State space models are considered for observations which have non-Gaussian distributions. We obtain accurate approximations to the loglikelihood for such models by Monte Carlo simulation. Devices are introduced which improve the accuracy of the approximations and which increase computational efficiency. The loglikelihood function is maximised numerically to obtain estimates of the unknown hyperparameters. Standard errors of the estimates due to simulation are calculated. Details are given for the important special cases where the observations come from an exponential family distribution and where the observation equation is linear but the observation errors are non-Gaussian. The techniques are illustrated with a series for which the observations have a Poisson distribution and a series for which the observation errors have a t-distribution.

Dynamic models extend state space models to non-normal observations. This paper suggests a specific hybrid Metropolis-Hastings algorithm as a simple device for Bayesian inference via Markov chain Monte Carlo in dynamic models. Hastings proposals from the (conditional) prior distribution of the unknown, time-varying parameters are used to update the corresponding full conditional distributions. It is shown through simulated examples that the methodology has optimal performance in situations where the prior is relatively strong compared to the likelihood. Typical examples include smoothing priors for categorical data. A specific blocking strategy is proposed to ensure good mixing and convergence properties of the simulated Markov chain. It is also shown that the methodology is easily extended to robust transition models using mixtures of normals. The applicability is illustrated with an analysis of a binomial and a binary time series, known in the literature.

This paper presents a new methodological approach for carrying out Bayesian inference about dynamic models for exponential family observations. The approach is simulationbased and involves the use of Markov chain Monte Carlo techniques. A Metropolis-Hastings algorithm is combined with the Gibbs sampler in repeated use of an adjusted version of normal dynamic linear models. Different alternative schemes based on sampling from the system disturbances and state parameters separately and in a block are derived and compared. The approach is fully Bayesian in obtaining posterior samples with state parameters and unknown hyperparameters. Illustrations with real datasets with sparse counts and missing values are presented. Extensions to accommodate more general evolution forms and distributions for observations and disturbances are outlined.

Abstract not found

A prevalent problem in statistical signal processing, applied statistics, and time series analysis is the calculation of the smoothed posterior distribution, which describes the uncertainty associated with a state, or a sequence of states, conditional on data from the past, the present, and the future. The aim of this paper is to provide a rigorous foundation for the calculation, or approximation, of such smoothed distributions, to facilitate a robust and efficient implementation. Through a cohesive and generic exposition of the scientific literature we offer several novel extensions such that one can perform smoothing in the most general case. Experimental results for: a Jump Markov Linear System; a comparison of particle smoothing methods; and parameter estimation using a particle implementation of the EM algorithm, are provided.

This paper is a statistical analysis of the manner in which the Federal Reserve determines the level of the Federal funds rate target, one of the most publicized and anticipated economic indicators in the nancial world. The analysis presents two econometric challenges: (1) changes in the target are irregularly spaced in time; (2) the target is changed in discrete increments of 25 basis points. The contributions of this paper are: (1) to give a detailed account of the changing role of the target in the conduct of monetary policy; (2) to develop new econometric tools for analyzing time-series duration data; (3) to analyze empirically the determinants of the target. The paper introduces a new class of models termed autoregressive conditional hazard processes, which allow one to produce dynamic forecasts of the probability of a target change. Conditional on a target change, an ordered probit model produces predictions of the magnitude by which the Fed will raise or lower the Federal funds ...

First version. Comments are very welcome This paper investigates the presence of bull and bear market states in stock price dynamics. A new definition of bull and bear market states based on sequences of stopping times tracing local peaks and troughs in stock prices is proposed. Duration dependence in stock prices is investigated through posterior mode estimates of the hazard function in bull and bear markets. We find that the longer a bull market has lasted, the lower is the probability that it will come to a termination. In contrast, the longer a bear market has lasted, the higher is its termination probability. Interest rates are also found to have an important effect on cumulated changes in stock prices: increasing interest rates are associated with an increase in bull market hazard rates and a decrease in bear market hazard rates.

Dynamic regression or state space models provide a flexible framework for analyzing non-Gaussian time series and longitudinal data, covering for example models for discrete longitudinal observations. As for non-Gaussian random coefficient models, a direct Bayesian approach leads to numerical integration problems, often intractable for more complicated data sets. Recent Markov chain Monte Carlo methods avoid this by repeated sampling from approximative posterior distributions, but there are still open questions about sampling schemes and convergence. In this article we consider simpler methods of inference based on posterior modes or, equivalently, maximum penalized likelihood estimation. From the latter point of view, the approach can also be interpreted as a nonparametric method for smoothing time-varying coefficients. Efficient smoothing algorithms are obtained by iteration of common linear Kalman filtering and smoothing, in the same way as estimation in generalized linear models w...

Introduction This chapter surveys dynamic or state space models and their relationship to non-- and semiparametric models that are based on the roughness penalty approach. We focus on recent advances in dynamic modelling of non--Gaussian, in particular discrete--valued, time series and longitudinal data, make the close correspondence to semiparametric smoothing methods evident, and show how ideas from dynamic models can be adopted for Bayesian semiparametric inference in generalized additive and varying coefficient models. Basic tools for corresponding inference techniques are penalized likelihood estimation, Kalman filtering and smoothing and Markov chain Monte Carlo (MCMC) simulation. Similarities, relative merits, advantages and disadvantages of these methods are illustrated through several applications. Section 2 gives a short introductory review of results for the classical situation of Gaussian time series observations. We start with Whittaker's (1923) "method of graduati

In likelihood-based approaches to robustify state space models, Gaussian error distributions are replaced by non-normal alternatives with heavier tails. Robustified observation models are appropriate for time series with additive outliers, while state or transition equations with heavytailed error distributions lead to filters and smoothers that can cope with structural changes in trend or slope caused by innovations outliers. As a consequence, however, conditional filtering and smoothing densities become analytically intractable. Various attempts have been made to deal with this problem, reaching from approximate conditional mean type estimation to fully Bayesian analysis using MCMC simulation. In this article we consider penalized likelihood smoothers, this means estimators which maximize penalized likelihoods or, equivalently, posterior densities. Filtering and smoothing for additive and innovations outlier models can be carried out by computationally efficient Fisher scoring steps ...