Analysis of Financial Time Series, Third Edition

12.5 Linear Regression with Time Series Errors

We are ready to consider some specific applications of MCMC methods. Examples discussed in the next few sections are for illustrative purposes only. The goal here is to highlight the applicability and usefulness of the methods. Understanding these examples can help readers gain insights into applications of MCMC methods in finance.

The first example is to estimate a regression model with serially correlated errors. This is a topic discussed in Chapter 2, where we use SCA to perform the estimation. A simple version of the model is

where yt is the dependent variable, xit are explanatory variables that may contain lagged values of yt, and zt follows a simple AR(1) model with {at} being a sequence of independent and identically distributed normal random variables with mean zero and variance σ2. Denote the parameters of the model by , where , and let be the vector of all regressors at time t, including a constant of unity. The model becomes

12.6

where n is the sample size.

A natural way to implement Gibbs sampling in this case is to iterate ...