STATISTICS & PFfgABlB~rY ELSEVIER Statistics & Probability Letters 28 (1996) 99-106 An algorithm for estimating parameters of state-space models Lilian Shiao-Yen Wu”, Jeffrey S. Pai b, J.R.M. Hosking”, * a IBM Research Division, Yorktown Heights, NY 10598, USA bInstitute of Statistics and Econometrics, University of Basel, Basel, Switzerland Received January 1995; revised May 1995 Abstract We describe an algorithm for estimating the parameters of time-series models expressed in state-space form. The algorithm is based on the EM algorithm, and generalizes an algorithm given by Shumway and Stoffer (1982). Keywords: EM algorithm; Kalman filter; Maximum likelihood; Time series 1. Introduction Kalman’s state-space model (Kalman, 1960; Kalman and Bucy, 1961) is a natural and useful framework for analyzing, smoothing and forecasting multivariate time-series data, decomposing time series into structural components, and combining multiple sources of data for forecasting. See, for example, Harrison and Stevens (1976), Shumway and Stoffer (1982), Harvey (1989) and Shumway and Katzoff (199 1). Within this framework, data irregularities such as outliers and missing data can be easily handled. In the original engineering applications for which the Kalman filter was developed, parameters of the model were based on differential equations and were assumed known. The model was later applied to business, economic, and biological data which are not based on well-understood differential equations, and in these applications additional proced- ures for estimating parameters in the models were needed. Parameter estimation procedures include the Newton-Raphson, the Gauss-Newton (scoring) and Davidson-Fletcher-Powell variable metric methods (Gupta and Mehra, 1974). Shumway and Stoffer (1982) showed how the EM (expectation-maximization) algorithm of Dempster et al. (1977) could be used for parameter estimation and discussed its advantages and disadvantages over previous nonlinear optimization methods. The Shumway-Stoffer algorithm is simple to apply since at each iteration the optimal solution for the unknown parameters can be obtained from explicit regression formulas. Shumway and Stoffer also showed how the algorithm can be modified for missing observations, and they gave an interesting example in which a time-series forecast is based on two time series of different lengths. Stoffer and Wall (1991) have shown that the nonparametric bootstrap can be used to assess the precision of the parameter estimates. * Corresponding author. 0167-7152/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0167-7152(95)00098-4