  Markov chain models are used in various fields, such 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.
