrom exports to imports is 0.01, and the test statistic is estimated as="2.93 withp-value 0.086, whereas theOMOfrom imports to exports is 0.06 and the test statistic is="1.88 with ap-value 0.17. Namely, at 0.05 critical level, there is no signi"cant one-way causality between Japanese exports and imports. It also shows that at 0.1 critical value, there exists a comparatively weak one-way e!ect from exports to imports.
f The one-way e!ect from M2#CD and call rates to GDP is signi"cant and
the corresponding one-way e!ect in the frequency domain has a peak at
frequency 0.4n [see plot (b1)], implying that the highest e!ect comes from
about 1 yr and a quarter period. Both of the one-way e!ects from M2#CD
and call rates to exports and to imports are not signi"cant at 0.05 signi"cance level. These"ndings seem to indicate that money supply is ine!ective to the
Japanese external trades in this period of the #oating exchange-rate of
Japanese Yen.
f Our Wald test shows that the e!ect from interest rates and exports, and from interest rates and imports to money supply are signi"cant at 0.05 critical value. The former e!ect is greater than that of the latter (see (b5) and (b6)). The one-way e!ect of interest rates and imports to money supply is very stable in all the frequency region (0,n]. The evidence also indicates that the one-way e!ect of exports to money is not signi"cant and we have
M
Y?M"0.03, MR?M"4.53, ME9?M"1.89, whereas
M
Y`R?M"7.77, MR`E9?M"5.86.
These results imply that for some cases a policy mix is needed and is perhaps more e!ective than pursuing a single-policy objective.
f The e!ect from exports and imports to money supply and interest rates is
strong and signi"cant [see plot (c1)]. The e!ect in the reverse direction, the e!ect of money supply and interest rates to exports and imports [see (c2)], is comparatively large in value (OMO"4.82) but not signi"cant (="1.15 with ap-value"0.23). The magnitude of the estimatedOMOitself does not tell us whether a one-way e!ect is statistically signi"cant or not, and a test is needed in judging the signi"cance. As a whole, the"ndings imply that in the recent Japanese economy, the external trades have a signi"cant one-way e!ect on the monetary side of the Japanese economy.
Fig. 2. Con"dence surface of two-dimensional vector-valued causal measures for 0}0.975 con"dence coe$cient.
Japanese economic growth in the period we dealt with, the cause is mainly from exports but it is not long run. The empirical results also indicate the cases for which policy mix might be more e!ective.
In the remainder of this section, we illustrate the con"dence-set construction given in (3.15). Based on the trivariate model (4.1) for the variable
Z"(>, M, R)H, and for the case of vector-valuedOMO,G"(G
1,G2)H, we have the estimated matrix
H(hK, tK)~1"
A
0.00263 !0.00118!0.00118 0.37053B, (4.7)
whereG
1is theOMOfrom>andRtoM, andG2is theOMOfromMandRto >, respectively. We have the Wald statistic=(2)"307.89, which means that the
two one-way e!ects are simultaneously signi"cant. Following the argument of Section 3, we can determine the con"dence set of the twoOMO,G
1andG2; the 0.9 con"dence set of the two one-way e!ect causal measures is given by
0.139(G
1!7.77)2!0.126(G1!7.77)(G2!3.93)
#19.638(G
2!3.93)2)4.605. (4.8)
To visualize the nested ellipses graphically, we give three plots in Figs. 2 and 3 [the scale ofcaxis in Fig. 3(a) is set to be four times as much as that in Fig. 2.]. Fig. 2 depicts the con"dence surface for con"dence coe$cientscranging from 0 to 0.975. For c"0, the ellipse is degenerated to one point on the plane of
c"0. The degenerated point is nothing but the ML estimate (GK1,GK 2)" (7.77, 3.93). For a speci"c con"dence coe$cientc"c
Fig. 3. Con"dence surface of two-dimensional vector-valued causal measures for 0.90}0.95 con"-dence coe$cient.
determined as the section cut by the plane in the right angle to thecaxis at the height of c"c
1. Fig. 3(a) shows the nested ellipses which correspond to the con"dence sets forcin the interval [0.90, 0.95]. It is just the portion cut out from the body in Fig. 2 by the parallel planesc"0.90 and 0.95. One cut end of the portion (the smaller ellipse) shows the 90% con"dence set of the one-way e!ect measures determined by (4.8). Fig. 3(b) is the con"dence sets forcin the interval [0.90,0.95] projected onto the (G1,G2) plane.
5. Concluding remarks
Furthermore, the con"dence set construction of a scalar OMO as well as a
vector-valuedOMOwas also shown by the use of the Wald test statistic.
We presented how the theory of the one-way e!ect is put into practice and how to interpret empirical evidence in view of the theory. The empirical analyses were conducted for"ve quarterly macroeconomic series for the period of the
"rst quarter of 1975 through the fourth quarter of 1994 in Japan. Our Wald test shows that money causes income mildly but not vice versa. The one-way e!ects from interest rates to the other variables are comparatively strong and signi" -cant in general but not so in the reverse direction. Our"ndings seem to indicate that monetary policies are ine!ective to the external trades of Japan and that the growth of Japanese economy is mainly driven by exports.
In this paper, we did not pursue generality with respect to model speci"cation and inference procedures. Although in order to be speci"c we used Johansen's modelling and inference method, they are not essential to our causal analysis at all and could be relaxed in many directions. Speci"cally, if we assume a more general linear process forMe(t)Nin (3.1) instead ofi.i.d. normal random vectors, more general inference method with accompanying asymptotic theory would be required, and then such a limit theory as proposed in Hosoya (1997b) would become usefully incorporated. Moreover, although the analysis of this paper relies entirely upon &simple'causal relations, ignoring interaction with a third series,&partial'causal measures, which explicitly take into account the presence of a third series e!ect and its elimination, might be more desirable if we start from a well-de"ned full model of a macroeconomy. The problem of eliminating a third-series e!ect has been discussed in Granger (1969), Geweke (1984), and recently in Hosoya (1998). The last paper shows that the concept of the one-way e!ect propounded in this paper is also e!ective in order to construct partial causal measures. Statistical inference and empirical studies in those directions will be dealt with in other papers.
Acknowledgements
The research of the second author is partially supported by the Japanese
Ministry of Education Scienti"c Research Grant No.(C)(2)-09630023. The
authors are grateful to an associate editor and three referees for helpful com-ments on an earlier draft.
References
Doornik, J.A., Hansen, H., 1994. An omunibus test for univariate and multivariate normality. Mimeo. Nu$eld College, Oxford.
D'Agostino, R.B., 1970. Transformation to normality of the null distribution ofg
Engle, R.F., Granger, C.W.J., 1987. Co-integration and error correction: representation, estimation and testing. Econometrica 55 (2), 251}276.
Florens, J.P., Mouchart, M., 1982. A note on noncausality. Econometrica 50 (3), 583}592. Geweke, J., 1982. Measurement of linear dependence and feedback between multiple time series.
Journal of the American Statistical Association 77 (378), 304}313.
Geweke, J., 1984. Measures of conditional linear dependence and feedback between time series. Journal of the American Statistical Association 79 (388), 907}915.
Granger, C.W.J., 1963. Economic process involving feedback. Information and Control 6 (1), 28}48. Granger, C.W.J., 1969. Investigating causal relations by cross-spectrum methods. Econometrica 39
(3), 424}438.
Granger, C.W.J., Lin, J.L., 1995. Causality in the long run. Econometric Theory 11 (3), 530}536. Hosking, J.R.M., 1980. The multivariate portmanteau statistics. Journal of the American Statistical
Association 75, 602}608.
Hosoya, Y., 1977. On the Granger condition for non-causality. Econometrica 45 (7), 1735}1736. Hosoya, Y., 1989. Hierarchical statistical models and a generalized likelihood ratio test. Journal of
the Royal Statistical Society, Series B 51 (3), 435}447.
Hosoya, Y., 1991. The decomposition and measurement of the interdependency between second-order stationary processes. Probability Theory and Related Fields 88, 429}444.
Hosoya, Y., 1997a. Causal analysis and statistical inference on possibly non-stationary time series. In: Kreps, D.M., Wallis, K.F. (Eds.), Advances in Economics and Econometrics: Theory and Application, Seventh World Congress, Vol. III. Cambridge University Press, Cambridge, pp. 1}33.
Hosoya, Y., 1997b. A limit theory for long-range dependence and statistical inference on related models. The Annals of Statistics 25 (1), 105}137.
Hosoya, Y., 1998. Elimination of a third-series e!ect in statistical causal analysis. Annal Report of the Economic Society, Tohoku University 59 (4), 136}155.
Johansen, S., 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamic and Control 12 (213), 231}254.
Johansen, S., 1991. Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59 (6), 1551}1580.
Johansen, S., 1995. Likelihood-based inference in cointegrated auto-regressive models. Oxford University Press, Oxford.
Johansen, S., Juselius, K., 1990. Maximum likelihood estimation and inference on cointegration with application to the demand for money. Oxford Bulletin of Economics and Statistics 52 (2), 169}210.
LuKtkepohl, H., Reimers, H.E., 1992. Granger causality in cointegrated VAR processes. Economics Letters 40 (3), 263}268.
Magnus, J.R., Neudecker, H., 1988. Matrix Di!erential Calculus with Applications in Statistics and Econometrics. Wiley, New York.
Mosconi, R., Giannini, C., 1992. Non-causality in cointegration systems: representation, estimation and testing. Oxford Bulletin of Economics and Statistics 54 (3), 399}417.
Osterwald-Lenum, M., 1992. A note with quantiles of the asymptotic distribution of the maximum likelihood cointegration rank test. Oxford Bulletin of Economics and Statistics 54, 461}471. Sims, C.A., 1972. Money, income and causality. American Economic Review 62 (4), 540}552. Sims, C.A., 1980. Macroeconomics and reality. Econometrica 48, 1}48.
Sims, C.A., Stock, J.H., Watson, M.W., 1990. Inference in linear time series models with some unit roots. Econometrica 58, 113}144.
Toda, H., Phillips, P.C.B., 1993. Vector autoregressions and causality. Econometrica 61 (6), 1367}1393.
Yao, F., 1996. Econometric analysis of nonlinear and nonstationary relationships: inference and computational methods. Ph.D. Thesis. The Graduate School of Tohoku University, Sendai. Yao, F., Hosoya, Y., 1995. Empirical causality analysis of Japanese macro economic data. Statistical
Analysis of Time Series: Theory and Application, The Institute of Statistical Mathematics Cooperative Research Report, No. 79, pp. 85}96.
