General Autoregressive Modelling of Markov Chains

The reduction of the number of parameters in high-order Markov chain already inspired several articles. In particular, Raftery proposed an autoregressive modelling which utilizes the same transition matrix, with a coefficient, for every lag. In this paper, we show that a model of the same type, but utilizing different matrices, gives best results and is not harder to estimate, even when the number of data is small. A numerical illustration confirms the theoretical results.