Discretizing nonlinear, non-Gaussian Markov processes with exact conditional moments

Abstract

Approximating stochastic processes by finite-state Markov chains is useful for reducing computational complexity when solving dynamic economic models. We provide a new method for accurately discretizing general Markov processes by matching low order moments of the conditional distributions using maximum entropy. In contrast to existing methods, our approach is not limited to linear Gaussian autoregressive processes. We apply our method to numerically solve asset pricing models with various underlying stochastic processes for the fundamentals, including a rare disasters model. Our method outperforms the solution accuracy of existing methods by orders of magnitude, while drastically simplifying the solution algorithm. The performance of our method is robust to parameters such as the number of grid points and the persistence of the process. and seminar participants at Duke, McGill, UCSD, University of Technology Sydney, and the 2016 Computing in Economics and Finance Conference for helpful comments and feedback. We are especially grateful to four anonymous referees for constructive comments and suggestions that significantly improved the paper. Matlab codes are posted in a supplementary file on the journal website,

Wealth Distribution with Random Discount Factors

“…12 To convert this process into a finite-state Markov chain, I use the discretization method proposed by Farmer and Toda (2017)-which is more accurate and generally applicable than other discretization methods-with a 9-point even-spaced grid and treat this Markov chain as the true process. Finally, I set the death probability p = 0.025 so that the mean life span (which should be interpreted as the average years in the labor market) is 40.…”

The discretization filter: A simple way to estimate nonlinear state space models

“…Gospodinov and Lkhagvasuren (2014) developed a method that builds on the Rouwenhorst method to better approximate persistent Gaussian VARs by matching low order conditional moments. Most recently, Farmer and Toda (2017) developed a method for approximating general nonlinear, non‐Gaussian first‐order Markov processes by matching conditional moments using maximum entropy.…”

“…The asymptotic theory I developed in Section 5 shows that if the Farmer and Toda (2017) method with a trapezoidal quadrature rule is used to construct the transition matrix, the discretization error of the likelihood function is of the order . While this is only a rate condition, I use it to recommend a rule of thumb choice for the number of points M used to construct the discretization.…”

“…Second, I construct the transition matrix as in Farmer and Toda (2017). Their method finds the discrete distribution which is “closest” to a coarse approximation of the original continuous distribution in terms of Kullback–Leibler distance.…”

Huggett economies with multiple stationary equilibria

“…This paper is also related to papers that obtain closed-form solutions to dynamic general equilibrium models, such as Labadie (1989), Burnside (1998), Tsionas (2003), and de Groot (2015, among others. Most of these papers provide solutions to asset pricing models, which have been applied to evaluate the solution accuracy of numerical methods (Collard and Juillard, 2001;SchmittGrohé and Uribe, 2004;Farmer and Toda, 2016). My solution to the income fluctuation problem may be useful for evaluating the accuracy of numerical algorithms to solve heterogeneous-agent models.…”