Abstract
The accuracy of the solution of dynamic general equilibrium models has become a major issue. Recent papers, in which second-order approximations have been substituted for first-order, indicate that this change may yield a significant improvement in accuracy. Second order approximations have been used with considerable success when solving for the decision variables in both small and large-scale models. Additionally, the issue of accuracy is relevant for the approximate solution of value functions. In numerous dynamic decision problems, welfare is usually computed via this same approximation procedure. However, Kim and Kim (Journal of International Economics, 60, 471–500, 2003) have found a reversal of welfare ordering when they moved from first- to second-order approximations. Other researchers, studying the impact of monetary and fiscal policy on welfare, have faced similar challenges with respect to the accuracy of approximations of the value function. Employing a base-line stochastic growth model, this paper compares the accuracy of second-order approximations and dynamic programming solutions for both the decision variable and the value function as well. We find that, in a neighborhood of the equilibrium, the second-order approximation method performs satisfactorily; however, on larger regions, dynamic programming performs significantly better with respect to both the decision variable and the value function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aruoba B., Fernandez-Villaverde J., Rubio-Ramirez J.F. (2004). Comparing Solution Methods for Dynamic Equilibrium Economies. Mimeo, University of Pennsylvania
Aström K.J. (1970). Introduction to stochastic control theory. In: Bellmann R., (eds) Mathematics in Science and Engineering (Vol 70). New York, Academic
Bauer F., Grüne L., Semmler W. (2006). Adaptive spline interpolations for hamilton–Jacobi–Bellman equations. Applied Numerical Method 56: 1196–1210
Becker S. (2005). Numerische Lösungen dynamischer Gleichgewichtsmodelle. Diploma Thesis, University of Bayreuth.
Benigno P., Woodford M. (2006a). Optimal taxation in an RBC model: A linear quadratic approach. Journal of Economic Dynamics and Control 30(9–10): 1445–1485
Benigno P., Woodford M. (2006b). Linear quadratic approximation of optimal policy problems. Mimeo, Colombia University
Christiano L., Fisher J. (2000). Algorithms for solving models with occasionally binding constraints. Journal of Econmic Dynamics and Control 24: 1179–1231
Collard F., Juillard M. (2001). Accuracy of stochastic perturbation Method: The case of asset Pricing Models. Journal of Economic Dynamics and Control 25, 979–999
Den Haan W.J., Marcet A. (1994). Accuracy in simulations. The Review of Economics Studies 61, 3–17
Gong G., Semmler W. (2006). Stochastic dynamic macroeconomics: Theory and empirical evidence. Oxford, New York, Oxford University Press
Grüne L. (1997). An adaptive grid scheme for the discrete Hamilton–Jacobi–Bellman equation. Numerische Mathematik 75, 319–337
Grüne L., & Semmler W. (2004). Using dynamic programming with adaptive grid scheme for optimal control problems in economics. Journal of Economic Dynamics and Control 28, 2427–2456
Jin H., & Judd K.L. (2005) Perturbation methods for general dynamic stochastic models. Working Paper, Stanford University, http://bucky.stanford.edu/defaultmain.htm.
Judd K.L. (1996). Approximation, perturbation, and projection methods in economic analyis. In: Amman H., Kendrick D., Rust J., (eds), Handbook of computational economics. Amsterdam, Elsevier
Judd K.L. (1998). Numerical Methods in Economics. Cambridge, MIT
Kim J., Kim S.H. (2006). Two pitfalls of linearization methods. Mimeo, Tufts University
Kim J., Kim S.H. (2003). Spurious welfare reversals in international business cycle models. Journal of International Economics 60, 471–500
Santos M.S., Vigor Aguiar J. (1998). Analysis of a numerical dynamic programming algorithm applied to economic models. Econometrica 66, 409–426
Schmitt-Grohe S., Uribe M. (2004a). Solving dynamic general equilibrium models using a second-order approximation to the policy function. Journal of Economic Dynamics and Control 28, 755–775
Schmitt-Grohe S., Uribe M. (2004b). Optimal fiscal and monetary policy under sticky prices. Journal of Economic Theory 114, 198–230
Schmitt-Grohe S. (2005). Perturbation methods for the numerical analysis of DSGE models, Lecture Notes, preliminary draft, Duke University, www.econ.duke.edu/~grohe/teaching/perturbationmethods.htm.
Schmitt-Grohe S., & Uribe M. (2005). Matlab codes for second-order approximation. http://www.econ.duke.edu/uribe/2nd_order.htm.
Smetts F., & Wouters R. (2004). Comparing shocks and frictions in US and EURO-Area business cycles. ECB Working paper series, no. 391
Taylor J., Uhlig H. (1990). Solving nonlinear stochastic growth models: A comparison of alternative solution methods. Journal of Business and Economic Statistics 8, 1–18
Author information
Authors and Affiliations
Corresponding author
Additional information
We want to thank Jinill Kim and Harald Uhlig for discussions and a referee of the Journal for extensive comments on a previous version of our paper.
Rights and permissions
About this article
Cite this article
Becker, S., Grüne, L. & Semmler, W. Comparing accuracy of second-order approximation and dynamic programming. Comput Econ 30, 65–91 (2007). https://doi.org/10.1007/s10614-007-9087-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10614-007-9087-1