Bayesian sequential inference for nonlinear multivariate diffusions

In this paper, we adapt recently developed simulation-based sequential algorithms to the problem concerning the Bayesian analysis of discretely observed diffusion processes. The estimation framework involves the introduction of m−1 latent data points between every pair of observations. Sequential MCMC methods are then used to sample the posterior distribution of the latent data and the model parameters on-line. The method is applied to the estimation of parameters in a simple stochastic volatility model (SV) of the U.S. short-term interest rate. We also provide a simulation study to validate our method, using synthetic data generated by the SV model with parameters calibrated to match weekly observations of the U.S. short-term interest rate.

[1]  S. E. Ahmed,et al.  Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference , 2008, Technometrics.

[2]  M. Pitt,et al.  Likelihood analysis of non-Gaussian measurement time series , 1997 .

[3]  N. Shephard,et al.  Stochastic Volatility: Likelihood Inference And Comparison With Arch Models , 1996 .

[4]  P. J. Green,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[5]  J. Q. Smith,et al.  1. Bayesian Statistics 4 , 1993 .

[6]  A. Gallant,et al.  Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes , 2002 .

[7]  D. Mayne,et al.  Monte Carlo techniques to estimate the conditional expectation in multi-stage non-linear filtering† , 1969 .

[8]  G. Roberts,et al.  On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm , 2001 .

[9]  A. Pedersen.  A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations , 1995 .

[10]  N. G. Best,et al.  Dynamic conditional independence models and Markov chain Monte Carlo methods , 1997 .

[11]  Christopher S. Jones,et al.  A Simple Bayesian Method for the Analysis of Diffusion Processes , 1998 .

[12]  M. Pitt,et al.  Likelihood based inference for diffusion driven models , 2004 .

[13]  P. Fearnhead,et al.  Improved particle filter for nonlinear problems , 1999 .

[14]  M. J. Bayarri,et al.  Non-Centered Parameterisations for Hierarchical Models and Data Augmentation , 2003 .

[15]  W. Wong,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[16]  Bjørn Eraker.  MCMC Analysis of Diffusion Models With Application to Finance , 2001 .

[17]  Nando de Freitas,et al.  Sequential Monte Carlo Methods in Practice , 2001, Statistics for Engineering and Information Science.

[18]  Nicholas G. Polson,et al.  State Space and Unobserved Component Models: Practical filtering for stochastic volatility models , 2004 .

[19]  P. Protter,et al.  The Monte-Carlo method for filtering with discrete-time observations , 2001 .

[20]  A. Gallant,et al.  ESTIMATION OF CONTINUOUS-TIME MODELS FOR STOCK RETURNS AND INTEREST RATES , 1997, Macroeconomic Dynamics.

[21]  Darren J. Wilkinson,et al.  Bayesian Sequential Inference for Stochastic Kinetic Biochemical Network Models , 2006, J. Comput. Biol..

[22]  Campbell R. Harvey,et al.  An Empirical Comparison of Alternative Models of the Short-Term Interest Rate , 1992 .

[23]  Bernard W. Silverman,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[24]  S. Chib,et al.  Understanding the Metropolis-Hastings Algorithm , 1995 .

[25]  M. Pitt,et al.  Filtering via Simulation: Auxiliary Particle Filters , 1999 .

[26]  Gareth O. Roberts,et al.  Non-centred parameterisations for hierarchical models and data augmentation. , 2003 .

[27]  D. Wilkinson,et al.  Bayesian Inference for Stochastic Kinetic Models Using a Diffusion Approximation , 2005, Biometrics.

[28]  Michael A. West,et al.  Combined Parameter and State Estimation in Simulation-Based Filtering , 2001, Sequential Monte Carlo Methods in Practice.

[29]  N. Shephard,et al.  Likelihood INference for Discretely Observed Non-linear Diffusions , 2001 .

[30]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[31]  Yadolah Dodge,et al.  Celebrating Statistics: Papers in honour of Sir David Cox on his 80th birthday , 2005 .

[32]  Neil Shephard.  Are there discontinuities in financial prices , 2005 .

[33]  B. Øksendal.  Stochastic Differential Equations , 1985 .

[34]  Siddhartha Chib,et al.  Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes: Comment. , 2002 .

[35]  M. West.  Approximating posterior distributions by mixtures , 1993 .

[36]  Simon J. Godsill,et al.  On sequential Monte Carlo sampling methods for Bayesian filtering , 2000, Stat. Comput..

[37]  T. Andersen,et al.  Estimating continuous-time stochastic volatility models of the short-term interest rate , 1997 .

[38]  M. Sørensen,et al.  Martingale estimation functions for discretely observed diffusion processes , 1995 .

[39]  Y. Amit.  On rates of convergence of stochastic relaxation for Gaussian and non-Gaussian distributions , 1991 .