Markov chain Monte Carlo test of toric homogeneous Markov chains

Markov chain models are used in various fields, such as behavioral sciences or econometrics. Although the goodness of fit of the model is usually assessed by large sample approximation, it is desirable to use conditional tests if the sample size is not large. We study Markov bases for performing conditional tests of the toric homogeneous Markov chain model, which is the envelope exponential family for the usual homogeneous Markov chain model. We give a complete description of a Markov basis for the following cases: (i) two-state, arbitrary length, (ii) arbitrary finite state space and length of three. The general case remains to be a conjecture. We also present a numerical example of conditional tests based on our Markov basis.