Michael K. Pitt

Auxiliary Variable Based Particle Filters

We model a time series {y t , t = 1, ..., n} using a state-space framework with the {y t |α t } being independent and with the state {α t } assumed to be Markovian. The task will be to use simulation to estimate f(α t |F t ), t = 1, ..., n, where F t is contemporaneously available information. We assume a known measurement density f(y t |α t ) and the ability to simulate from the transition density f(α t+1|α t ). Sometimes we will also assume that we can evaluate f(α t+1|α t ).

Constructing First Order Stationary Autoregressive Models via Latent Processes

First order stationary autoregressive (AR(1)) models are introduced for which there exists a linear relation between the expectations of the observations, and where it is readily possible to arrange the marginal distributions to be other than normal.

Constructing Stationary Time Series Models Using Auxiliary Variables With Applications

Here we present a novel method for modeling stationary time series. Our approach is to construct the model with a specified marginal family and build the dependence structure around it. We show that the resulting time series is linear with a simple autocorrelation structure. We construct models that parallel existing structures, namely state-space models, autoregressive conditional heteroscedasticity (ARCH) models, and generalized ARCH models. We use Bayesian techniques to estimate the resulting models. We also demonstrate that the models perform well compared with competing methods for the applications considered, count models and volatility models.

Importance Sampling Squared for Bayesian Inference in Latent Variable Models

We consider Bayesian inference by importance sampling when the likelihood is analytically intractable but can be unbiasedly estimated. We refer to this procedure as importance sampling squared (IS2), as we can often estimate the likelihood itself by importance sampling. We provide a formal justification for importance sampling when working with an estimate of the likelihood and study its convergence properties. We analyze the effect of estimating the likelihood on the resulting inference and provide guidelines on how to set up the precision of the likelihood estimate in order to obtain an optimal tradeoff between computational cost and accuracy for posterior inference on the model parameters. We illustrate the procedure in empirical applications for a generalized multinomial logit model and a stochastic volatility model. The results show that the IS2 method can lead to fast and accurate posterior inference under the optimal implementation.

Modelling Stochastic Volatility with Leverage and Jumps: A Simulated Maximum Likelihood Approach via Particle Filtering

In this paper we provide a unified methodology for conducting likelihood-based inference on the unknown parameters of a general class of discrete-time stochastic volatility (SV) models, characterized by both a leverage effect and jumps in returns. Given the nonlinear/non-Gaussian state-space form, approximating the likelihood for the parameters is conducted with output generated by the particle filter. Methods are employed to ensure that the approximating likelihood is continuous as a function of the unknown parameters thus enabling the use of standard Newton-Raphson type maximization algorithms. Our approach is robust and efficient relative to alternative Markov Chain Monte Carlo schemes employed in such contexts. In addition it provides a feasible basis for undertaking the nontrivial task of model comparison. Furthermore, we introduce new volatility model, namely SV-GARCH which attempts to bridge the gap between GARCH and stochastic volatility specifications. In nesting the standard GARCH model as a special case, it has the attractive feature of inheriting the same unconditional properties of the standard GARCH model but being conditionally heavier-tailed; thus more robust to outliers. It is demonstrated how this model can be estimated using the described methodology. The technique is applied to daily returns data for S&P 500 stock price index for various spans. In assessing the relative performance of SV with leverage and jumps and nested specifications, we find strong evidence in favour of a including leverage effect and jumps when modelling stochastic volatility. Additionally, we find very encouraging results for SV-GARCH in terms of predictive ability which is comparable to the other models considered.

Adaptive Metropolis---Hastings sampling using reversible dependent mixture proposals

This article develops a general-purpose adaptive sampler for sampling from a high-dimensional and/or multimodal target. The adaptive sampler is based on reversible proposal densities each of which has a mixture of multivariate

Likelihood analysis of non-Gaussian measurement time series

t

Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator

t densities as its invariant density. The reversible proposals are a combination of independent and correlated components that allow the sampler to traverse the sample space efficiently as well as allowing the sampler to keep moving and exploring the sample space locally. We employ a two-chain approach, using a trial chain to adapt the proposal in the main chain. Convergence of the main chain and a strong law of large numbers are obtained under checkable conditions, and without imposing a diminishing adaptation condition. The mixtures of multivariate