Multivariate Markov Chain Approximations

Abstract

To solve equilibrium models numerically, it is necessary to discretize vector autoregressive processes (VAR) into a finite number of states. Univariate Markov chain approximations are well studied, however, few papers address the multivariate case. This paper presents three approaches to the problem: quadrature, moment matching and bin estimation.

Quadrature uses numerical integration schemes over the conditional distribution of the error terms. Moment matching replicates the first moments of the VAR. Bin estimation segments the data into bins and estimates the transition probabilities with maximum likelihood.

A comparative study in a standard asset pricing model shows that quadrature has difficulties when the model involves only few states, bin estimation fares better, while moment matching delivers the smallest errors. However, an experiment demonstrates the convincing results of moment matching to be double-edged. The introduction of a disaster state permits to alter model results despite matching all first and second moments.

Keywords: Markov chain approximation, Asset Pricing, Macroeconomics, Dynamic Programming

JEL Classification: C63, E44, G12