Academic journal article Federal Reserve Bank of Atlanta, Working Paper Series

A Moment-Matching Method for Approximating Vector Autoregressive Processes by Finite-State Markov Chains

Academic journal article Federal Reserve Bank of Atlanta, Working Paper Series

A Moment-Matching Method for Approximating Vector Autoregressive Processes by Finite-State Markov Chains

Article excerpt

1 Introduction

Nonlinear dynamic macroeconomic and asset pricing models often imply a set of integral equations that do not admit explicit solutions. The finite-state Markov chain approximation methods developed by Tauchen (1986a) and Tauchen and Hussey (1991) prove to be an effective tool for reducing the complexity of solving these equations where the state variables follow autoregressive processes (Burnside, 1999). However, it is well known that these methods do not perform well for highly persistent autoregressive (AR) processes or processes with characteristic roots close to unity (see, e.g., Tauchen, 1986a, Tauchen and Hussey, 1991, and Floden, 2008). Although, the methods can generate a better approximation at the cost of a finer state space, this type of approach is not always feasible, especially in the multivariate case. The latter is important, as persistent multivariate structural shocks have become an increasingly popular device in accounting for business cycle fluctuations (e.g., Curdia and Reis, 2010, and Caldara. Fernandez-Villaverde, Rubio-Ramirez and Yao, 2012).

The poor approximation of the methods by Tauchen (1986a) and Tauchen and Hussey (1991) for strongly autocorrelated processes has spurred a renewed research interest given the prevalence of highly persistent shocks in dynamic macroeconomic models. Rouwenhorst (1995) proposes a Markov-chain approximation of an AR(1) process constructed by targeting its first two conditional moments. Some recent advances in the literature on Markov-chain approximation methods include Adda and Cooper (2003), Floden (2008) and Kopecky and Suen (2010). While these methods provide substantial improvements in approximating the first-order univariate autoregressions, their extension to vector autoregressions (and higher-order autoregressive processes), which is of great practical interest to macroeconomists, is not readily available and possibly highly non-trivial. As a result, the method by Tauchen (1986a) continues to be employed almost exclusively by researchers for approximating multivariate processes by finite-state Markov chains. The only alternative method that is available for approximating multivariate processes is the method proposed by Galindev and Lkhagvasuren (2010) for models with correlated AR(1) shocks. Although this method can be applied to vector autoregressions (VAR) by decomposing the latter into a set of interdependent AR(1) shocks, the state space generated by the method is not finite, except for the special case of equally-persistent underlying shocks. Therefore, to the best of our knowledge, a general method for approximating VAR processes by a finitestate Markov chain with appealing approximation properties over the whole parameter region of interest (including highly persistent parameterizations) is not yet available in the literature.

This paper fills this gap and proposes a moment-matching method for approximating vector autoregressions by a finite-state Markov chain. The main idea behind this method is to construct the Markov chain by targeting conditional moments of the underlying continuous process as in Rouwenhorst (1995), rather than directly calculating the transition probabilities using the distribution of the continuous process as in the existing methods. More specifically, we obtain the Markov-chain transitional probabilities by mixing a set of probability mass functions associated with the conditional distributions of finite-state univariate processes. To target the conditional moments in constructing the Markov chain, we use key elements of the Markov chains generated by the methods of Tauchen (1986a) and Rouwenhorst (1995). Therefore, the proposed method extends the multivariate methods of Tauchen (1986a) and Tauchen and Hussey (1991) to highly persistent cases and Rouwenhorst's (1995) scalar method (see Kopecky and Suen, 2010) to vector cases, while still maintaining a finite number of states.

Our method yields accurate approximations without relying on a large number of grid points for the state variables. …

Search by... Author
Show... All Results Primary Sources Peer-reviewed

Oops!

An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.