Multivariate Markov Chain Bootstrapping and Contiguity Constraint, Working Paper no. 15 of the Department of Economics and Management of the University of Brescia, p. 1-53

This paper extends Markov chain bootstrapping to the case of multivariate continuous-valued processes. To this purpose we follow the approach of searching an optimal partition of the state space of an observed (multivariate) time series. The optimization problem is based on a distance indicator calculated on the transition probabilities of the Markov chain. Such criterion searches to group those states showing similar transition probabilities. A second methodological contribution is represented by the addition of a contiguity constraint, which is introduced to force the states to group only if they have “near” values (in the state space). This requirement meets two important aspects: firstly, it allows a more intuitive interpretation of the results; secondly, it contributes to control the complexity of the problem, which explodes with the cardinality of the states. The computational complexity of the optimization problem is also addressed through the introduction of a Tabu Search algorithm. The bootstrap method is applied to two empirical cases: the bivariate process of prices and traded volumes of electricity in the Spanish market; the trivariate process composed by prices and traded volumes of a US company stock (McDonald’s) and prices of the Dow Jones Industrial Average index. A comparison between our proposal and another Tabu Search procedure previously advanced in the literature is also performed. The analysis of the empirical studies and of the outcomes of the comparisons confirms good consistency properties for the bootstrap method here proposed.