Empirical Identification Of Multiple Time Series

. In the univariate case the problem of empirical identification consists in determining the order parameters p, d and q of ARIMA (p, d, q) processes. In this paper we introduce some new techniques for handling the corresponding problem for a multiple time series X(t) with the main emphasis on AR and MA models. Types of joint nonstationarity (or rather almost nonstationarity) are defined and a method of analyzing such structures based on the ordered eigenvalues of the function C(t) =K(t)K‐1(0) is discussed, where K(t) is the covariance function of X(t). It is proposed that the further identification procedure should be based on two X2 statistics and on the estimated trace and eigenvalues of C(t), the matrix correlation function p(t) and the matrix partial correlation function P(t). The suitability for identification purposes of each of these functions is examined in terms of such properties as scale‐invariance, existence of normalized eigenvalues and standard errors. The methods introduced are illustrated on a 5‐dimensional economic time series first studied by Quenouille and on a 4‐dimensional smulated MA series.