Optimal Dimension of Transition Probability Matrices for Markov Chain Bootstrapping

 <font size="1">While the large portion of the literature on Markov chain (possibly of orderhigher than one) bootstrap methods has focused on the correct estimation ofthe transition probabilities, little or no attention has been devoted to theproblem of estimating the dimension of the transition probability matrix.Indeed, it is usual to assume that the Markov chain has a one-step memoryproperty and that the state space could not to be clustered, and coincideswith the distinct observed values. In this paper we question the opportunityof such a standard approach.In particular we advance a method to jointly estimate the order of the Markovchain and identify a suitable clustering of the states. Indeed in several reallife applications the "memory" of manyprocesses extends well over the last observation; in those cases a correctrepresentation of past trajectories requires a significantly richer set thanthe state space. On the contrary it can sometimes happen that some distinctvalues do not correspond to really "differentstates of a process; this is a common conclusion whenever,for example, a process assuming two distinct values in t is not affected inits distribution in t+1. Such a situation would suggest to reduce thedimension of the transition probability matrix.Our methods are based on solving two optimization problems. More specificallywe consider two competing objectives that a researcher will in general pursuewhen dealing with bootstrapping: preserving the similarity between theobserved and the bootstrap series and reducing the probabilities of getting aperfect replication of the original sample. A brief axiomatic discussion isdeveloped to define the desirable properties for such optimal criteria. Twonumerical examples are presented to illustrate the method.</font>