Detecting Co‐Movements in Non‐Causal Time Series
| Date | 01 June 2019 |
| Author | Sean Telg,Gianluca Cubadda,Alain Hecq |
| Published date | 01 June 2019 |
| DOI | http://doi.org/10.1111/obes.12281 |
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©2018 TheAuthors. OxfordBulletin of Economics and Statistics published by Oxford University and John Wiley & Sons Ltd.
This is an open accesses article under the terms of the Creative Commons Attribution License which permits use, distribution and reproduction in any medium,
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OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 81, 3 (2019) 0305–9049
doi: 10.1111/obes.12281
Detecting Co-Movements in Non-Causal Time
Series*
Gianluca Cubadda,†Alain Hecq‡ and Sean Telg‡
†Dipartimento di Economia e Finanza, Universita’ di Roma “Tor Vergata”, Via Columbia 2,
00133 Roma, Italy (e-mail: gianluca.cubadda@uniroma2.it)
‡Department of Quantitative Economics, Maastricht University, P.O. Box 616, 6200 MD
Maastricht, The Netherlands (e-mail: a.hecq@maastrichtuniversity.nl;
j.telg@maastrichtuniversity.nl)
Abstract
This paper introduces the notion of common non-causal features and proposes tools to
detect them in multivariatetime series models. We arguethat the existence of co-movements
might not be detected using the conventional stationary vector autoregressive(VAR) model
as the common dynamics are present in the non-causal (i.e. forward-looking) component
of the series. We show that the presence of a reduced rank structure allows to identify
purely causal and non-causal VARprocesses of order P>1 even in the Gaussian likelihood
framework. Hence, usual test statistics and canonical correlation analysis can be applied,
where either lags or leads are used as instruments to determine whether the common
features are present in either the backward- or forward-looking dynamics of the series.The
proposed definitions of co-movements are also valid for the mixed causal—non-causal
VAR, with the exception that a non-Gaussian maximum likelihood estimator is necessary.
This means however that one loses the benefits of the simple tools proposed. An empirical
analysis on Brent and West Texas Intermediate oil prices illustrates the findings. No short
run co-movements are found in a conventional causal VAR, but they are detected when
considering a purely non-causal VAR.
I. Introduction
A ’feature’ is defined as a dominant characteristic observed in univariate time series. One
might be interested in testing whether certain time series contain the same feature. We
JEL Classification numbers: C12, C32, E32.
*Previous versions of this paper have been presented at conferences of the Society for Nonlinear Dynamics
and Econometrics (Oslo, March 2015), the International Association for Applied Econometrics-IAAE (Tessaloniki,
June 2015), the Netherlands Econometric Study Group (Maastricht, May 2015), the International Symposium on
Forecasting (Santander, June 2016), the Computational and Financial Econometrics (Sevilla, December 2016), the
Brazilian Econometric Society (Iguassu, December 2016), the Scientific Meeting of the Italian Statistical Society
(Palermo, June 2018) as well as at seminars at the University of Balear Islands (Palma, November 2015), SKKU
(Seoul, January 2016), Aix-Marseille (Marseille, March 2016), KU Leuven (Leuven, March 2016), FGV (Rio de
Janeiro, May 2016). We thank the participants for helpful comments and suggestions. Additionally, weare greatly in
debt to Lenard Lieb for fruitful discussions and two anonymous referees for valuable comments and suggestions.
698 Bulletin
say that a feature is common if a linear combination of multiple series no longer has the
feature, while it is present in each of the series individually. There are ample examples
of common features in the literature: e.g. cointegration, common ARCH, co-breaking
and common nonlinearity. In this paper, we focus on the presence of common cyclical
features as introduced by Engle and Kozicki (1993) and Vahid and Engle (1993). That is,
we are interested in extracting commonalities in the dynamics of multivariate time series
models.This is commonly done by imposing short run restrictions, which are known to have
direct implications for forecasting accuracy and parameter efficiency as less coefficients
have to be estimated. Common features restrictions are also directly implied by economic
theories such as the present value model or the permanent income hypothesis (Guill´en et al.,
2015; Issler and Vahid, 2001). Impulse response functions are collinear when common
transmission mechanisms are present and economies with strong nexus generally show
statistical similarities in their business cycle fluctuation phases.
This paper contributes to this literature by studying the existence of common cyclical
features in economic and financial variables for both causal and non-causal vector autore-
gressive (VAR) models. In our terminology, the conventional vector autoregressive model
popularized by Sims (1980) is called causal, as the variables of interest only depend on
their own past values. The non-causal VAR has been investigated in the literature to a
much lesser extent and is defined as the analogue model in reverse time (see e.g. Davis
and Song, 2012; Lanne and Saikkonen, 2013; Gouri´eroux and Jasiak, 2017). In this frame-
work, variables are modelled to depend on their future values. From an empirical point
of view, non-causal VAR models are worth investigating as they are simple linear models
that are able to generate different dynamics than their causal counterparts. For instance,
Gouri´eroux and Zako¨ıan (2016) show that non-causal models are able to mimic speculative
bubbles, i.e. processes that experience (locally) a rapid increase or decrease followed by a
sudden crash or recovery. A more general model that depends on both past and future values
is called a mixed causal–non-causal model1and combines the specific characteristics of
both the backward- and forward-looking VAR. It is able to generate even richer dynamics
and therefore captures characteristics of macroeconomic and financial variables that could
previously only be generated using highly nonlinear and complex models (see Hecq, Lieb
and Telg, 2016; Gouri´eroux and Jasiak, 2017).
Serial correlation common features (SCCF hereafter, see Engle and Kozicki, 1993) is a
well-known approach to test for co-movements in an n−dimensional stationary time series
process Ytgenerated by a causal VAR model of order p:
(L)Yt≡(In−1L−…−pLp)Yt="t,
where Lis the lag operator, and "tare i.i.d. innovations with mean vector E("t)=0 and
positive definite covariance matrix E("t"
t)=. The presence of SCCF implies that each
VAR(p) coefficient matrix can be written as the product of two reduced rank matrices such
that
(In−⊥
1L−…−⊥
pLp)Yt="t,
1More precisely, a non-causal model is defined as a model that has a unique stationary solution in terms of current
and future error terms. For the mixed causal–non-causal model, this is the two-sided movingaverage representation,
i.e. past, current and future disturbances.
©2018 The Authors. Oxford Bulletin of Economics and Statistics published by Oxford University and JohnWiley & Sons Ltd.
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