A Gibbs sampling scheme to the product partition model: an application to change-point problems

Abstract

This paper extends previous results for the classical product partition model applied to the identification of multiple change points in the means and variances of time series. Prior distributions for these two parameters and for the probability p that a change takes place at a particular period of time are considered and a new scheme based on Gibbs sampling to estimate the posterior relevances of the model is proposed. The resulting algorithm is applied to the analysis of two Brazilian stock market data. The computational experiments seem to indicate that the algorithm runs fast in common PC-like machines and it may be a useful tool for analyzing change-point problems. Scope and purpose The problem of change-point identification is encountered in many subject areas, including disease mapping, medical diagnosis, industrial control, and finance. A Bayesian way to tackle the problem is through the well-known product partition model (PPM) introduced by Hartigan in the early 1990s. Nowadays, the PPM is still attracting researchers’ attention because of its flexibility and the spreading use of the powerful personal computers that make it possible to deal with its inherent computational complexity. In this paper, the PPM is tailored to the identification of change points both in the means and variances of time series, assuming that, given these parameters, the data are normally distributed. We extend some previous works by considering a non-degenerate prior distribution to the probability p of having a change point at a particular period of time. An original Gibbs sampling scheme is also developed to compute the product estimates and, consequently, to attack the difficult resulting model which is applied to the identification of change points in the expected returns (means) and volatilities (variances) of two important stock market data in Brazil. The computational results seem to indicate that method is effective and efficient, making it possible useful inferences. In addition, the method is simple and easy to implement.