Statistics and Probability Letters 78 (2008) 2963–2970 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Feasible parameter regions for alternative discrete state space models ✩ Paul D. Feigin a , Phillip Gould b,c , Gael M. Martin b,* , Ralph D. Snyder b a Faculty of Industrial Engineering and Management, Technion — Israel Institute of Technology, Israel b Department of Econometrics and Business Statistics, Monash University, Australia c Monash University Accident Research Centre, Australia article info Article history: Received 2 October 2007 Received in revised form 9 May 2008 Accepted 11 May 2008 Available online 24 May 2008 JEL classification: C22 C46 C52 abstract This paper provides a comparison of a parameter-driven and an observation-driven discrete state space model. The two models are shown to have non-overlapping feasible regions for dispersion and first-order autocorrelation, with the region for the parameter- driven model being much larger than that of the observation-driven model, as well as providing a much better representation of the empirical moments of observed count series. © 2008 Elsevier B.V. All rights reserved. 1. Introduction Models for time series of counts can be categorized as either ‘observation-driven’ or ‘parameter-driven’; see Cox (1981). In the former case, serial correlation is modelled directly via lagged values of the count variable, with strategies adopted to ensure that the integer nature of the data is preserved (e.g. the binomial thinning operation used in the integer-valued autoregressive (INAR) models of Al-Osh and Alzaid (1987) and McKenzie (1988)). In the case of parameter-driven models, correlation is introduced indirectly by specifying the parameter(s) of the conditional distribution to be a function of a correlated latent stochastic process. The random parameter model is equivalent, in turn, to a dual source of error (DSOE) discrete state space model, in which both measurement and state equation contain a source of randomness; see, for example, West et al. (1985), Zeger (1988), Harvey and Fernandes (1989), West and Harrison (1997), Davis et al. (2000), Durbin and Koopman (2001) and McCabe et al. (2006). 1 An intermediate class of models includes the generalized linear autoregressive moving average (GLARMA) models of Shephard (1995) and Davis et al. (1999, 2003), the autoregressive conditional Poisson (ACP) model of Heinen (2003) and the autoregressive conditional ordered probit (ACOP) model of Jung et al. (2006). In these models, autocorrelation is modelled indirectly by allowing (functions of) the parameter(s) of the conditional distribution to be both serially correlated and dependent on lagged counts. Such models are thus, in style, parameter-driven. However, in contrast with a DSOE model, the latent parameter(s) in these models, conditional on lagged values of the counts and initial values for the latent parameters, are deterministic. As a consequence, such models can be referred to as single source of error (SSOE) models and would typically be classified as observation-driven. ✩ This research has been partially supported by Australian Research Council Discovery Grant No. DP0450257 and Israel Science Foundation Grant No. 1046/04. The authors would like to thank a referee for some very constructive and insightful comments on an earlier draft of the paper. * Corresponding address: Department of Econometrics and Business Statistics, P.O. Box, 11E, Monash University, Victoria, 3800, Australia. E-mail address: gael.martin@buseco.monash.edu.au (G.M. Martin). 1 The term ‘dual’, rather than ‘multiple’ is used here in order to emphasize the fact that randomness characterizes both the measurement and state equations. These equations could, of course, be defined for vectors, in which case the dual sources of error encompass multiple scalar error terms. 0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.05.021