Merging Simulation and Projection Approaches to Solve High-Dimensional Problems with an Application to a New Keynesian model

Sep 18, 2026

A numerical algorithm is presented for solving dynamic economic models by merging stochastic simulation and projection. Stochastic simulation approximates the ergodic measure, while the method covers its support using a fixed grid. Projection techniques then solve the model accurately on that grid. The core novelty is replacing a large cloud of simulated points with a smaller set of representative points. Three construction techniques are compared: clustering, epsilon-distinguishable sets, and a locally adaptive variant. The approach targets high-dimensional state spaces and is illustrated using neoclassical growth and large-scale New Keynesian models with a zero lower bound.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages51

Reading Time2 h 9 min

AccessFree to download

Abstract

Introduction

Stochastic simulation methods

Projection techniques

Combined approach and example (Figures 1a-1d)

Merging Simulation and Projection Approaches to Solve High-Dimensional Problems with an Application to a New Keynesian model Lilia Maliar and Serguei Maliar March 9, 2014 Abstract We introduce a numerical algorithm for solving dynamic economic models that merges stochastic simulation and projection approaches: we use simulation to approximate the ergodic measure of the solution, we cover the support of the constructed ergodic measure with a Öxed grid, and we use projection techniques to accurately solve the model on that grid. The construction of the grid is the key novel piece of our analysis: we replace a large cloud of simulated points with a small set of "representative" points. We present three alternative techniques for constructing representative points: a clustering method, an epsilon-distinguishable set method, and a locally-adaptive variant of the epsilon-distinguishable set method. As an illustration, we solve one-and multi-agent neoclassical growth models and a large-scale new Keynesian model with a zero lower bound on nominal interest rates. The proposed solution algorithm is tractable in problems with high dimensionality (hundreds of state variables) on a desktop computer. JEL classification : C61, C63, C68, E31, E52 Key Words : ergodic set; "-distinguishable set; clusters; adaptive grid; discrepancy; large-scale model; new Keynesian model; ZLB; stochastic simulation Ken Judd was also an author on this paper. See page 43 1 Introduction We introduce a numerical algorithm for solving dynamic economic models that merges stochastic simulation and projection approaches: we use simulation to approximate the ergodic measure of the solution, we cover the support of the constructed ergodic measure with a Öxed grid, and we use projection techniques to accurately solve the model on that grid. The construction of the grid is the key novel piece of our analysis: wereplace a large cloud of simulated points with a small set of "representative" points. We present three alternative techniques for constructing representative points: a clustering method, an epsilon-distinguishable set method, and a locally-adaptive variant of the epsilon-distinguishable set method. As an illustration, we solve one-and multi-agent neoclassical growth models and a large-scale new Keynesian model with a zero lower bound on nominal interest rates. The proposed solution algorithm is tractable in problems with high dimensionality (hundreds of state variables) on a desktop computer. One broad class of numerical methods for solving dynamic economic models buildson stochastic simulation. First, this class includes iterative methods for solving rational expectations models; see, e.g., Marcet (1988), Smith (1993), Maliar and Maliar (2005) and Judd et al. (2011a) . Second, it includes learning-based analysis; see, e.g., Marcet and Sargent (1989), Bertsekas and Tsitsiklis (1996), Pakes and McGuire (2001) and Powell (2011) . Finally, it includes methods that use simulation to reduce information sets of decision makers; see, e.g., Krusell and Smith (1998), and Benkard et al. (2008) . The key advantage of stochastic simulation methods is that the geometry of the set on which the solution is computed is adaptive. Namely, these methods solve dynamic economic modelson a set of points produced by stochastic simulation, avoiding thus the cost of Önding solutions in areas of the state space that are e§ectively never visited in equilibrium; see Judd et al. (2011a) for a discussion. However, a set of simulated points itself is not ane¢ cient choice either as a grid for approximating a solution (it contains many closelylocated and hence, redundant points) or as a set of nodes for approximating expectation functions (the accuracy of Monte Carlo integration is low) . Another broad class of numerical methods for solving dynamic economic models relies on projection techniques; see, e.g., Wright and Williams (1984), Judd (1992), Christiano and Fisher (20

FAQs

What problem does the proposed algorithm address?

It addresses solving dynamic economic models with high-dimensional state spaces efficiently, using an approach that remains tractable on a desktop computer.

How does the method combine simulation and projection?

Simulation approximates the solution’s ergodic measure, a fixed grid covers its support, and projection techniques then compute accurate solutions on that grid.

What is the main novelty in the paper’s grid construction?

The grid replaces a large cloud of simulated points with a smaller set of representative points, constructed via clustering, epsilon-distinguishable sets, or a locally adaptive variant.

Sep 18, 2026

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