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COPYRIGHT 2001 MIT Press Journals
I. Introduction
FEW empirical relations have the status of "stylized facts" in economics. One of them is that macroeconomic aggregates comove, an observation that has been the source of speculation of economic theory since its birth. In modern theories of the business cycle, people have asked whether comovements can be explained by large aggregate shocks, monetary or real, or whether an explanation should be found in nonlinear propagation mechanisms. Every macroeconomic textbook starts by a statement on comovements between aggregates. However, this fact is paradoxically one of the least well documented, and the literature there is exhibits much confusion of meaning and terminology. Comovement is a lose term, possibly describing different phenomena and, consequently, with many different interpretations. What are the stylized facts really, and what should macroeconomics be trying to explain? Appropriate measures of comovement between time-series processes should be developed to provide a meaningful answer to this question. The informal discussion on comovements usually refers to something close to a notion of correlation. However, the traditional way with which the time-series literature has dealt with measurement of comovements is based on a notion of rank reduction (Ahn & Reinsel, 1988), which has a different meaning. In this category belong the following three concepts: (i) the idea of co-integration (Engle & Granger, 1987): two processes are co-integrated if the spectral density at frequency zero has rank one; (ii) codependence (Gourieroux & Peaucelle, 1992), which refers to linear combinations of correlated processes which are of lower autoregressive order than others; and (iii) common features (Engle & Kozicki, 1993), that is, linear combinations that are unpredictable with respect to past information and common cycles (Vahid & Engle, 1993) which are defined as common features in first differences for processes that are co-integrated. This class of concepts presents several problems. First, high cross-correlation neither implies nor is implied by co-integration, common cycles, or common features (Quah, 1993; Forni & Reichlin, 2001). Second, these measures are binary. For example, two processes are either co-integrated or not, but we can't establish different degrees of association. Finally, to test for rank reduction, we need to estimate the parameters of a VAR, which may be problematic when the number of time series is large. For all these reasons, although the notion of rank reduction is certainly interesting to characterize some aspects of the dynamic properties of multivariate time series, it is not the appropriate one for the study of comovements.
But what is an appropriate measure of comovement? In this paper, we show that existing textbook quantities such as coherence and coherency, which are widely used in time-series literature, are not appropriate as comovement indices. We propose a related but different measure, dynamic correlation, that arises quite naturally from basic frequency domain notions. Dynamic correlation can be decomposed by frequency and frequency band and can then be used to study business-cycle as well as long-run questions. Dynamic correlation between two processes over a band turns out to be identical to static correlation of the same processes, after suitable prefiltering. Moreover, long-run dynamic correlation is related in a simple way to stochastic co-integration.
We use our notion of dynamic correlation to construct a multivariate index of comovement, which we name cohesion. The latter provides a summary measure of the degree of comovement within a group of variables or between two groups of variables and can be used, for instance, as a metric to construct dynamic clusters.
To illustrate our proposed measure and to provide further motivation, we estimate cohesion of output data in U.S. states and European nations and study the following questions. Are output fluctuations within Europe more correlated than are output fluctuations within the United States, and are results the same for business-cycle frequencies and the long run? Are states or countries that comove more strongly closely located from a geographical point of view? Does border matter for output synchronization and at what frequency range?
II. Dynamic Correlation
A. The Definition and the Basic Motivation
Consider two zero-mean real stochastic processes, x and y. Let [S.sub.x]([Lambda]) and [S.sub.y]([Lambda]), - [Pi] [is less than or equal to] [Lambda] [is less than] [Pi], be the spectral-density density functions of x and y and [C.sub.xy]([Lambda]) be the cospectrum. The measure we propose, dynamic correlation, is
(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
To motivate this measure, we introduce the spectral decomposition of the processes [x.sub.t] and [y.sub.t]; that is,
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where d[Z.sub.x]([Lambda]) and d[Z.sub.y]([Lambda]) are (complex) orthogonal increment processes (See, for example, Brockwell and Davis (1987, ch. 4)). Expression 2 says that [x.sub.t] and [y.sub.t] can be expressed as infinite sums of waves of different frequencies, each having a random amplitude. As is well known, the spectral and cross-spectral density functions of [x.sub.t] and [y.sub.t] are related to the above representation in the following way:
(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Although the waves appearing in representation (2) are complex, it is easily seen that, if [x.sub.t] is real, then d[Z.sub.x]([Lambda]) = d[Z.sub.x](-[Lamda]), so that
(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where d[U.sub.x] and d[V.sub.x] denote, respectively, the real and the imaginary part of d[Z.sub.x]. Hence, we have the alternative representation
(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where the component at frequency [Lambda] is cos ([Lambda]t)d[U.sub.x]([Lambda]) -- sin ([Lambda]t)d[V.sub.x]([Lambda]), which is real. Similarly, the real wave decomposition of [y.sub.t] is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
As is easily obtained from equation (3) and (4), our proposed measure, [[Rho].sub.xy]([Lambda]), is nothing else than the correlation coefficient between the real waves of frequency [Lambda] appearing in the above representation: that is,...
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