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TANJUNGPINANG, KEPULAUAN RIAU] Date: 11 January 2016, At: 18:42
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Cross-Correlation Matrices for Tests of
Independence and Causality Between Two
Multivariate Time Series
Michael W. Robbins & Thomas J. Fisher
To cite this article: Michael W. Robbins & Thomas J. Fisher (2015) Cross-Correlation Matrices for Tests of Independence and Causality Between Two Multivariate Time Series, Journal of Business & Economic Statistics, 33:4, 459-473, DOI: 10.1080/07350015.2014.962699
To link to this article: http://dx.doi.org/10.1080/07350015.2014.962699
Accepted author version posted online: 25 Sep 2014.
Published online: 27 Oct 2015. Submit your article to this journal
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Cross-Correlation Matrices for Tests
of Independence and Causality Between
Two Multivariate Time Series
Michael W. ROBBINS
RAND Corporation, Pittsburgh, PA 15213 ([email protected])
Thomas J. FISHER
Department of Statistics, Miami University, Oxford, OH 45056 ([email protected])
An often-studied problem in time series analysis is that of testing for the independence of two (potentially multivariate) time series. Toeplitz matrices have demonstrated utility for the related setting of time series goodness-of-fit testing—ergo, herein, we extend those concepts by defining a nontrivial block Toeplitz matrix for use in the setting of independence testing. We propose test statistics based on the trace of the square of the matrix and determinant of the matrix; these statistics are connected to one another as well as known statistics previously proposed in the literature. Furthermore, the log of the determinant is argued to relate to a likelihood ratio test and is proven to be more powerful than other tests that are asymptotically equivalent under the null hypothesis. Additionally, matrix-based tests are presented for the purpose of inferring the location or direction of the causality existing between the two series. A simulation study is provided to explore the efficacy of the proposed methodology—the methods are shown to offer improvement over existing techniques, which include the famous Granger causality test. Finally, data examples involving U.S. inflation, trade volume, and exchange rates are given. Supplementary materials for this article are available online.
KEY WORDS: Granger causality; Residual diagnostics; Vector ARMA.
1. INTRODUCTION
In the data-rich environment of modern econometrics, a time series is nearly always accompanied by a multitude of corre-sponding time series, which provide a wealth of supplementary information for analyses. For context, consider that the list of interrelated economic indicators is limitless. Thereby, diagnos-tic tools for multivariate time series are of immeasurable import. Motivated by the seminal portmanteau test of Box and Pierce (1970), the statistic of Hosking (1980) enabled diagnosis of the appropriateness of a fitted vector autoregressive moving average (VARMA) model. However, a drawback of Hosking’s test is that it provides no feedback as to which aspects of the fitted model are appropriate and which are not. For instance, time series that are independent of one another may test as being well-fit when modeled jointly. This is particularly problematic consid-ering that a consequence of a data-rich environment is the need for parsimony: some supplemental information is inevitably su-perfluous. Therefore, an ancillary diagnostic to goodness-of-fit testing is that of testing for independence of two (or more) time series. Machinery for these two realms of diagnostic testing has developed in parallel, with notable similarities inherent across methodologies. In this article, we propose a new test for the independence of two (multivariate) time series, which is of the flavor of extant methodology used for goodness-of-fit diagnos-tics. Further, it is a classical problem in econometrics to detect the presence (and direction) of temporal causality across time series. We illustrate simple modifications of our tests of inde-pendence, which yield new statistics for testing one-directional causality among time series.
Prior to introducing our proposed method, we summarize pertinent techniques. The statistic of Box and Pierce (1970) is calculated as the sum (across various lags) of the squared sample autocorrelations of time series residuals, and much of the ex-tensional work is based on the supposition that a weighted sum may be preferable. To begin, Ljung and Box (1978) included a standardizing weight in the summands of the squared sample autocorrelations. Pe˜na and Rodr´ıguez (2002,2006) defined a Toeplitz matrix based on the aforementioned sample autocor-relations and presented a goodness-of-fit statistic derived via the determinant of the resulting matrix. Fisher and Gallagher (2012) proposed a weighted Box–Pierce type test and illustrated that it shares the same underlying probability structure as the determinant-based statistic. Further, Hosking (1980) extended the Box and Pierce test to multivariate time series; as an exten-sion of the work of Hosking, Mahdi and McLeod (2012) define a block Toeplitz matrix via multivariate sample autocorrelations that is akin to that of Pe˜na and Rodr´ıguez (2002).
Motivated by the Box–Pierce test, Haugh (1976) introduced a portmanteau test used for gauging independence between two univariate time series that is calculated as a sum of squared residual cross-correlations. Since the Haugh test was found to lack power (see, Pierce1977, among others), Hong (1996) of-fered a modification wherein a weighted sum (the weighting scheme is determined via a kernel) is used. El Himdi and Roy
© 2015American Statistical Association Journal of Business & Economic Statistics
October 2015, Vol. 33, No. 4 DOI:10.1080/07350015.2014.962699
459
(1997) extended the Haugh method to test for independence of two multivariate time series; the modifications of Hong (1996) are applied to the resulting test within Bouhaddioui and Roy (2006).
Testing for independence of two time series is fundamentally similar to the detection of causality. The well-known Granger causality test (Granger1969) has been modified to account for multivariate series (L¨utkepohl2005; Pfaff2008a). Further, El Himdi and Roy (1997) made simple adjustments to their test of noncorrelation to propose a test for the detection of causality in the sense of Granger (1969).
Herein, we define a nontrivial block Toeplitz matrix in the vein of Pe˜na and Rodr´ıguez (2002) and Mahdi and McLeod (2012) for use in the setting of testing for independence of time series, and we define trace-based and determinant-based test statistics using the resulting matrix. Through computations mo-tivated by those within Fisher and Gallagher (2012), we are able to illustrate that the resulting technique relates to a weighted version of the method of El Himdi and Roy (1997), thereby en-abling direct comparisons to the techniques of Bouhaddioui and Roy (2006). By scrutinizing the mathematical underpinnings of the statistic based on the log-determinant of our matrix, we il-lustrate that the log-based statistic offers an increase in power over the others proposed.
Following introductory concepts and notation presented in Section2, our proposed methodology is presented in Sections3
and4. Evaluations and comparisons are made via a Monte Carlo study provided in Section5, and an application of the proposed methodology is given in Section 6. Supplemental proofs and theory are provided in an appendix.
2. PRELIMINARIES
Consider two multivariate time series, X(1)t and X(2)t , for t =1, . . . , n, where Xt(i) is a length-pi vector for i=1,2.
Specifically, X(ti) = {X1(i,t), X(2i,t), . . . , X(pii),t}
T. We assume that
each series has a VARMA formulation. That is, we impose
t is Gaussian white noise.
Assuming weak stationarity, we define the multivariate (cross) covariances (at lag-h) as
Ŵ(hi,j)=EX(i)
for the observed data, and as
C(hi,j) =EZ(ti)
We are interested in testing the null hypothesis,
H0: SeriesX is nondiagonal) is possible under the null. First, we test against the so-called global alternative hypothesis,
Hm1:C (1,2)
h =0for at least onehthat satisfies−m≤h≤m,
and we will consider other alternatives in Section 4. In the paradigm of goodness-of-fit portmanteau tests, the valuem rep-resents the (absolute) maximum lag that will be considered.
Since the asymptotic distribution of estimates ofC(i,j)
h is more
tractable than that ofŴ(hi,j), we (as is commonly done with time series portmanteau tests) consider the processes of residuals. This process is defined by
Z(ti)=(Xt(i)−µˆ(i))− respectively; these estimators are known to be√n-consistent. The consistency of the parameters implies thatZ(ti)consistently approximatesZ(ti). Therefore, we estimateC(i,j)
Extant approaches for the above objective are paraphrased as follows. To extract the sampling distribution ofC(1,2)
h , El Himdi
and Roy (1997) isolated to the quantity
Q(h)=nvecCh(1,2)TC(20,2)−1⊗C(10,1)−1vecC(1h,2), (3) forh= −m, . . . , m, where vec(A) represents the columns ofA stacked on top of one another for some matrixA, and where
⊗represents the Kronecker product. To test against the global alternative, they used
which is illustrated to observe an asymptotic chi-squared distri-bution underH0. Akin to Hong (1996), Bouhaddioui and Roy
(2006) proposed test statistics of the form
QHκm=
n−1
h=1−n
κ2(h/m)Q(h), (5)
wherein theQ(h) terms are weighted at specific lags via a kernel function,κ(u) (which may be nonzero only foru∈[−1,1]), to increase power; several different kernel functions are suggested. These statistics are illustrated to be asymptotically normal upon standardization.
3. TOEPLITZ MATRICES AND TESTS OF INDEPENDENCE
For the purpose of goodness-of-fit testing, Pe˜na and Rodr´ıguez (2002,2006), Mahdi and McLeod (2012), and Fisher and Gallagher (2012) developed test statistics based on a Toeplitz matrix, which is composed of residual autocorrelation terms. In light of that construct, we develop a Toeplitz matrix, which yields test statistics that connect to those of El Himdi and Roy (1997) and Bouhaddioui and Roy (2006).
Consider the sample residual autocovariances of the expanded seriesχt = {X
underH0. Thereby, the null hypothesis estimate of the residual
covariance (forχt) is
Additionally, the lower triangular Cholesky decomposition of ( ˆC∗
0 are the lower triangular Cholesky
decom-positions of (C(1,1)
0 )−
1and (C(2,2) 0 )−
1, respectively. Further, the
alternative hypothesis estimates of the residual autocovariances (forχt) are the matrices
presumed under both the null and alternative hypotheses) and that the models have been (marginally) well fit.
The residual autocorrelations are now defined via the matrix
may be used to represent the residual cross-correlations of the two series. Also, note thatC(1,2)
Using the residual autocorrelations defined above, we now construct a block Toeplitz matrix as follows:
We are interested in testing ifℜm is statistically different from
the identity matrix.
3.1 Trace-Based Statistics
As in Fisher and Gallagher (2012), we let ℜm denote the
probability limit ofℜmand note that testing whetherℜm=Iis equivalent to testing whether each eigenvalue ofℜmequals one.
Lettingλ1, λ2, . . . λd denote the eigenvalues ofℜm, whered =
(m+1)(p1+p2) is the dimension of the matrix, a quantity that
will likely observe large values under the alternative hypothesis (and small values under the null) is
Qm =n
The above computations imply that
Qm=2 m
h=−m
(m+1− |h|)Q(h). (11)
As our first proposed test statistics, we now define
Q†m=
which are slightly modified versions ofQm. In the above,
Q∗(h)= n
n− |h|Q(h), (13)
which incorporates the modification of Ljung and Box (1978). Since this modification is known to increase power, most of our discussion focuses onQ∗min place ofQ†m.
The statisticQ∗mis a weighted version of the statistic proposed by El Himdi and Roy (1997). It follows from the convergence results provided therein that
Q∗(h)−→D χp21p2, (14) underH0for−m≤h≤m, where
D
−→denotes convergence in distribution and whereχ2
νdenotes a chi-squared random variable
with ν degrees of freedom. Further, Q∗(h) is asymptotically
uncorrelated withQ∗(k) so long ash=k. The convergence of Q∗
m, which is a direct consequence of the result stated in (14) and
distributional properties of quadratic forms, is stated as follows.
Theorem 1. Assume that the observed data,X(1)t andX(2)t for t =1, . . . , n, obey the formulation given in (1) and thatQ∗
mis
as defined in (12). It holds that
where theX2kare independent and identically distributedχp21p2 random variables, and whereak=(m+1− |k|)/(m+1).
The statistic Q∗
m connects directly to QHmκ from (5) while
usingκ(u)=√1− |u|1(|u|≤1). A similar kernel, the Bartlett
ker-nel,κ(u)=(1− |u|)1(|u|≤1) is considered in Bouhaddioui and
Roy (2006). Our trace-based statistic also has connections to the Fejer kernel. The weights match the firstmterms from the Fejer kernel when smoothing the sample periodogram. This decreas-ing sequence is known to lead to asymptotic unbiasedness of the sample periodogram. Further, we choose to scale bym+1 withinQ∗m so as to ensure that the maximum weight given at any lag is 1, which enables the most direct comparison toQHκ
m.
3.2 Procedures Based on the Determinant
After defining a Toeplitz matrix based on residual autocor-relations, Pe˜na and Rodr´ıguez (2002) provided a test statistic based on the matrix determinant. In that vein, we propose the following determinant-based statistic for testing for indepen-dence ofX(1)t andX(2)t .
The Toeplitz matrices of Pe˜na and Rodr´ıguez (2002,2006) and Mahdi and McLeod (2012) have direct representations as sample autocorrelation matrices. Herein, we regard our matrix as a pseudo-sample autocorrelation matrix since the matrix is distorted through the placement of zeros. Consequentially, the matrix is no longer assured to be nonnegative definite, hence the determinant may be negative (and we must take the absolute value). The frequency of this occurrence is studied in Section
5; the phenomenon is problematic only whennis small andm
is comparatively large. However, ℜm, the probability limit of
ℜm, is an autocorrelation matrix (Brockwell and Davis1991)
so long as all model assumptions are correct. Thereby, ℜm is
nonnegative definite under bothH0andHm1.
Further, Pe˜na and Rodr´ıguez (2006) and Mahdi and McLeod (2012) defined test statistics based on the log of the determinant of their respective Toeplitz matrices. Here, we let
m= −nlog||ℜm||/(m+1). (16)
The large sample behavior of the above statistics is illustrated in the following.
Theorem 2. Assume that the conditions of Theorem 1 hold, and letQ∗
m,Dm, andm be as defined in (12), (15), and (16),
respectively. It follows that
m−Q∗m=Op(n−1), and Dm−m=Op(n−1),
ifH0is true, and consequentially,
where theXk2are independent and identically distributedχp2 1p2
random variables, whereak=(m+1− |k|)/(m+1).
Within the proof of Theorem 2, we use the following lemma, which is itself proven in the Appendix.
Lemma 1. Letλ1, . . . , λdrepresent the eigenvalues ofℜm. It
Proof of Theorem 2. We begin by establishing the result form.
This result is similar to the key finding of Mahdi and McLeod (2012); however, our line of proof differs from that taken by those authors, although our method of proof is also applicable in their setting.
nis nonzero only when at least one
λk<0. When considering a random matrix that converges in
probability to a matrix of constants, the eigenvalues of the for-mer will converge in probability to the eigenvalues of the latter (Eaton and Tyler1991). Since the eigenvalues ofℜmare
nonneg-ative under bothH0andHm1, we see limn→∞P(an|An|>0)=0
for any desired convergence ratean. Hence,Anis asymptotically
negligible under both hypotheses.
Next, we use the above finding and the result from Lemma 1 to establish
whereQmis defined in (8). The convergence of the remainder
term in the final line is at a rate ofn−1as a consequence of the
observation that√n(1−λk)=Op(1) for allk, which follows
from the√n-consistency of theλkas estimators of unity under
H0. Next, the relations in (11) and (12) give the result.
To establish the result forDm, consider that
which, after applying the limit distribution ofm, concludes the
proof.
The expansions ofmused in the above proof can also be
used to illustrate thatmprovides a nonnegligible increase in
power over the other statistics we have proposed (so long as critical values derived from the asymptotic distribution is used for all tests). This is stated formally as follows.
Theorem 3. As n→ ∞, it holds that 0<m−Q∗m=
Op(n), and likewise, 0<m−Dm=Op(n) whenHm1 is true.
Thus, under the conditions of Theorems 1 and 2, the statisticm
is asymptotically more powerful than theQ∗
mandDmstatistics.
Proof. From (18), we can express the discrepancy between
Anis negligible in comparison toBnunderHm1 as well. Further,
Bn is asymptotically negligible when H0 is true, enabling us
to use the same critical values for both test statistics. When the alternative is true, the eigenvalues will thereby converge to constants, at least one of which is not unity. Consequentially, An+Bn=Op(n) and is positive asn→ ∞underHm1. Hence,
we conclude thatm is a more powerful test statistic thanQ†m
in the limit. SinceQ∗
m−Q
†
m=Op(1) under the alternative, we
conclude thatmoffers an improvement in power overQ∗mas
well.
3.3 Approximations of the Limit Distribution
Recall that theH0limit distribution of the matrix-based
statis-tics (see Theorems 1 and 2) is a weighted sum of 2m+1 iid χ2 random variables each withp1p2 degrees of freedom.
Fol-lowing the arguments in Box (1954), we can approximate this distribution via a gamma random variable with shape and scale based on cumulant matching. We note the first two cumulants
of this distribution are:
The shape and scale are calculated asK2
1/K2 andK2/K1,
re-spectively.
Should one wish to side-step asymptotic approximations en-tirely, the distribution can be extracted through purely com-putational methods. Imhof (1961) provided the foundational work for such approaches. Lin and McLeod (2006) provided the Monte Carlo algorithm for determining the critical points andp-values for the distribution of statistics based on a Toeplitz matrix in the time series goodness-of-fit realm. A similar result should hold in the case of testing for independence, but we use the gamma approximation herein.
3.4 Connections to Likelihood Ratio Tests
The statistics proposed above can be motivated through likelihood arguments. Define the vector
t= {Z
matrix oft(assumingH0holds) can be approximated viaC
∗
m,
which is a block diagonal matrix where each diagonal block is given byC∗
0. Likewise, the covariance matrix oft(assuming
the alternative is true) can be approximated byCm, which is a
version ofℜm that containsCh in place of eachRh. Further, letL∗m denote the lower triangular Cholesky decomposition of
(C∗
m)−
1
, and note thatL∗
m is block-diagonal where each block
is given byL∗
0.
In multivariate settings, Gaussian likelihood ratio tests are often calculated based on the log of the ratio of||1||to||0||,
where0is a covariance matrix estimated under the null and1
is a covariance matrix estimated under the alternative. Noting that
||Cm||
||C∗m||= ||
Cm(C∗m)−1|| = ||(Lm∗)TCmL∗m|| = ||ℜm||,
we motivate the statisticmas being a pseudo-likelihood ratio
statistic (whose underlying process is given byQ∗
m).
4. TOEPLITZ MATRICES AND TESTS OF CAUSALITY
It may be desirable to consider an alternative hypothesis that is more restrictive in terms of which lags are allowed to con-tain nonzero cross-correlations. In that vein, El Himdi and Roy (1997) considered the objective of testing for cross-correlation at a specific lagm. To elaborate, they testH0against the single
lag alternative
H1,m:C(1m,2)=0.
As a test statistic, they proposedQ∗(m), which is defined in (13)
with limiting behavior illustrated in (14).
Toeplitz matrices may also be used to test the single lag alternative. For instance, considerH1,0—the alternative of
in-stantaneouscausality, which is equivalent to applying the global test withm=0. Noting thatℜ0=R0, we may test againstH1,0
The following proposition, which is proven in the Appendix, implicates that statistics that useℜ(m) will be devoid of some
of the issues encountered by those which useℜm.
Proposition 1. Under the conditions of Theorem 1,ℜ(m) is
nonnegative definite. Therefore, |ℜ(m)| ≥0, where | · |
indi-cates a matrix determinant.
Note that the test based on the trace ofℜ(m)2is equivalent to
Q(m) as defined in (3). We consider the matrixℜ∗(m), which is
defined likeℜ(m) but with the off-diagonal blocks being
mul-tiples by √n/(n− |m|), since Q∗(m)=n[tr{ℜ∗(m)2
} −p1−
p2], whereQ∗(m) is defined in (13). Our proposed statistics for
testingH0againstH1,minclude
D∗(m)=n(1− ||ℜ∗(m)||) and ∗(m)= −nlog||ℜ∗(m)||.
(19) Furthermore, the above statistics are asymptotically equivalent toQ∗(m) underH0, which yields the following.
Corollary 1. Assume that the conditions of Theorem 1 hold. It follows that
D∗(m)−→D X2,
and ∗(m)−→D X2,
if H0 is true, where X2 is aχp21p2 random variable. Further,
it holds that∗(m) is a more powerful statistic asymptotically
thanD∗(m) andQ∗(m).
The proof of the convergence results in the above corollary directly mimics the proof of Theorem 2. Likewise, the proof of the final statement in the above follows the same rationale as the proof of Theorem 3.
Although Theorem 3 states that∗
m offers more power than
other statistics that observe the same limit distribution (under
H0), theQHmκ statistic may prove most powerful in the global
setting by using a kernel function that is optimal under a specific cross-correlation structure. However, with respect to the single lag test statistics, we emphasize the optimality of the ∗(m) statistic over other previously proposed tests designed for the single lag setting (since in that setting there is no selection of a kernel).
Continuing, we may wish to ascertain whether X(1)t causes X(2)t , or vice versa. Therefore, we consider the alternatives of one-way causality as follows:
which implies thatX(1)t causesX(2)t . To test against these alter-natives, El Himdi and Roy (1997) proposed
QHm+=
To apply the methodology of Section 3 to tests involving one-way causality, we define
incorporate the additional restrictions that are implicated by the one-way alternatives. Further, define
Computations similar to those used in proving (9), (10), and Theorem 2 will provide that each of the statistics in (21) are asymptotically equivalent to
and that the statistics defined in (22) are asymptotically equiva-lent to
statistics (instead ofm+1) to assure the maximum weight ap-plied to any lag is one. One could also develop versions of (23) and (24) that incorporate a kernel-based lag-weighting structure in the vein of (5); however, such pursuits are not entertained further here.
The following corollary is now evident, which like Corollary 1, can be proven via the arguments used to illustrate Theorems 2 and 3.
Corollary 2. LetTm±denote any of the quantities in (21)–(24). Assuming that the conditions of Theorem 1 hold, it follows that underH0,
where theXk2are independent and identically distributedχp21p2 random variables, and whereak=(m+1−k)/m.
Table 1. Univariate, bivariate, and trivariate VARMA model parameter descriptions. VAR models are also used from the definitions below. For
The asymptotic distribution of the one-way statistics can be approximated in the same manner as that of the statisticsQ∗
m,
In this section, the finite sample performance of each of the methods discussed in Sections 2–4 is scrutinized via simula-tion. For added rigor, we provide comparisons to the Wald-type test of instantaneous causality and multivariate Granger causal-ity test described in Pfaff (2008a) (for further discussion, see L¨utkepohl2005); these procedures necessitate that two multi-variate time series initially be modeled jointly as one VAR. We use the version of these tests implemented within thevars software package (Pfaff2008b). In the simulations presented below, we explore an assortment of sample sizes, null and al-ternative distribution models, andα-levels. To summarize, the findings demonstrate that the proposed methods provide im-provements (in terms of power and observed Type I error) over extant techniques—the result of Theorem 3 is demonstrated in the process.
5.1 Experiment Description
Data are generated from different VARMA models based on the set of parameters inTable 1: a univariate series based on the parameters in Hong (1996) (Model A), a bivariate based on parameters from El Himdi and Roy (1997) (Model B), and a trivariate VARMA based on Reinsel, Basu, and Yap (1992) (Model C).
Any dependence structure between series and the nominal level (α) is specified in the specific studies. Further, the scheme detailed as follows is used for prewhitening:
1. The two series,X(1)t andX(2)t , are generated based on the model specifications in Table 1, with normal innovations containing possible cross-correlation as specified below.
2. For each marginal seriesX(ti), autoregressions are fit of or-derp≤12 chosen via Akaike information criterion (AIC) based on the joint series{X(1)t ;X(2)t }. Cross-correlations of the residuals series{Z(1)t ;Z(2)t }are calculated by computing the appropriateR(1,2)
h values.
3. The test statistics under study were calculated.
4. For each model, each series lengthn, each maximum lagm, and each nominal level, the empirical frequencies of rejec-tion of the null hypothesis of noncorrelarejec-tion were obtained from 10,000 realizations based on the theoretical asymptotic distribution.
To generate innovations with cross-correlation, we simulate an error series of dimensionp1+p2following a VAR or VMA
process. The error series is then deconstructed into two series of dimensionp1 andp2, which yield the data seriesX(1)t and
X(2)t , respectively. Our VAR(1) error series are generated based on the parameters and ones, respectively, δ is a perturbation parameter that en-ables variation of the strength of the interseries autocorrelation, and(1)and(2)are marginal covariance matrices taken from
Table 1. We note that due to prewhitening (Step 2 above), the underlying residual cross-correlation is difficult to predict and may behave in an MA-type fashion, that is, the recursive de-cay of an AR-type correlation structure will not necessarily be present in the residual cross-correlations.
In several of our simulations, we wish to control the cross-correlation more directly; this can be accomplished by gener-ating errors from a VMA model. Herein, we aim to impose cross-correlation between the error series at lags 0 and 4. There-fore, we generate error series using a VMA(4) process with parameters
Table 2. Empirical size and power (in %) of the global tests at nominal level 5% based on 10,000 realizations for various truncation levels (m) and sample sizes (n) of the log-determinant,m, determinant,Dm, squared trace,Q∗
m, Haugh-type,QHm∗, and kernel weighted,QH κ
musing the Daniell (Dan) and Bartlett-Priestley (B.P.) tests. Both marginal series follow the ARMA(1,1) from Model A, and under the alternative, cross-correlation is imposed using errors obeying (26) withδ=0.1. The unlabeled column of each study reports the number of times (out of
10,000) the matrixℜmhad negative eigenvalues present
Empirical size Empirical power
n m m Dm Q∗m QHm∗ Dan B.P. m Dm Q∗m QHm∗ Dan B.P.
4 0 7.8 6.3 5.7 5.4 7.7 7.5 0 15.7 13.7 12.6 11.5 15.8 15.0
100 5 0 8.6 6.5 5.5 5.3 8.0 7.7 0 17.1 14.1 12.2 10.7 15.7 15.0
8 0 11.7 7.7 5.4 5.0 7.7 7.4 5 20.0 14.9 11.7 9.2 14.1 14.1
12 48 16.7 10.1 5.2 5.0 6.9 6.8 114 26.8 18.1 10.8 8.3 14.1 13.0
4 0 5.6 5.1 5.0 5.0 6.9 6.8 0 32.0 30.8 30.1 31.1 33.4 32.0
300 6 0 6.4 5.8 5.4 5.1 7.5 7.3 0 34.3 32.6 31.7 26.8 32.6 33.7
9 0 6.9 5.9 5.3 5.2 7.0 6.8 0 34.5 31.6 30.1 21.9 33.2 33.4
17 0 9.3 7.0 5.3 5.1 6.8 6.8 1 35.8 30.3 26.3 17.2 31.4 28.9
4 0 5.5 5.3 5.2 5.0 7.1 7.0 0 51.4 50.6 50.2 54.3 51.4 50.7
500 6 0 5.6 5.3 5.1 5.3 6.9 6.7 0 55.2 54.1 53.6 47.5 51.2 55.1
10 0 6.1 5.5 5.2 5.3 6.6 6.6 0 54.9 53.1 51.8 38.1 56.3 55.2
19 0 7.4 6.0 5.1 5.2 6.3 6.2 0 50.2 45.8 42.9 27.0 49.3 45.0
and j =0(p1+p2)×(p1+p2) for j =1,2,3. The residual
se-quences Z(1)t and Z(2)t of two series should exhibit cross-correlation at lag 0 and lag 4 with a magnitude depending onδ. The above model can be modified to isolate all cross-correlation at lag 4 by making0block-diagonal.
5.2 Results for Global Testing
To begin, we study the so-called global tests (which test for a general lack of independence between the two series) out-lined in Sections2and3. The first study considers the empirical size and power at nominal levelα=5% when both series fol-low the ARMA(1,1) process of Model A (Table 2). To examine performance in higher dimensions, we also provide results in which each series is a VAR(1) process from Models B and C, respectively, usingα=1% (Table 3). In both power studies, the
error series is generated following the moving average model shown in (26) with cross-correlation at lags 0 and 4 based on the perturbation parameterδwith values 0.10 and 0.05, respec-tively. We find that the dimension of the series can influence the rate of convergence of the statistics to their null distribution; thereby, the latter setting will necessitate larger values ofnfor comparable results.
The proposed statistics based on the trace of the squared cor-relation matrix,Q∗
m, the determinant,Dm, and log-determinant,
m, are compared to the kernel weighted statistic,QHmκ, from
Bouhaddioui and Roy (2006) and the Haugh-type statistic, QHm∗ =mh=−mQ∗(h) (which is the slightly modified version ofQHm) as introduced by El Himdi and Roy (1997). For brevity,
we only include the Bouhaddioui and Roy (2006) tests using the Daniell and Bartlett-Priestley kernels since they appeared to be the best performing in the simulations in Hong (1996) and
Table 3. Empirical size and power (in %) of the global tests at nominal level 1%. The first series follows a VAR(1) from Model B and the second follows a VAR(1) from Model C. Under the alternative, errors are generated via (26) withδ=0.05. Otherwise, the setup and format of
this table are the same asTable 2
Series 1 Generated via Model B, Series 2 Generated via Model C
Empirical size Empirical power
n m m Dm Q∗m QHm∗ Dan B.P. m Dm Q∗m QHm∗ Dan B.P.
4 0 1.7 0.9 1.1 0.9 2.1 2.0 0 5.8 3.4 3.9 8.4 5.4 5.1
500 6 0 2.4 0.9 1.2 1.0 1.9 1.9 0 10.5 5.0 6.0 7.0 5.7 7.6
10 0 4.3 0.9 1.2 1.1 1.8 1.9 0 16.8 5.3 6.8 5.7 9.2 9.4
19 0 16.1 0.6 1.1 1.1 1.7 1.5 1 38.6 3.7 6.0 3.8 9.0 8.1
4 0 1.4 1.1 1.2 1.1 2.1 2.0 0 14.1 11.5 12.0 32.4 10.8 11.6
1000 7 0 1.7 1.0 1.1 0.9 1.9 1.8 0 26.3 19.3 20.6 21.9 17.2 24.0
12 0 2.7 1.0 1.2 1.2 1.9 1.7 0 34.6 20.2 23.0 14.8 29.1 29.1
24 0 7.4 0.6 1.0 1.2 1.4 1.5 0 44.2 13.0 17.7 8.7 24.2 20.4
4 0 1.4 1.1 1.2 1.2 2.3 2.2 0 46.9 44.6 44.8 87.4 27.9 35.4
2000 8 0 1.5 1.1 1.2 1.2 1.8 2.0 0 76.7 72.6 73.2 69.0 66.7 77.7
14 0 1.7 0.9 1.1 0.9 1.5 1.5 0 78.4 70.5 72.0 49.9 80.5 78.1
29 0 3.5 0.7 1.0 1.1 1.5 1.3 0 73.6 50.7 56.0 26.8 66.6 58.1
Figure 1. Power of global tests atα=1% (left) andα=0.01% (right) asδparameter varies withn=500 andm=4. Series 1 follows a VAR(1) from Model B and Series 2 follows a VAR(1) from Model C. Error series are generated using (25) with correlation decaying with the lag.
Bouhaddioui and Roy (2006). In addition to the frequency of rejections, we also record the number of times our constructed matrix, ℜm in (7), has negative eigenvalues (therefore, the
absolute value of the determinant may be required). With re-spect to truncation levels, we considerm=4 and those based on the rates [lnn], [3n0.2], and [3n0.3] from Hong (1996) and
Bouhaddioui and Roy (2006). We see the testQ∗
mand that from El Himdi and Roy (1997)
have satisfactory size across bothmandn. The statistics from Bouhaddioui and Roy (2006) are close to the nominal level but tend to be a little liberal. Themstatistic has liberal Type I error
rates for smallernand largerm, but we see the Type I errors improve asnincreases—theDmstatistic seems to offer a more
stable (but less powerful) option. Likewise, the occurrence of
ℜm having negative eigenvalues is only seen whenn is small
andmis large. It is noteworthy that even though the size ofm
is generally greater thanDmandQ∗m, all three can demonstrate
improved Type I error rates compared to the statistics using the kernel function in many of the cases. As explanation, we find that, generally, the Gamma approximation seems to be a better approximation than standardizing and using a normal distribu-tion cutoff; this observadistribu-tion is in line with the findings of Gal-lagher and Fisher (in press). Overall the proposed methods are comparable to those from the literature and are more powerful in some cases; however, the comparative power of lag-weighted portmanteau tests (e.g., theQ∗
m,QHm∗, andQHmκtests) is highly
dependent upon location (by lag) of the true cross-correlation. To offer detailed comparisons of the three matrix-based tics proposed herein, we explore the power of those three statis-tics as a function of the perturbation parameterδwhile fixingn
andm. For this study, the error series was generated following the VAR(1) model from (25) where the amount of cross-correlation is determined byδ; results are provided inFigure 1. As impli-cated by Theorem 3, themstatistic is more powerful thanQ∗m;
however, in accordance with (18), the “more incorrect”H0 is,
the greater discrepancy between the two statistics will be. Since in this simulation setup, smaller values ofαwill require larger values of δ (i.e., a more incorrect H0) to maintain detection
power, we useα=1% andα=0.01%. As seen inFigure 1, the power of theDmstatistic is comparable to that of statistic based
on the trace of the squared matrix. However, the power function of the statistic using the log of the determinant demonstrates
an increase in power (around 9% at δ=0.08 when α=1% and 14% atδ=0.10 whenα=0.01%), which is in line with Theorem 3. Furthermore, although not visualized in the graph, we note that atδ=0 all the statistics reject the null hypothesis close to the nominal level.
5.3 Cross-Correlation at a Specific Lag
Here, we study the performance of the proposed determinant-based statistics in Section4versus the quadratic form test from El Himdi and Roy (1997). Recall that the statistic based on the trace is equivalent to El Himdi and Roy (1997) for the single lag case. As in El Himdi and Roy (1997), we can use the test for a correlation at a single lag over multiple lags to determine the location of causality. To investigate, we consider a simulation similar to that fromFigure 1to explore the residual cross-correlation as well as the aforementioned statistics. The first series is a VAR(1) from Model B, the second is a VAR(1) from Model C, and the error series are derived using the VAR formulation of (25).
The left panel ofFigure 2provides the empirical power of the log-determinant-based test, ∗(m) and the quadratic form test,Q∗(m), at nominal levelα=5% withn=100. Following prewhitening, residual cross-correlation exists mainly at lags 1, 2, and 3; therefore, the detection rates are approximately at the nominal level for the other lags. The test based on the determi-nant,D∗(m), is excluded here as its performance matches that
ofQ∗(m). Overall we see both statistics report satisfactory size
close to the nominal level of 5% and that the test based on the log-determinant improves over that from El Himdi and Roy (1997) in terms of power. Note how quickly the cross-correlation of the residual series appears to decay; this suggests that statistics that weight large lag cross-correlations weakly would perform well. The right panel ofFigure 2illustrates investigations into the performance of the single-lag tests under H0 (δ=0) for
variousnwhile imposingm=0. The Wald-type instantaneous causality test from Pfaff (2008a) is also included as well as the determinant test (D∗(0)). The Wald method is applied by first
fitting a VAR to both series jointly where the order is determined by AIC. It is seen that all tests provide acceptable Type I errors and that the rejection rates become nearly identical for largen.
Figure 2. Power of ˆ∗(m) (gray) andQ∗(m) (black) withn=100 for causality at individual lags at nominal size 5% when the first series is generated from a VAR(1) of Model B, the second a VAR(1) from Model C, and the error series obeys (25) withδ=0.20 (left); size of four tests of instantaneous causality under the same model for variousn(right) with 95% bounds given for the acceptable range around the target value of α=1%.
We also present a study similar to the study onδinFigure 1. Specifically, two VAR(1) series from Model B fromTable 1are generated, and we set the cross-correlation between the error two series to occur only at lag 0 based on the covariance matrix
0=
B0 δD
δD B0
,
whereB0is defined inTable 1andDis a diagonal matrix with
{1,2}along the diagonal. This is quite similar to the alternative hypothesis in the power simulations in Bouhaddioui and Roy (2006). Letting theδterm vary for sample sizen=100, the sin-gle lag statistics are calculated form=0 in addition to the Wald test described above;α=1% andα=0.01% are used. Results are shown in Figure 3. There is very little difference between the quadratic form statistic,Q∗(m) and the determinant-based statistic,D∗(m) in terms of power; whereas, the statistics based
on the logarithm of the determinant,∗(m) shows substantial improvement (around 5% at δ=0.1 for α=1% and 12% at δ=0.14 forα=0.01%) over the closest competitor for sev-eral δ values; in accordance with Corollary 1, this method is always the most powerful.
5.4 One-Sided Causality
Via studies analogous to those within Section5.2, we explore the performance of the one-sided statistics, which may be used to determine causality of one series on another. First, we look at the empirical size and powers in a study, which mimics the format ofTables 2and3. Here, the weighted statistics based on a kernel function are excluded since they have not been formally introduced into the literature, although it appears the results from Bouhaddioui and Roy (2006) are valid in a one-sided construction. The three proposed statistics,+
m,Dm+, and
Q∗+
m , are compared to the one-sided causality test from El Himdi
and Roy (1997),QH∗+
m and the Granger causality test described
in Pfaff (2008a), GR. Here, the Granger test is applied by fitting a VAR(m) jointly to both series; so long asmis greater than or equal to the true VAR order for both series (underH0), the
underlying model will be captured (although perhaps overfit), and the test is theoretically viable. We isolate our discussion to tests of positive lag correlations while noting that similar results hold for the negative lags.
Table 4reports the empirical size and power of the one-sided tests atα=1% of the statistics when the two series are gen-erated in the same manner as in the simulations presented in
Table 3, except that for the power study, the underlying
inno-Figure 3. Power for single lag tests, atα=1% (left) andα=0.01% (right), as theδparameter varies while usingm=0. Each series follows a VAR(1) from Model B withn=100.
Figure 4. Power, atα=5% (left) andα=0.1% (right), of the one-sided tests asδparameter varies. The first series follows a VARMA(1,1) from Model B, the second series follows a VARMA(1,1) Model C withn=500, and cross-correlation is isolated to lag 4. Test statistics are calculated usingm=6.
vation sequence is generated using the VAR(1) model shown in (25). It should be noted that compared toTable 3, the size of the log-determinant statistic has improved. For this particular alternative hypothesis, the log-determinant test is the most powerful in many cases while exhibiting acceptable Type I er-rors. Furthermore, we note that the constructed matrix never has negative eigenvalues. The Granger test displays poor Type I errors for larger values ofm, making power comparisons to the other tests inappropriate. The Granger test is more powerful than the test of El Himdi and Roy (1997) but appears less power-ful than the proposed methods. Generalized comparisons of the matrix-based methods to the Granger test are complicated by differences in the underlying (lag-wise) weighting structure of the tests. However, the matrix-based methods seem preferable when correlation is centered toward lower lag values since those statistics weight the lower lags more heavily.
Finally, we look at the power of the proposed methods as a function of δ for the one-sided tests. The El Himdi-Roy and Granger tests are excluded on account of their aforemen-tioned discrepancies from the matrix-based methods. The data series are generated as a bivariate VARMA(1,1) and a trivariate
VARMA(1,1) from Models B and C, respectively, with sample size n=500. Cross-correlation is isolated to lag 4 (with the parameter δ governing the strength) by the use of the VMA formulation discussed following (26). Further, the three pro-posed statistics for the one-sided hypothesis are calculated with m=6.
We note that, as seen inFigure 4, theD+
mstatistic outperforms
the trace-based method in this setting. Consistent with the other studies herein and Corollary 2, we see the logarithm based statis-tic is the most powerful (around a 5% increase atδ=0.18 for α=5% and a 9% increase forδ=0.26 andα=0.1% over the
D+
mstatistic) and that the statistic based on the log-determinant
appears to dominate the one based on the trace of the square (around a 7% increase atδ=0.18 forα=5% and a 16% in-crease forδ=0.26 andα=0.1% over theQ∗+m statistic).
6. DATA ANALYSIS
Here, we use the methodology of the previous sections to address questions of particular interest within the economic
Table 4. Empirical size and power (in %) for the one-sided tests based on 10,000 realizations for different truncation levels (m) and sample sizes (n) of the log-determinant (+
m), determinant (D+m), squared trace (Q∗+m), El Himdi-Roy (QHm∗+), and Granger (GR) tests. Data are generated in the same manner as in the simulations provided inTable 3with the exception that the error series are generated using the VAR(1) model of (25) when data are simulated underHm1+. The unlabeled column of each study reports the number of times (out of 10,000) the matrix
ℜ+
mhad negative eigenvalues present
Empirical size Empirical power
n m +
m D+m Q∗+m QHm∗+ GR +m D+m Q∗+m QHm∗+ GR
4 0 1.1 0.9 0.9 0.8 1.2 0 29.8 27.2 26.8 20.5 23.3
500 6 0 1.3 0.8 0.9 0.8 1.2 0 29.2 25.1 24.9 14.8 17.6
10 0 1.7 1.0 1.0 0.9 1.6 0 27.1 20.1 20.2 9.3 13.9
19 0 3.7 0.8 0.8 0.8 5.6 0 28.1 12.9 13.5 5.7 24.8
4 0 1.1 1.0 1.0 1.0 1.2 0 76.3 74.9 74.6 65.4 67.7
1000 7 0 1.1 0.9 0.9 0.9 0.9 0 73.0 70.2 70.2 48.0 50.4
12 0 1.4 1.0 1.0 0.8 2.3 0 65.4 60.1 60.3 31.2 51.2
24 0 2.2 0.9 1.0 0.8 7.4 0 55.2 41.2 42.1 17.3 66.2
4 0 1.3 1.2 1.2 1.1 1.2 0 99.5 99.4 99.4 98.7 98.7
2000 8 0 1.4 1.2 1.2 1.1 1.4 0 99.1 99.0 99.0 93.3 93.7
14 0 1.3 1.1 1.1 0.9 3.2 0 97.4 96.9 96.9 80.9 96.0
29 0 2.0 1.1 1.2 1.0 9.4 0 92.2 88.5 88.8 53.1 98.0
Table 5. Empiricalp-values for tests of independence with various values ofm
m m Dm Q∗m QHm∗ Daniell Bart.-Prie.
Inflation versus exchange rates
1 0.000417 0.000532 0.000587 0.001016 0.000011 0.000003
2 0.000352 0.000501 0.000599 0.003392 0.000000 0.000001
4 0.000685 0.001144 0.001477 0.005135 0.000012 0.000072
6 0.000653 0.001356 0.001772 0.006762 0.000145 0.000305
10 0.000519 0.001763 0.002764 0.004630 0.000686 0.000997
15 0.000205 0.001596 0.003182 0.012219 0.001215 0.001575
20 0.000119 0.002175 0.005291 0.057400 0.001717 0.003500
Imports/exports versus exchange rates
1 0.401585 0.415667 0.411030 0.211478 0.848043 0.790215
2 0.265605 0.291941 0.290050 0.197133 0.530804 0.377778
4 0.080519 0.115158 0.117805 0.013915 0.258177 0.164003
6 0.018617 0.043139 0.044571 0.030823 0.112154 0.042974
10 0.015380 0.067845 0.068006 0.308076 0.030458 0.045079
15 0.034496 0.217566 0.220956 0.780234 0.068707 0.194350
20 0.036477 0.380745 0.419541 0.859244 0.199349 0.423512
literature. Specifically, we analyze data pertaining to the rate of inflation, imports/exports, and exchange rates within the U.S. economy. We consider monthly data ranging from January 1971 to September 2012 (n=501). Data were obtained from the OECD iLibrary at stats.oecd.org. Seasonal adjustments were previously applied to the data as necessary.
We let
X(1)t =INFt, X(2)t =
(1−B) ln EXPt
(1−B) ln IMPt
,
and
X(3)t =
⎡ ⎢ ⎣
(1−B) ln CANt
(1−B) ln JAPt
(1−B) ln UKt
⎤ ⎥ ⎦,
where INF, EXP, and IMP denote inflation (%), exports, and imports, respectively. Further, CAN, JAP, and UK represent the exchange rate between the U.S. dollar and the Canadian dollar, Japanese yen, and British pound, respectively. Log transfor-mations and differencing are performed as needed to achieve stationarity, whereB denotes the backshift operator. Each se-ries is marginally fit based on AIC as follows: an AR(13) for X(1)t , a VAR(6) toX(2)t , and VAR(3) toX(3)t , respectively. The test of Hosking (1980) suggests that each model is adequate for the given series. Of significant importance is determining the need for a cointegration approach to modeling these multivariate series.
To evaluate whether these series behave independently of one another, we apply the matrix-based methodology of Section3. That is, we calculate them,Dm, andQ∗mstatistics, which are
defined in (16), (15), and (12), respectively. We also consider the Haugh-type statisticQHm∗ from El Himdi and Roy (1997) and the QHmκ statistic of (5) for the Daniell and Bartlett-Priestley kernels. Results are provided inTable 5.
Results for inflation versus imports/exports are excluded from
Table 5since each test of this pair of series rejects at significance levelα=0.0001. Also, we see clear evidence of a relationship
between inflation and exchange rates. The kernel-based tests for these series are consistently the most significant, which is likely a result of causality being centered at or near lag 0. Further, evi-dence of a relationship between trade volume (imports/exports) and exchange rates is more dubious. When examining test statis-tics for these two series, we see circumstances where the matrix-based statistics are more significant than the kernel-matrix-based ones; this is in line with the findings of our simulation study, thereby indicating the possibility of long-run causality. Further, in all circumstances, we see evidence of the purported lack of power of the Haugh-type statisticQH∗
m.
Now that a lack of independence between these three series has been established, it is prudent to investigate the nature of the inherent causality by considering unidirectional and single lag tests—such causal relationships between these economic indicators have been thoroughly scrutinized in the literature. For example, the Marshall–Lerner condition, which presumes that trade volume reacts to changes in exchange rates, has been a particularly controversial topic (see Rose1991; Abeysinghe and Yeok1998; Boyd, Caporale, and Smith2001; Mahmud, Ul-lah, and Yucel2004, among others). The presence of causality between trade quantities and inflation has also been analyzed, with several researchers providing evidence that inflation influ-ences exports and imports (Lovasy1962; Longmire and Morey
1983). Further, the relationship between inflation and exchange rates has also been highly debated. Several authors argue that inflation reacts to changes in dollar values (Kim1998), whereas others argue against the presence of such an effect (Hafer1989; Manning and Andrianacos1993).
To examine the interseries relationships, we calculated the
∗(m) statistic of (19) to test for causality at specific lags for each combination of series. Comparisons are made to theQ∗(m) test of El Himdi and Roy (1997), and results are shown in
Figure 5. Additionally, the one-way test statistics of Section4
were applied with various values ofm. TheQ∗−
m andQ∗+m tests of
El Himdi and Roy (1997) and the aforementioned Granger tests are included for comparison against the matrix-based method-ology. Results for the one-way tests are given inTable 6.
Figure 5. Tests for causality at individual lags across the combinations of the three series. The lagmis indicated on the horizontal axis.
Table 6. Empiricalp-values for tests of one-directional causality with various values ofm
m −
m Dm− Q∗−m QHm∗− Granger− +m D+m Q∗+m QHm∗+ Granger+
Inflation versus imports/exports
1 0.0000 0.0000 0.00000 0.00000 0.00000 0.0310 0.0314 0.0316 0.0316 0.2095 2 0.0000 0.0000 0.00000 0.00000 0.00000 0.0417 0.0426 0.0428 0.0776 0.5035 4 0.0000 0.0000 0.00000 0.00000 0.00031 0.0081 0.0088 0.0097 0.0046 0.1264 6 0.0000 0.0000 0.00000 0.00000 0.00057 0.0061 0.0069 0.0083 0.0214 0.1990 10 0.0000 0.0000 0.00000 0.00004 0.00116 0.0100 0.0121 0.0152 0.0373 0.3546 15 0.0000 0.0000 0.00000 0.00003 0.00000 0.0124 0.0165 0.0214 0.0474 0.0153 20 0.0000 0.0000 0.00000 0.00011 0.00000 0.0148 0.0216 0.0290 0.1510 0.0030
Inflation versus exchange rates
1 0.4391 0.4397 0.4394 0.4394 0.2530 0.0078 0.0080 0.0082 0.0082 0.0733 2 0.4885 0.4902 0.4904 0.5016 0.5670 0.0076 0.0081 0.0083 0.0180 0.0414 4 0.3636 0.3694 0.3728 0.0970 0.3723 0.0245 0.0267 0.0269 0.1312 0.0812 6 0.1708 0.1803 0.1843 0.0829 0.0284 0.0540 0.0595 0.0601 0.1548 0.0699 10 0.0605 0.0720 0.0771 0.0103 0.0131 0.1317 0.1487 0.1508 0.3716 0.2271 15 0.0247 0.0361 0.0412 0.0335 0.0002 0.1701 0.2036 0.2088 0.3110 0.1024 20 0.0206 0.0362 0.0433 0.1120 0.0030 0.1826 0.2350 0.2458 0.4137 0.3089
Imports/exports versus exchange rates
1 0.0786 0.0804 0.0815 0.0815 0.2093 0.1824 0.1847 0.1849 0.1849 0.0287 2 0.2136 0.2195 0.2206 0.4383 0.4418 0.0639 0.0676 0.0680 0.0398 0.0028 4 0.3156 0.3316 0.3293 0.3125 0.4870 0.0097 0.0123 0.0132 0.0014 0.0008 6 0.2497 0.2772 0.2764 0.2840 0.2886 0.0026 0.0042 0.0046 0.0070 0.0033 10 0.2085 0.2631 0.2631 0.3733 0.0359 0.0085 0.0168 0.0175 0.2217 0.0043 15 0.2302 0.3332 0.3225 0.5916 0.0079 0.0635 0.1224 0.1241 0.7103 0.0008 20 0.2521 0.4155 0.3972 0.6838 0.0115 0.1525 0.2945 0.3164 0.7819 0.0000
The results indicate the presence of instantaneous causal-ity except when comparing trade volume to exchange rates. Additionally, there is evidence of a bi-directional relationship between inflation and trade volume. We note that the dominant mechanism appears to be that changes in the rate of inflation causes changes in trade volume. However, there is no convincing evidence of directional causality between inflation and exchange rates. There is some evidence that fluctuations in exchange rates cause fluctuation in inflation, primarily in the short term. How-ever, the data also suggest that the opposite occurs in the long term. Finally, there is weak evidence that a unidirectional rela-tionship between trade and exchange rates, with the latter poten-tially influencing the former. All of these findings are consistent with what has been stipulated in the economic literature.
The results presented inFigure 5andTable 6also indicate that the log-determinant methods (∗(m),−
mand+m) offer stronger
significance levels than their asymptotic brethren,Q∗(m),Q∗−
m ,
andQ∗+
m . This is particularly noteworthy considering the large
sample size used in this example. The results involving the Granger statistics should be interpreted with caution, as these tests necessitate fitting of VAR models of different orders than the other tests. Similarly, as indicated byTable 4, the Granger tests likely yield erroneousp-values for largem. Further,QHm∗− andQHm∗+, the one directional tests of El Himdi and Roy (1997), appear to lose detection power asmincreases. This loss of power appears to be mitigated through the use of the matrix-based methodology presented in Section4.
7. CONCLUSIONS AND DISCUSSION
In the previous sections, block Toeplitz matrices were pro-posed for testing correlation among two multivariate time series. Statistics based on the trace of the squared matrix, determinant, and log-determinant were suggested. We provided theoretical re-sults that illustrated that all three methods follow the same under-lying distribution (asymptotically) and that the log-determinant statistic is asymptotically the most powerful. In the case of cor-relation at a specific lag, the proposed log-determinant statistic is clearly more powerful than the only other extant mechanism that is viable in the same setting (the test of El Himdi and Roy
1997).
Our simulation studies demonstrated under the global alter-native, the proposed methods may outperform those which exist in the literature (however, the choice of kernel negates the possi-bility of a uniformly most powerful test). Further, there are some concerns about the Type I errors rates for the logarithm-based test, mainly for largemand smalln. Our simulations confirmed the theory presented within Sections 3 and4 in terms of the asymptotic power of our proposed methods. Finally, the utility of the proposed methodology was illustrated through application to a research question that is of great interest to practitioners; therein, preexisting economic theory was confirmed through use of our methods.
The results of this article have several logical extensions for further development. In the theme of Wong and Li (1996), a statistic could be developed to test for independence of the squared residuals of two ARMA series. The proposed block Toeplitz matrices can be extended to test for the independence of more than two series. Further, a statistic similar to El Himdi
and Roy (1997) or Bouhaddioui and Roy (2006) could be con-structed using the results from Section4as a measure of cross-correlation. Finally, we note that the statistics for testing causal-ity presented here enable weighting toward lower lags; this is a feature not enabled by the Granger and other extant causal-ity tests. Also, this facet of the proposed methods was shown to be beneficial within our simulation and application studies since prewhitening often removes or greatly reduces cross-correlation at higher lags. Therefore, it is implicated that kernel-weighted causality tests will have utility.
APPENDIX: PROOFS
Proof of Lemma 1. The result is obvious forℓ=1 since tr(ℜm)=d.
To show the result for higher orders, letℜm=ℜm−Id and R0=
R0−Ip1+p2 (these are versions ofℜmand R0that have zeros across the diagonal). For notational convenience, we also let Rh=Rh for h=0. We examine the elements ofℜℓ
mfrom a blockwise perspective, lettingA(ℓ)
i,j be a square matrix of dimensionp1+p2, which denotes the (i×j) block ofℜℓ spectively. Using (A.1), the above is easy to see whenℓ=1; induction provides the result for higher even powers. Next it can be shown that
A(2ℓ−1)
for some matricesX∗and Y∗ of dimensionsp
2×p1 and p1×p2, respectively. Equations (A.1) and (A.2) directly yield the above. It immediately follows that
m represent the autocovariance matrix for the se-ries {Z(1)t ,Z(2)t , . . . ,Z(1)t+m,Z(2)t+m}, which must be nonnegative def-inite (Brockwell and Davis 1991, sec. 7.2). Define the vec-tor a= {a1,0, . . . ,0,am}T for any vectors a
1 and am of dimen-sion p1+p2, and let ˜a= {a1,am}T. It follows that ˜aTC†ma˜ = aTC†
ma≥0. Therefore, C†m is nonnegative definite. Let L†m denote the lower Cholesky decomposition ofC†
m. The representation
0illustrates thatℜ(m) is nonnegative
def-inite.
[Received February 2014. Revised August 2014.]
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