Stationary multivaria time series analysis

Abstract

Multivariate time series analysis became popular in the early 1950s when the need to analyse time series simultaneously arose in the field of economics. This study provides an overview of some of the aspects of multivariate time series analysis in the case of stationarity. The VARMA (vector autoregressive moving average) class of multivariate time series models, including pure vector autoregressive (VAR) and vector moving average (VMA) models is considered. Methods based on moments and information criteria for the determination of the appropriate order of a model suitable for an observed multivariate time series are discussed. Feasible methods of estimation based on the least squares and/or maximum likelihood are provided for the different types of VARMA models. In some cases, the estimation is more complicated due to the identification problem and the nonlinearity of the normal equations. It is shown that the significance of individual estimates can be established by using hypothesis tests based on the asymptotic properties of the estimators. Diagnostic tests for the adequacy of the fitted model are discussed and illustrated. These include methods based on both univariate and multivariate procedures. The complete model building process is illustrated by means of case studies on multivariate electricity demand and temperature time series. Throughout the study numerical examples are used to illustrate concepts. Computer program code (using basic built-in multivariate functions) is given for all the examples. The results are benchmarked against those produced by a dedicated procedure for multivariate time series. It is envisaged that the program code (given in SAS/IML) could be made available to a much wider user community, without much difficulty, by translation into open source platforms.