Convergence Properties of the Likelihood of Computed Dynamic Models

Jesús Fernández-Villaverde, Juan Francisco Rubio-Ramírez, and Manuel Santos
Working Paper 2004-27
November 2004

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This paper studies the econometrics of computed dynamic models. Since these models generally lack a closed-form solution, economists approximate the policy functions of the agents in the model with numerical methods. But this implies that, instead of the exact likelihood function, the researcher can evaluate only an approximated likelihood associated with the approximated policy function. What are the consequences for inference of the use of approximated likelihoods? First, we show that as the approximated policy function converges to the exact policy, the approximated likelihood also converges to the exact likelihood. Second, we prove that the approximated likelihood converges at the same rate as the approximated policy function. Third, we find that the error in the approximated likelihood gets compounded with the size of the sample. Fourth, we discuss convergence of Bayesian and classical estimates. We complete the paper with three applications to document the quantitative importance of our results.

JEL classification: C63, C68, E37

Key words: likelihood function, dynamic models, numerical approximation

The authors thank Jim Nason, Tom Sargent, Frank Schorfheide, Tao Zha, and participants at several seminars for useful comments. The views expressed here are the authors’ and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors’ responsibility.

Please address questions regarding content to Jesús Fernández-Villaverde, Department of Economics, University of Pennsylvania, 160 McNeil Building, 3718 Locust Walk, Philadelphia, Pennsylvania 19104, 215-898-1504,; Juan Francisco Rubio-Ramírez, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street, NE, Atlanta, Georgia 30309-4470, 404-498-8057;; or Manuel Santos, BAC 551, Department of Economics, Arizona State University, Tempe, Arizona 85287-3806, 480-965-6335.