Smooth particle filters for likelihood evaluation and maximisation

In this paper,a method is introduced for approximating the likelihood for the unknown parameters of a state space model.The approximation converges to the true likelihood as the simulation size goes to infinity. In addition,the approximating likelihood is continuous as a function of the unknown parameters under rather general conditions.The approach advocated is fast, robust and avoids many of the pitfalls associated with current techniques based upon importance sampling.We assess the performance of the method by considering a linear state space model, comparing the results with the Kalman filter, which delivers the true likelihood. We also apply the method to a non-Gaussian state space model, the Stochastic Volatility model, finding that the approach is efficient and effective. Applications to continuous time finance models are also considered. A result is established which allows the likelihood to be estimated quickly and efficiently using the output from the general auxiliary particle filter

Likelihood based inference for diffusion driven models

This paper provides methods for carrying out likelihood based inference for diffusion driven models, for example discretely observed multivariate diffusions, continuous time stochastic volatility models and counting process models. The diffusions can potentially be non-stationary. Although our methods are sampling based, making use of Markov chain Monte Carlo methods to sample the posterior distribution of the relevant unknowns, our general strategies and details are different from previous work along these lines. The methods we develop are simple to implement and simulation efficient. Importantly, unlike previous methods, the performance of our technique is not worsened, in fact it improves, as the degree of latent augmentation is increased to reduce the bias of the Euler approximation. In addition, our method is not subject to a degeneracy that afflicts previous techniques when the degree of latent augmentation is increased. We also discuss issues of model choice, model checking and filtering. The techniques and ideas are applied to both simulated and real data.Bayes estimation, Brownian bridge, Non-linear diffusion, Euler approximation, Markov chain Monte Carlo, Metropolis-Hastings algorithm, Missing data, Simulation, Stochastic differential equation.

Smooth Particle Filters for Likelihood Evaluation and Maximisation

In this paper, a method is introduced for approximating the likelihood for the unknown parameters of a state space model. The approximation converges to the true likelihood as the simulation size goes to infinity. In addition, the approximating likelihood is continuous as a function of the unknown parameters under rather general conditions. The approach advocated is fast, robust and avoids many of the pitfalls associated with current techniques based upon importance sampling. We assess the performance of the method by considering a linear state space model, comparing the results with the Kalman filter, which delivers the true likelihood. We also apply the method to a non-Gaussian state space model, the Stochastic Volatility model, finding that the approach is efficient and effective. Applications to continuous time finance models are also considered. A result is established which allows the likelihood to be estimated quickly and efficiently using the output from the general auxilary particle filter. keywords: Importance Sampling ; Filtering ; Particle filter ; Simulation ; SIR ; State space

Bayesian inference for a semi-parametric copula-based Markov chain

This paper presents a method to specify a strictly stationary univariate time series model with particular emphasis on the marginal characteristics (fat tailedness, skewness etc.). It is the first time in time series models with specified marginal distribution, a non-parametric specification is used. Through a Copula distribution, the marginal aspect are separated and the information contained within the order statistics allow to efficiently model a discretely-varied time series. The estimation is done through Bayesian method. The method is invariant to any copula family and for any level of heterogeneity in the random variable. Using count times series of weekly rearm homicides in Cape Town, South Africa, we show our method efficiently estimates the copula parameter representing the first-order Markov chain transition density

Modelling stochastic volatility with leverage and jumps: a simulated maximum likelihood approach via particle filtering

In this paper we provide a unified methodology in order to conduct likelihood-based inference on the unknown parameters of a general class of discrete-time stochastic volatility models, characterized by both a leverage effect and jumps in returns. Given the non-linear/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 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 non-trivial task of model comparison. The technique is applied to daily returns data for various stock price indices. We find strong evidence in favour of a leverage effect in all cases. Jumps are an important component in two out of the four series we consider