Chapter 12

Markov Chain Monte Carlo Methods with Applications

Advances in computing facilities and computational methods have dramatically increased our ability to solve complicated problems. The advances also extend the applicability of many existing econometric and statistical methods. Examples of such achievements in statistics include the Markov chain Monte Carlo (MCMC) method and data augmentation. These techniques enable us to make some statistical inference that was not feasible just a few years ago. In this chapter, we introduce the ideas of MCMC methods and data augmentation that are widely applicable in finance. In particular, we discuss Bayesian inference via Gibbs sampling and demonstrate various applications of MCMC methods. Rapid developments in the MCMC methodology make it impossible to cover all the new methods available in the literature. Interested readers are referred to some recent books on Bayesian and empirical Bayesian statistics (e.g., Carlin and Louis, 2000; Gelman, Carlin, Stern, and Rubin, 2003).

For applications, we focus on issues related to financial econometrics. The demonstrations shown in this chapter represent only a small fraction of all possible applications of the techniques in finance. As a matter of fact, it is fair to say that Bayesian inference and the MCMC methods discussed here are applicable to most, if not all, of the studies in financial econometrics.

We begin the chapter by reviewing the concept of a Markov process. Consider a stochastic process {Xt}, where each Xt assumes a value in the space . The process {Xt} is a Markov process if it has the property that, given the value of Xt, the values of Xh, h > t, do not depend on the values Xs, s < t. In other words, {Xt} is a Markov process if its conditional distribution function satisfies

If {Xt} is a discrete-time stochastic process, then the prior property becomes

Let A be a subset of . The function

is called the transition probability function of the Markov process. If the transition probability depends on h − t, but not on t, then the process has a stationary transition distribution.