Numerically Stable and Accurate Stochastic Simulation Approaches for Solving Dynamic Economic Models - A Generalized Stochastic Simulation Algorithm (GSSA) - Abstract

Sep 18, 2026

Numerically stable and accurate stochastic simulation methods are developed for solving dynamic economic models. Instead of standard least-squares approximation, the paper evaluates numerically stable alternatives such as singular value decomposition and Tikhonov regularization least-squares, least-absolute deviations, and principal component regression, suited for ill-conditioned problems. It replaces conventional Monte Carlo integration with accurate quadrature and monomial integration, and tests the generalized stochastic simulation algorithm (GSSA) across representative-agent growth, rare-disasters, and multicountry models, while providing MATLAB code.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages38

Reading Time1 h 36 min

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Introduction

Motivation: numerical methods for dynamic stochastic models

From standard approaches to generalized stochastic simulation (GSSA)

Quantitative Economics 2 (2011), 173–210 1759-7331/20110173 Numerically stable and accurate stochastic simulation approaches for solving dynamic economic models Kenneth L. Judd Hoover Institution, Stanford University and NBER Lilia Maliar University of Alicante and Hoover Institution, Stanford University Serguei Maliar University of Alicante and Hoover Institution, Stanford University We develop numerically stable and accurate stochastic simulation approaches for solving dynamic economic models. First, instead of standard least-squares approximation methods, we examine a variety of alternatives, including leastsquares methods using singular value decomposition and Tikhonov regularization, least-absolute deviations methods, and principal component regression method, all of which are numerically stable and can handle ill-conditioned problems. Second, instead of conventional Monte Carlo integration, we use accurate quadrature and monomial integration. We test our generalized stochastic simulation algorithm (GSSA) in three applications: the standard representative–agent neoclassical growth model, a model with rare disasters, and a multicountry model with hundreds of state variables. GSSA is simple to program, and MATLAB codes are provided. Keywords. Stochastic simulation, generalized stochastic simulation algorithm, parameterized expectations algorithm, least absolute deviations, linear programming, regularization. JEL classification. C63, C68 . 1. Introduction Dynamic stochastic economic models do not generally admit closed-form solutions and must be studied with numerical methods.1 Most methods for solving such models fall [Kenneth L. Judd:](Kenneth L. Judd: kennethjudd@mac.com)[ kennethjudd@mac.com](Kenneth L. Judd: kennethjudd@mac.com) [Lilia Maliar:](Lilia Maliar: maliarl@stanford.edu)[ maliarl@stanford.edu](Lilia Maliar: maliarl@stanford.edu) [Serguei Maliar:](Serguei Maliar: maliars@stanford.edu)[ maliars@stanford.edu](Serguei Maliar: maliars@stanford.edu) We thank a co-editor and two anonymous referees for very useful comments that led to a substantial improvement of the paper. LiliaMaliar and Serguei Maliar acknowledge support from the Hoover Institution at Stanford University, the Ivie, the Generalitat Valenciana under Grants BEST/2010/142 and BEST/2010/141, respectively, the Ministerio de Ciencia e Innovación de España, and FEDER funds under project SEJ-2007- 62656 and under the programs José Castillejo JC2008-224 and Salvador Madariaga PR2008-190, respectively. 1 For reviews of such methods, see Taylor and Uhlig (1990), Rust (1996), Gaspar and Judd (1997), Judd (1998), Marimon and Scott (1999), Santos (1999), Christiano and Fisher (2000), Miranda and Fackler (2002), Copyright © 2011 Kenneth L. Judd, Lilia Maliar, and Serguei Maliar. Licensed under the Creative Commons Attribution-NonCommercial License 3.0. Available at [http://www.qeconomics.org](http://www.qeconomics.org). DOI: 10.3982/QE14 174 Judd, Maliar, and Maliar Quantitative Economics 2 (2011) into three broad classes: projection methods, which approximate solutions on a prespeciﬁed domain using deterministic integration; perturbation methods, which ﬁnd solutions locally using Taylor expansions of optimality conditions; and stochastic simulation methods, which compute solutions on a set of simulated points using Monte Carlo integration. All three classes of methods have their relative advantages and drawbacks, and the optimal choice of a method depends on the application. Projection methods are accurate and fast when applied to models with few state variables; however, their cost increases rapidly as the number of state variables increases. Perturbation methods are practical to use in high-dimensional applications, but the range of their accuracy is uncertain.2 Stochastic simulation algorithms are simple to program although they are generally less accurate than projection methods and often numerically unstable.3 In th

FAQs

Why do dynamic stochastic economic models require numerical methods?

They generally do not admit closed-form solutions, so solutions must be studied with numerical methods.

What distinguishes the paper’s GSSA approach from conventional methods?

GSSA emphasizes numerically stable approximation alternatives and uses accurate quadrature/monomial integration instead of conventional Monte Carlo integration.

Which applications are used to test the generalized stochastic simulation algorithm (GSSA)?

The algorithm is tested in a representative-agent neoclassical growth model, a rare-disasters model, and a multicountry model with hundreds of state variables.

Sep 18, 2026

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