Statistical Software for State Space Methods

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.Keywords: ARMA model, Kalman filter, state space methods, unobserved components, software tools. Promising trendsIn 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). Ten years later, it is gratifying to see how much progress has been made in the further dissemination of these methods. Not only have state 2 Statistical Software for State Space Methods 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.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. A riveting account of this application of state space methods to space travel can be found in the NASA (National Aeronautics and Space Administration) report of McGee and Schmidt (1985), a document that has clearly been written on an old-fash...

State Space Models inR

“…Suppose that we want to describe the series, on a logarithmic scale, by a state space model containing a quarterly seasonal component and a local linear trend, in the form of an integrated random walk (i.e., in the notation of expression (4) in Commandeur et al (2011), σ 2 ξ = 0). The model, and the MLE of the unknown parameters, can be obtained by dlm as follows.…”

REGCMPNT- AFortranProgram for Regression Models with ARIMA Component Errors

“…We require the φ j (B) to have all their zeros outside the unit circle, and the θ j (B) to have all their zeros on or outside the unit circle. Common versions of the ∆ j (B) would be (i ) the identity operator (∆ j (B) = 1), corresponding to stationary components (such as the observation disturbance t in Equation 1 of Commandeur et al 2011); (ii ) a nonseasonal (1 − B) or seasonal (1 − B s ) difference, or a product of these; or (iii) a seasonal summation operator, 1 + B + · · · + B s−1 (see Equation 5 in Commandeur et al 2011 or equation (7) in the model of Section 5 below). The ∆ j (B) typically have all their zeros on the unit circle, and usually must have no common zeros, as common zeros can create problems for signal extraction results (Bell 1984(Bell , 1991Kohn and Ansley 1987).…”

“…Model (1) extends the pure ARIMA components model given as Equation 18 of Commandeur et al (2011) in two ways. The first extension involves the regression mean function x t β (also mentioned in Section 2.2 of Commandeur et al 2011).…”

“…The first extension involves the regression mean function x t β (also mentioned in Section 2.2 of Commandeur et al 2011). REGCMPNT allows models to include regression variables for several types of regression effects commonly used in modeling seasonal economic time series.…”

Neglected serial correlation tests in UCARIMA models

“…Once the parameters have been estimated, filtered values of the unobserved components can be extracted by means of the Kalman smoother or its Wiener-Kolmogorov counterpart. These estimation and filtering issues are well understood (see Harvey 1989;Durbin and Koopman 2012 for textbook treatments), and the same can be said of their efficient numerical implementation (see Commandeur et al 2011 and the references therein).…”