Variable Length Markov Chain with Exogenous Covariates

TL;DR

This paper introduces a variable length Markov chain model that incorporates exogenous covariates via a logistic function, improving prediction accuracy when covariates influence transitions.

Contribution

It presents a new model combining variable length Markov chains with exogenous covariates and proves its consistency in identifying true contexts and coefficients.

Findings

The method is consistent as sample size increases.

Outperforms traditional variable length Markov chains when covariates are relevant.

Performs comparably when covariates are not influential.

Abstract

Markov Chains with variable length are useful stochastic models for data compression that avoid the curse of dimensionality faced by that full Markov Chains. In this paper we introduce a Variable Length Markov Chain whose transition probabilities depend not only on the state history but also on exogenous covariates through a logistic model. The goal of the proposed procedure is to obtain the context of the process, that is, the history of the process that is relevant for predicting the next state, together with the estimated coefficients corresponding to the significant exogenous variables. We show that the proposed method is consistent in the sense that the probability that the estimated context and the coefficients are equal to the true data generating mechanism tend to 1 as the sample size increases. Simulations suggest that, when covariates do contribute for the transition probability, the proposed procedure outperforms variable length Markov Chains that do not consider covariates while yielding comparable results when covariates are not present.