Estimation and inference in multivariate Markov chains

Abstract

The literature of Markov chains has recently focused on modeling multiple categorical data sequences. The usual procedure for handling these multivariate Markov chains (MMC), with $m$ categorical data and $s$ states, consists of expanding the state space by considering $m^{s}$ new states. This model rapidly becomes intractable even with moderate values of $m$ and $s$ due to the excessive number of parameters to estimate. Ching and Fung (2002) found a way to cope with the intractability of the conventional MMC. They also suggested a method of estimation that proved to be inefficient. Zhu and Ching (2010) proposed another method of estimation based on minimizing the prediction error with equality and inequality restrictions. However, both these procedures treat the estimation problem as a mechanic method, without addressing the statistical inference problem. In this article we try to overcome this shortcoming and, at the same time, we propose a new approach to estimate MMC (under Ching et al. hypothesis) which avoids imposing equality and inequality restrictions on the parameters. We illustrate the model and the estimation method with two applications on financial time series data.