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• Roy Cerqueti

While the large portion of the literature on Markov chain (possibly of order
higher than one) bootstrap methods has focused on the correct estimation of
the transition probabilities, little or no attention has been devoted to the
problem of estimating the dimension of the transition probability matrix.
Indeed, it is usual to assume that the Markov chain has a one-step memory
property and that the state space could not to be clustered, and coincides
with the distinct observed values. In this paper we question the opportunity
of such a standard approach.
In particular we advance a method to jointly estimate the order of the Markov
chain and identify a suitable clustering of the states. Indeed in several real
life applications the "memory" of many
processes extends well over the last observation; in those cases a correct
representation of past trajectories requires a significantly richer set than
the state space. On the contrary it can sometimes happen that some distinct
values do not correspond to really "different
states of a process; this is a common conclusion whenever,
for example, a process assuming two distinct values in t is not affected in
its distribution in t+1. Such a situation would suggest to reduce the
dimension of the transition probability matrix.
Our methods are based on solving two optimization problems. More specifically
we consider two competing objectives that a researcher will in general pursue
when dealing with bootstrapping: preserving the similarity between the
observed and the bootstrap series and reducing the probabilities of getting a
perfect replication of the original sample. A brief axiomatic discussion is
developed to define the desirable properties for such optimal criteria. Two
numerical examples are presented to illustrate the method.

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Paper provided by Macerata University, Department of Finance and Economic Sciences in its series Working Papers with number 53-2009.

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Date of creation: Apr 2009
Date of revision: Apr 2009
Handle: RePEc:mcr:wpdief:wpaper00053

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Related research

Keywords: order of Markov chains; similarity of time series; transition probability matrices; multiplicity of time series; partition of states of Markov chains; Markov chains; bootstrap methods;

Find related papers by JEL classification:
C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Semiparametric and Nonparametric Methods
C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Statistical Simulation Methods
C61 - Mathematical and Quantitative Methods - - Mathematical Methods and Programming - - - Optimization Techniques; Programming Models; Dynamic Analysis

This paper has been announced in the following NEP Reports:

• NEP-ALL-2009-05-02 (All new papers)
• NEP-ECM-2009-05-02 (Econometrics)
• NEP-ETS-2009-05-02 (Econometric Time Series)
• NEP-ORE-2009-05-02 (Operations Research)

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This page was last updated on 2009-12-3.