Multiple Time Series: Determining the Order of Approximating Autoregressive Schemes.

TLDR: An approach to the solution of these problems of time series analysis through a criterion called CAT, an abbreviation for criterion autoregressive transfer-function, introduced in Parzen (1969) is described.

Abstract: : Three aims of the time series analysis can be distinguished of a finite sample Y(t), t = 1,2,...,T of a univariate or multivariate time series: (1) Spectral analysis, (2) Model identification, and (3) Prediction. In this paper we consider the case in which a joint autoaggressive scheme is a multiple time series which is stationary, normal, and zero mean. We describe an approach to the solution of these problems of time series analysis through a criterion called CAT (an abbreviation for criterion autoregressive transfer-function). CAT enables one to choose the order of an approximating autoregressive scheme which is 'optimal' in the sense that its transfer function is a minimum overall mean square error estimator (called ARTFACT) of the infinite autoregressive transfer function ARTF) of the filter which transforms the time series to its innovations (white noise). Algorithms for choosing the order of an ARTFACT (autoregressive transfer function approximation converging to the truth) enables one to carry out the approach to empirical multiple time series analysis introduced in Parzen (1969), in particular autoregressive spectral estimation of the spectral density matrix of a stationary multiple time series. Such estimators for univariate time series have been very successfully applied in geophysics (see Ulrych and Bishop (1975)) where they are called 'maximum entropy spectral estimators.' This paper provides a basis for an extension of these procedures to multiple time series.