EMPIRICAL IDENTIFICATION OF MULTIPLE TIME SERIES

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Abstract
In the univariate case the problem of empirical identification consists in determining the order parametersp,dandqof ARIMA (p, d, q) processes. In this paper we introduce some new techniques for handling the corresponding problem for a multiple time seriesX(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 functionC(t) =K(t)K‐1(0) is discussed, whereK(t) is the covariance function ofX(t). It is proposed that the further identification procedure should be based on twoX2statistics and on the estimated trace and eigenvalues ofC(t), the matrix correlation functionp(t) and the matrix partial correlation functionP(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.

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