MCMC Methods for Continuous-Time Financial Econometrics

Abstract

Publisher Summary This chapter describes various Markov Chain Monte Carlo (MCMC) methods for exploring the posterior distributions generated by continuous-time asset pricing models. The MCMC methods are particularly well suited for continuous-time finance applications for several reasons. MCMC is a unified estimation procedure, which simultaneously estimates both parameters and latent variables. MCMC directly computes the distribution of the latent variables and parameters given the observed data and allows the researcher to quantify estimation and model risk. Estimation risk is the inherent uncertainty present in estimating parameters or state variables, while model risk is the uncertainty over model specification. The simplest MCMC algorithm is called the Gibbs sampler, which requires one to conveniently draw from the complete set of conditional distributions. In many cases, implementing the Gibbs sampler requires drawing random variables from standard continuous distributions such as normal, t, beta, or gamma or discrete distributions such as binomial, multinomial, or Dirichlet. The Griddy Gibbs sampler is an approximation that can be applied to approximate the conditional distribution by a discrete set of points. The Metropolis–Hastings algorithm allows the functional form of the density to be nonanalytic, where one only has to evaluate the true density at two given points. Random-walk Metropolis is the original algorithm considered by Metropolis et al. in 1953, and it is the mirror image of the independence Metropolis–Hastings algorithm.