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

Post-crash distributions inferred from S

This paper is concerned with the Bayesian estimation of non-linear stochastic differential equations when only discrete observations are available. The estimation is carried out using a tuned MCMC method, in particular a blocked Metropolis-Hastings algorithm, by introducing auxiliary points and using the Euler-Maruyama discretisation scheme. Techniques for computing the likelihood function, the marginal likelihood and diagnostic measures (all based on the MCMC output) are presented. Examples using simulated and real data are presented and discussed in detail.

Summary. Probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions. We propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distributions subject to calibration. Calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize. Sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only. A simple theoretical framework allows us to distinguish between probabilistic calibration, exceedance calibration and marginal calibration. We propose and study tools for checking calibration and sharpness, among them the probability integral transform histogram, marginal calibration plots, the sharpness diagram and proper scoring rules. The diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the Stateline wind energy centre in the US Pacific Northwest. In combination with cross-validation or in the time series context, our proposal provides very general, nonparametric alternatives to the use of information criteria for model diagnostics and model selection.

Kim, Shephard, and Chib (1998) provided a Bayesian analysis of stochastic volatility models based on a fast and reliable Markov chain Monte Carlo (MCMC) algorithm. Their method ruled out the leverage effect, which is known to be important in applications. Despite this, their basic method has been extensively used in the financial economics literature and more recently in macroeconometrics. In this paper we show how the basic approach can be extended in a novel way to stochastic volatility models with leverage without altering the essence of the original approach. Several illustrative examples are provided.

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 ...

Abstract: Bayesian inference in a time series model provides exact, out-of-sample predictive distributions that fully and coherently incorporate parameter uncertainty. This study compares and evaluates Bayesian predictive distributions from alternative models, using as an illustration five alternative models of asset returns applied to daily S&P 500 returns from 1972 through 2005. The comparison exercise uses predictive likelihoods and is inherently Bayesian. The evaluation exercise uses the probability integral transform and is inherently frequentist. The illustration shows that the two approaches can be complementary, each identifying strengths and weaknesses in models that are not evident using the other. JEL classification: C11, C53 Key words: forecasting, GARCH, inverse probability transform, Markov-mixture, predictive likelihood, S&P 500 returns, stochastic volatility The authors gratefully acknowledge financial support from NSF grant SBR-0720547. The views expressed here are the authors ’ and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors ’ responsibility.

In a recent article Gneiting, Balabdaoui and Raftery (JRSSB, 2007) propose the criterion of sharpness for the evaluation of predictive distributions or density forecasts. They motivate their proposal by an example in which standard evaluation procedures based on probability integral transforms cannot distinguish between the ideal forecast and several competing forecasts. In this paper we show that their example has some unrealistic features from the perspective of the time-series forecasting literature, hence it is an insecure foundation for their argument that existing calibration procedures are inadequate in practice. We present an alternative, more realistic example in which relevant statistical methods, including information-based methods, provide the required discrimination between competing forecasts. We conclude that there is no need for a subsidiary criterion of sharpness.

Summary. We discuss tools for the evaluation of probabilistic forecasts and the critique of statistical models for ordered discrete data. Our proposals include a non-randomized version of the probability integral transform, marginal calibration diagrams and proper scoring rules, such as the predictive deviance. In case studies, we critique count regression models for patent data, and assess the predictive performance of Bayesian age-period-cohort models for larynx cancer counts in Germany.

A continuous time econometric modelling framework for multivariate financial market event (or `transactions') data is developed in which the model is specified via the vector stochastic intensity.