On the Causality between Multiple Locally Stationary Processes

Junichi Hirukawa

doi:10.1155/2012/261707

Abstract

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.

Highlights

In discussion of the relations between time series, concepts of dependence and causality are frequently invoked
For the d(Z)-dimensional locally stationary process {Zt,T }, we introduce H, the Hilbert space spanned by Zt(,jT), j = 1, . . . , d(Z), t = 0, ±1, . . . and call H (Zt,T ) the closed subspace spanned by Zs(,jT), j = 1, . . . , d(Z), s ≤ t
We impose the following assumption on Gt,T

Summary

If we set

Φt,T (z) is a analytic function in the unit disc with Φt,T (0)Φt,T (0)∗ = Gt,T and maximal, such that the time varying spectral density ft,T (λ) has a factorization ft,T (λ). From this lemma, it is seen that time varying spectral density is decomposed into two parts ft(,xTx)(λ). The average measure of linear dependence is given by the following integral functional of time varying spectral density. We consider the testing problem for existence of linear dependence; H: K(X,Y ) {f (u, λ)} dλdu = 0. The asymptotic variance of ST becomes the integral functional of the time varying spectral density

If we take

Then we see that