Abstract
In this paper we review the state space approach to time series analysis and establish the notation that is adopted in this special volume of the Journal of Statistical Software. We first provide some background on the history of state space methods for the analysis of time series. This is followed by a concise overview of linear Gaussian state space analysis including the modelling framework and appropriate estimation methods. We discuss the important class of unobserved component models which incorporate a trend, a seasonal, a cycle, and fixed explanatory and intervention variables for the univariate and multivariate analysis of time series. We continue the discussion by presenting methods for the computation of different estimates for the unobserved state vector: filtering, prediction, and smoothing. Estimation approaches for the other parameters in the model are also considered. Next, we discuss how the estimation procedures can be used for constructing confidence intervals, detecting outlier observations and structural breaks, and testing model assumptions of residual independence, homoscedasticity, and normality. We then show how ARIMA and ARIMA components models fit in the state space framework to time series analysis. We also provide a basic introduction for non-Gaussian state space models. Finally, we present an overview of the software tools currently available for the analysis of time series with state space methods as they are discussed in the other contributions to this special volume.
Highlights
In 2001, when considering the possible drawbacks of state space models, Durbin and Koopman wrote: “In our opinion, the only disadvantages are the relative lack in the statistical and econometric communities of information, knowledge, and software regarding these models.”, see Durbin and Koopman (2001, p. 52)
Have state space models been applied in a growing number of scientific fields, but – as is witnessed by this special volume of the Journal of Statistical Software that is completely dedicated to statistical software for state space methods – they have been implemented in STAMP, R, MATLAB, REGCMPNT, SAS, EViews, GAUSS, Stata, RATS, gretl, and SsfPack with links established with S-PLUS and Ox
Nowadays state space methods are used for fitting the autoregressive integrated moving average (ARIMA) models of Box and Jenkins (1976) as these can be put in state space form and analysed by the Kalman filter
Summary
In 2001, when considering the possible drawbacks of state space models, Durbin and Koopman wrote: “In our opinion, the only disadvantages are the relative lack in the statistical and econometric communities of information, knowledge, and software regarding these models.”, see Durbin and Koopman (2001, p. 52). State space methods originated in the field of control engineering, starting with the groundbreaking paper of Kalman (1960) They were initially (and still are) deployed for the purpose of accurately tracking the position and velocity of moving objects such as ships, airplanes, missiles, and rockets. When fitting an ARIMA model to a time series with missing observations in SPSS version 15.0, for example, the SPSS output informs the user that “A Kalman filtering algorithm was used for estimation”, confirming the fact that missing values are handled in a state space framework while this is much more difficult in the Box-Jenkins approach to time series analysis.
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