This paper analyses the recently suggested particle approach to filtering time series. We suggest that the algorithm is not robust to outliers for two reasons: the design of the simulators and the use of the discrete support to represent the sequentially updating prior distribution. Both problems are tackled in this paper. We believe we have largely solved the first problem and have reduced the order of magnitude of the second. In addition we introduce the idea of stratification into the particle filter which allows us to perform on-line Bayesian calculations about the parameters which index the models and maximum likelihood estimation. The new methods are illustrated by using a stochastic volatility model and a time series model of angles. Some key words: Filtering, Markov chain Monte Carlo, Particle filter, Simulation, SIR, State space. 1 1

this article is to develop new methods for inference and prediction in a simple class of stochastic volatility models in which logarithm of conditional volatility follows an autoregressive (AR) times series model. Unlike the autoregressive conditional heteroscedasticity (ARCH) and gener- alized ARCH (GARCH) models [see Bollerslev, Chou, and Kroner (1992) for a survey of ARCH modeling], both the mean and log-volatility equations have separate error terms. The ease of evaluating the ARCH likelihood function and the ability of the ARCH specification to accommodate the timevarying volatility found in many economic time series has fostered an explosion in the use of ARCH models. On the other hand, the likelihood function for stochastic volatility models is difficult to evaluate, and hence these models have had limited empirical application

this paper we exploit Gibbs sampling to provide a likelihood framework for the analysis of stochastic volatility models, demonstrating how to perform either maximum likelihood or Bayesian estimation. The paper includes an extensive Monte Carlo experiment which compares the efficiency of the maximum likelihood estimator with that of quasi-likelihood and Bayesian estimators proposed in the literature. We also compare the fit of the stochastic volatility model to that of ARCH models using the likelihood criterion to illustrate the flexibility of the framework presented. Some key words: ARCH, Bayes estimation, Gibbs sampler, Heteroscedasticity, Maximum likelihood, Quasi-maximum likelihood, Simulation, Stochastic EM algorithm, Stochastic volatility, Stock returns. 1 INTRODUCTION

rapolation techniques, especially exponential smoothing and exponentially weighted moving average methods ([20, 71]). Developments of smoothing and discounting techniques in stock control and production planning areas led to formalisms in terms of linear, state-space models for time series with time-varying trends and seasonal patterns, and eventually to the associated Bayesian formalism of methods of inference and prediction. From the early 1960s, practical Bayesian forecasting systems in this context involved the combination of formal time series models and historical data analysis together with methods for subjective intervention and forecast monitoring, so that complete forecasting systems, rather than just routine and automatic data analysis and extrapolation, were in use at that time ([19, 22]). Methods developed in those early days are still in use now in some companies in sales forecasting and stock control areas. There have been major developments in models and methods since t

Modelling and reconstruction methods are presented for noise reduction of autocorrelated signals in non-Gaussian, impulsive noise environments. A Bayesian probabilistic framework is adopted and Markov chain Monte Carlo methods are developed for detection and correction of impulses. Individual noise sources are modelled as Gaussian with unknown scale (variance), allowing for robustness to `heavy-tailed' impulse distributions, while the underlying signal is modelled as autoregressive (AR). Results are presented for both artificial and real data from voice and music recordings and comparisons are made with existing techniques. The new techniques are found to give improved detection and elimination of impulses in adverse noise conditions at the expense of some extra computational complexity.

This paper provides a novel approach to forecasting time series subject to discrete structural breaks. We propose a Bayesian estimation and prediction procedure that allows for the possibility of new breaks over the forecast horizon, taking account of the size and duration of past breaks (if any) by means of a hierarchical hidden Markov chain model. Predictions are formed by integrating over the hyper parameters from the meta distributions that characterize the stochastic break point process. In an application to US Treasury bill rates, we find that the method leads to better out-of-sample forecasts than alternative methods that ignore breaks, particularly at long horizons.

This paper shows how to combine MCMC and importance sampling to estimate efficiently the sequence of standard normal random variables used to form the goodness of fit statistics to test for the adequacy of a time series model. In particular, the methodology allows testing the adequacy of a very general state space model with unknown parameters and latent variables. The MCMC is run for only a small percentage of the data rather than at each time point as in Kim and Shephard (1994) and functionals at other time points are estimated as weighted averages. The effectiveness of the methodology is studied by an extensive simulation for an autoregressive model which allows for complex interventions. The methodology is also applied to two real examples. The first example determines the goodness of fit of an autoregressive model of zinc concentration. The second example determines the goodness of fit of a stochastic volatility model for U.S. Treasury bill data. Using the methods in the paper we also show how to compute the marginal likelihood of a time series model subject to interventions. Such marginal likelihoods are used for Bayesian model comparison as in Kass and Raftery (1996) and Chib (1995). Geweke (1994) proposed a combination of MCMC and importance sampling to calculate the marginal likelihood of a time series when there are no interventions in the model and our approach extends that of Geweke (1994) to allow for interventions. The connection between our work and the simulated filtering literature is discussed briefly at the end of section 2. The paper is organized as follows. Section 2 introduces the methodology and section 3 describes the test statistics. Section 4 studies using simulation the effectiveness of the methodology when applied to several autoregressive ...

In the vehicle routing problem with time windows a number of identical vehicles must be routed to and from a depot to cover a given set of customers, each of whom has a specified time interval indicating when they are available for service. Each customer also has a known demand, and a vehicle may only serve the customers on a route if the total demand does not exceed the capacity of the vehicle. The most effective solution method proposed to date for this problem is due to Kohl, Desrosiers, Madsen, Solomon, and Soumis. Their algorithm uses a cutting-plane approach followed by a branchand -bound search with column generation, where the columns of the LP relaxation represent routes of individual vehicles. We describe a new implementation of their method, using Karger's randomized minimum-cut algorithm to generate cutting planes. The standard benchmark in this area is a set of 87 problem instances generated in 1984 by M. Solomon; making using of parallel processing in both the cutting-pla...

This paper develops a new approach to change-point modeling that allows the number of change-points in the observed sample to be unknown. The model we develop assumes regime durations have a Poisson distribution. It approximately nests the two most common approaches: the time varying parameter model with a change-point every period and the change-point model with a small number of regimes. We focus considerable attention on the construction of reasonable hierarchical priors both for regime durations and for the parameters which characterize each regime. A Markov Chain Monte Carlo posterior sampler is constructed to estimate a version of our model which allows for change in conditional means and variances. We show how real time forecasting can be done in an efficient manner using sequential importance sampling. Our techniques are found to work well in an empirical exercise involving US GDP growth and in‡ation. Empirical results suggest that the number of change-points is larger than previously estimated in these series and the implied model is similar to a time varying parameter (with stochastic volatility) model.

This paper discusses Bayesian inference in change-point models. The main existing approaches either attempt to be noninformative by using a Uniform prior over change-points or use an informative hierarchical prior. Both these approaches assume a known number of change-points. We show how they have some potentially undesirable properties and discuss how these properties relate to the imposition of a …xed number of change-points. We develop a new Uniform prior which allows some of the change-points to occur out-of sample. This prior has desirable properties, can reasonably be interpreted as “noninformative”and handles the case where the number of change-points We would like to thank Edward Leamer for useful conversations and also seminar participants at the Federal Reserve Bank of St. Louis and University of Kansas. The views expressed in this paper are those of the authors and do not necessarily re‡ect the views of the Federal Reserve Bank of New York or the Federal Reserve System. 1 is unknown. We show how the general ideas of our approach can be extended to informative hierarchical priors. With arti…cial data and two empirical illustrations, we show how these di¤erent priors can have a substantial impact on estimation and prediction even with moderately large data sets. 1