Recent Developments of Bayesian Econometrics:
Bayesian Inference Using MCMC Algorithms
Hiroki Tsurumi
December 16 2005

Over the last decade Bayesian statisticians and econometricians been using Markov Chain Monte Carlo (MCMC) algorithms at an increasing rate.
So prevalent is the application of MCMC algorithms, Bayesian inference, atleast at the graduate and research levels, cannot be discussed without understanding MCMC algorithms. There are ready-to-use computer software
for MCMC algorithms, a leading one being WinBugs. Many papers havebeen written using WinBugs. However, I believe that one should write one’s
own MCMC algorithms so that one needs not rely on canned software.
I will explain MCMC algorithms using recent published and unpublishedpapers. To do so, I have put together these lecture notes so that the reader
will first run a relevant GAUSS program and and then check the lecturenotes. With this “hands-on” or “do it by yourself” approach, I hope thatthe reader will learn MCMC algorithms. I will cover linear regression model,
convergence tests, probit and logit models, and various time series models.
Let me explain each major topic in the lecture notes.
1. linear regression modelThis section is written to understand Gibbs sampler and MetropolisHastings algorithms. Predictive densities (pp.16–23) and densities ofthe mean squared forecast errors are original materials. A proof that
the Metropolis-Hastings algorithm yields an invariant distribution isgiven in Appendix A, and in Appendix B three ways to draw random
numbers from an inverted Wishart distribution are discussed. All thecomputer programs are in GAUSS.
2. Convergence tests
This section is taken from a paper by Goldman, Valieva, and Tsurumi(2005) “Kolmogorov-Smirnov, Fluctuation, and Zg Tests for Convergence of Markov Chain Monte Carlo Draws.” See the pdf file:flks_modc.pdf.
The GAUSS program is ftks_pw.pro.

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3. Probit and Logit Models
Probit and logit models have been used predominantly in micro econometric applications under the title of qualitative choice models. Recently probit and logit models have been used in marketing to analyze
consumer behavioral patterns in choosing different brands and models. I shall discuss the basic probit model and nested multinomial logit
model. Applications are (i) consumer choice of organic produce (usingprobit) and (ii) medical treatment case (using nested multinomial logit
model.)
4. Asset Pricing Models
Focusing on the CKLS model, I will explain Griddy Gibbs and Efficient
Jump algorithms. Also, the CKLS model with ARMA-GARCH error
processes will be discussed.
5. Bayesian Unit Root Test
The posterior probability density function (pdf) of the maximum of
the absolute value of the inverted roots of an autoregressive process
is used to test a unit root. The text for this topic is Goldman, E.,
S. Radchencko, T. Nakatsuma and H. Tsurumi.“A Bayesian test ofstationarity in a regression model with an ARMA or ARMA-GARCH
error term,” Proceedings of the American Statistical Association
Section B, 2001, 61–68
6. Bayesian Cointegration Test and Pairwise Granger Causality
Test
The text for this topic is Chapter 3 of Bayes Keriryokeizai Bunseki:
Markov Chain Monte Carlo ho to sono oyou, Toyokeizai Shimpo,Tokyo, 2005
7. Threshold Autoregressive and Moving Average (TARMA) andFractional Autoregressive and Moving Average (FARMA) models
Threshold autoregressive models (TAR) have been popular in timeseries analysis. The standard TAR model is extended to threshold autoregressive and moving average (TARMA) models. Another popular2

time series models are long memory models or fractional autoregressive and moving average (FARMA) models. We estimate TARMA andFARMA models using MCMC algorithms.
8. Stochastic Volatility Models
There are two approaches to estimate SV models: (i) use of a proposal
density for the SV variable, and (ii) state-space representation for theSV variable. We discuss both of these approaches.

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