On the Causality between Multiple Locally Stationary Processes

When one would like to describe the relations between multivariate time series, the concepts of dependence and causality are of importance. These concepts also appear to be useful when one is describing the properties of an engineering or econometric model. Although the measures of dependence and causality under stationary assumption are well established, empirical studies show that these measures are not constant in time. Recently one of the most important classes of nonstationary processes has been formulated in a rigorous asymptotic framework by Dahlhaus in (1996), (1997), and (2000), called locally stationary processes. Locally stationary processes have time-varying spectral densities whose spectral structures smoothly change in time. Here, we generalize measures of linear dependence and causality to multiple locally stationary processes. We give the measures of linear dependence, linear causality from one series to the other, and instantaneous linear feedback, at time t and frequency λ. 1. Introduction In discussion of the relations between time series, concepts of dependence and causality are frequently invoked. Geweke [1] and Hosoya [2] have proposed measures of dependence and causality for multiple stationary processes (see also Taniguchi et al. [3]). They have also showed that these measures can be additively decomposed into frequency-wise. However, it seems to be restrictive that these measures are constants all the time. Priestley [4] has developed the extensions of prediction and filtering theory to nonstationary processes which have evolutionary spectra. Alternatively, in this paper we generalize measures of dependence and causality to multiple locally stationary processes. When we deal with nonstationary processes, one of the difficult problems to solve is how to set up an adequate asymptotic theory. To meet this Dahlhaus [5–7] introduced an important class of nonstationary processes and developed the statistical inference. We give the precise definition of multivariate locally stationary processes which is due to Dahlhaus [8]. Definition 1.1. A sequence of multivariate stochastic processes , is called locally stationary with mean vector and transfer function matrix if there exists a representation where(i) is a complex valued stochastic vector process on with and for , , where denotes the cumulant of th order, and is the period extension of the Dirac delta function.(ii)There exists a constant and a -periodic matrix valued function with and for all and . is assumed to be continuous in . We call the time-varying spectral density matrix of