Univariate Box/Jenkins-Modelle in der Zeitreihenanalyse

Abstract

Die Zeitreihenanalyse, die eine längere Tradition in den Ingenieur- und Wirtschaftswissenschaften aufweist, hat in den letzten zehn bis fünfzehn Jahren durch die Diskussion 'langer Wellen' auch in den Sozialwissenschaften erheblich an Bedeutung gewonnen. In Heft 3/1992 der vorliegenden Zeitschrift erschien vom gleichen Autor ein Artikel zur Komponentenzerlegung, mit dem eine umfassende Einführung in die methodologischen Probleme der Zeitreihenanalyse begonnen wird. Der vorliegende Beitrag als Fortsetzung dieser Arbeit führt in Modellstrategien ein, die als 'Box/Jenkins-Methode' bekannt geworden sind. Es werden vor allem die Basis-Modelle (ARMA-Modelle) vorgestellt, die in den letzten beiden Abschnitten erweitert werden durch die Berücksichtigung bestimmter Formen der Nicht-Stationarität (ARIMA-Modelle) und durch den Einbau sozialer Komponenten (SARIMA-Modelle). Für die Einführung in diese Basis-Modelle werden Grundkenntnisse der Inferenzstatistik und der Regressionsanalyse vorausgesetzt. (pmb)

'This is the second in a series of articles which introduces basic concepts and models of time series analysis. Whereas the first paper (HS 3/1992) presented elementary descriptive concepts and traditional techniques of decomposing a time series into trend, season, and irregular fluctuations, this second paper offers an introduction into the Box-Jenkins approach to modelling univariate processes. The basic concept underlying this methodology is the idea to treat observed time series data as (generally non-independent) realizations of a 'stochastic process'. This concept is discussed (after some introductory remarks) in section 2. In order to actually model a stochastic process, a number of restrictive assumptions need to be made regarding the 'stationary' of the process and the nature of the dependencies linking the time ordered realizations. The latter set of assumptions leads to three types of basic models which are outlined in subsequent sections: the autoregressive (AR), the moving-average (MA), and the mixed (ARMA) model. These models are constructed in a three step procedure: the identification (based on empirical autocorrelation and partial autocorrelation functions) of the model (section 7), the estimation of the model parameters (section 8), and the evaluation or diagnosis of the model (section 9). Since the assumption of stationarity is often unrealistic, Box and Jenkins have extended the repertoire of models in a way to include certain types of non-stationary processes, the socalled integrated processes, for which they invented their ARIMA models (section 10). In a further extension, seasonal components may be incorporated therby creating SARIMA models (section 11). The methods and models presented in this paper remain within the confines of unvariate analysis. Strategies for modelling possible relationships between two or more series (dynamic regression, transfer-function models) will be outlined in one of the forthcoming issues of HSR.' (author's abstract)