Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T05:14:29.309Z Has data issue: false hasContentIssue false
  • English
  • Français

A COMPARISON OF NUMERICAL METHODS FOR THE SOLUTION OF CONTINUOUS-TIME DSGE MODELS

Published online by Cambridge University Press:  23 May 2017

Juan Carlos Parra-Alvarez
Juan Carlos Parra-Alvarez*
Affiliation:
Aarhus University and CREATES
*
Address correspondence to: Juan Carlos Parra-Alvarez, Department of Economics and Business Economics, Fuglesangs Allé 4, 8210 Aarhus V, Denmark; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This study evaluates the accuracy of a set of techniques that approximate the solution of continuous-time Dynamic Stochastic General Equilibrium models. Using the neoclassical growth model, I compare linear-quadratic, perturbation, and projection methods. All techniques are applied to the Hamilton–Jacobi–Bellman equation and the optimality conditions that define the general equilibrium of the economy. Two cases are studied depending on whether a closed-form solution is available. I also analyze how different degrees of non-linearities affect the approximated solution. The results encourage the use of perturbations for reasonable values of the structural parameters of the model and suggest the use of projection methods when a high degree of accuracy is required.

Keywords

Continuous-Time DSGE ModelsLinear-Quadratic ApproximationPerturbation MethodProjection Method
Type
Articles
Information
Macroeconomic Dynamics , Volume 22 , Issue 6 , September 2018 , pp. 1555 - 1583
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

I would like to thank the editor and two anonymous referees for extremely useful comments and suggestions, as well as the participants of the 18th Annual Conference on Computing in Economics and Finance in Prague, the 6th CSDA International Conference on Computational and Financial Econometrics in Oviedo and the CDMA Workshop on DSGE models in St. Andrews for their helpful comments. I would also like to thank Olaf Posch, Bent Jesper Christensen, Jesús Fernández-Villaverde, Kenneth Judd, Martin Møller Andreasen, Willi Semmler and seminar participants at CREATES for useful suggestions and insights. CREATES (Center for Research in Econometric Analysis of Time Series) is funded by the Danish National Research Foundation.

References

REFERENCES

Achdou, Y., Buera, F.J., Lasry, J.M., Lions, P.L., and Moll, B. (2014) Partial differential equation models in macroeconomics. Philosophical Transactions of the Royal Society A 372, 20130397.Google Scholar
Ackerberg, D., Geweke, J., and Hahn, J. (2009) Comments on “convergence properties of the likelihood of computed dynamic models”. Econometrica 77 (6), 2009–2017.Google Scholar
Aldrich, E. M. and Kung, H. (2010) Computational Methods for Production-Based Asset Pricing Models with Recursive Utility. Working papers 10-90, Duke University, Department of Economics.Google Scholar
Anderson, E. W., McGrattan, E. R., Hansen, L. P., and Sargent, T. J. (1996) Mechanics of forming and estimating dynamic linear economies. In Amman, H. M., Kendrick, D. A., and Rust, J. (eds.), Handbook of Computational Economics, vol. 1, chap. 4, pp 171–252. Netherlands: Elsevier.Google Scholar
Aruoba, S. B., Fernández-Villaverde, J., and Rubio-Ramírez, J. F. (2006) Comparing solution methods for dynamic equilibrium economies. Journal of Economic Dynamics and Control 30 (12), 2477–2508.Google Scholar
Benigno, P. and Woodford, M. (2012) Linear-quadratic approximation of optimal policy problems. Journal of Economic Theory 147 (1), 1–42.Google Scholar
Bergstrom, A. (1984) Continuous time stochastic models and issues of aggregation over time. In Griliches, Z. and Intriligator, M. D. (eds.), Handbook of Econometrics, vol. 2, chap. 20, 1145–1212, Netherlands: Elsevier.Google Scholar
Binsbergen, J. v., Fernández-Villaverde, J., Koijen, R. S., and Rubio-Ramírez, J. (2012) The term structure of interest rates in a DSGE model with recursive preferences. Journal of Monetary Economics 59 (7), 634–648.Google Scholar
Blanchard, O. (2009) The state of macro. Annual Review of Economics 1 (1), 209–228.Google Scholar
Brennan, M. J. (1998) The role of learning in dynamic portfolio decisions. European Finance Review 1 (3), 295–306.Google Scholar
Brunnermeier, M. K. and Sannikov, Y. (2014) A macroeconomic model with a financial sector. American Economic Review 104 (2), 379–421.Google Scholar
Caldara, D., Fernández-Villaverde, J., Rubio-Ramírez, J., and Yao, W. (2012) Computing DSGE models with recursive preferences and stochastic volatility. Review of Economic Dynamics 15 (2), 188–206.Google Scholar
Candler, G. V. (2004) Finite difference methods for continuous time dynamic programming. In Marimon, R. and Scott, A. (eds.), Computational Methods for the Study of Dynamic Economies, chap. 8, pp. 172–194. Oxford, UK: Oxford University Press.Google Scholar
Chang, F. R. (2009) Stochastic Optimization in Continuous Time, 2nd ed. Cambridge, UK: Cambridge University Press.Google Scholar
Collard, F. and Juillard, M. (2001) Accuracy of stochastic perturbation methods: The case of asset pricing models. Journal of Economic Dynamics and Control 25 (6–7), 979–999.Google Scholar
Den Haan, W. J. and De Wind, J. (2010) How Well-Behaved are Higher-Order Perturbation Solutions? DNB working papers 240, Netherlands Central Bank, Research Department.Google Scholar
Den Haan, W. J. and Marcet, A. (1994) Accuracy in simulations. Review of Economic Studies 61 (1), 3–17.Google Scholar
Doraszelski, U. and Judd, K. L. (2012) Avoiding the curse of dimensionality in dynamic stochastic games. Quantitative Economics 3 (1), 53–93.Google Scholar
Fernández-Villaverde, J., Rubio-Ramírez, J. F., and Santos, M. S. (2006) Convergence properties of the likelihood of computed dynamic models. Econometrica 74 (1), 93–119.Google Scholar
Furlanetto, F. and Seneca, M. (2014) New perspectives on depreciation shocks as a source of business cycle fluctuations. Macroeconomic Dynamics 18, 1209–1233.Google Scholar
Gaspar, J. and Judd, K. (1997) Solving large-scale rational-expectations models. Macroeconomic Dynamics 1 (1), 45–75.Google Scholar
Gourio, F. (2012) Disaster risk and business cycles. American Economic Review 102 (6), 2734–2766.Google Scholar
Grinols, E. L. and Turnovsky, S. J. (1993) Risk, the financial market, and macroeconomic equilibrium. Journal of Economic Dynamics and Control 17 (1–2), 1–36.Google Scholar
Harvey, A. and Stock, J. H. (1989) Estimating integrated higher-order continuous time autoregressions with an application to money-income causality. Journal of Econometrics 42, 319–336.Google Scholar
Heer, B. and Maussner, A. (2009) Dynamic General Equilibrium Modeling: Computational Methods and Applications, 2nd ed. Heidelberg: Springer-Verlag.Google Scholar
Judd, K. L. (1996) Approximation, perturbation, and projection methods in economic analysis. In Amman, H. M., Kendrick, D. A., and Rust, J. (eds.), Handbook of Computational Economics, vol. 1, chap. 12, pp. 509–585. Netherlands: Elsevier.Google Scholar
Judd, K. L. (1998) Numerical Methods in Economics. Cambridge, MA: MIT Press.Google Scholar
Judd, K. L. and Guu, S. M. (1993) Perturbation methods for economic growth models. In Varian, H. R. (ed.), Economic and Financial Modeling with Mathematica, chap. 2, pp. 80–103. New York: Springer-Verlag.Google Scholar
Kimball, M. S. (2014) Effect of uncertainty on optimal control models in the neighbourhood of a steady state. Geneva Risk Insurance Review 39 (1), 2–39.Google Scholar
Kompas, T. and Chu, L. (2012) A comparison of parametric approximation techniques to continuous-time stochastic dynamic programming problems: Applications to natural resource modelling. Enviornmental Modelling and Software 38, 1–12.Google Scholar
Kydland, F. E. and Prescott, E. C. (1982) Time to build and aggregate fluctuations. Econometrica 50 (6), 1345–1370.Google Scholar
Ljungqvist, L. and Sargent, T. J. (2004) Recursive Macroeconomic Theory, 2nd ed. Cambridge, MA: MIT Press.Google Scholar
Magill, M. J. P. (1977a) Some new results on the local stability of the process of capital accumulation. Journal of Economic Theory 15 (1), 174–210.Google Scholar
Magill, M. J. P. (1977b) A local analysis of n-sector capital accumulation under uncertainty. Journal of Economic Theory 15 (1), 211–219.Google Scholar
McCandless, G. (2008) The ABCs of RBCs: An Introduction to Dynamic Macroeconomic Models. Cambridge, MA: Harvard University Press.Google Scholar
McGrattan, E. (2004) Application of weighted residual methods to dynamic economic models. In Marimon, R. and Scott, A. (eds.), Computational Methods for the Study of Dynamic Economies, chap. 6, pp. 114–142. Oxford, UK: Oxford University Press.Google Scholar
Miranda, M. J. and Fackler, P. L. (2002) Applied Computational Economics and Finance. Cambridge, MA: MIT Press.Google Scholar
Nuño, G. and Thomas, C. (2015) Monetary Policy and Sovereign Debt Vulnerability. Tech. rep., Banco de Espana. Unpublished.Google Scholar
Posch, O. (2011) Risk premia in general equilibrium. Journal of Economic Dynamics and Control 35 (9), 1557–1576.Google Scholar
Posch, O. and Trimborn, T. (2013) Numerical solution of dynamic equilibrium models under poisson uncertainty. Journal of Economic Dynamics and Control 37 (12), 2602–2622.Google Scholar
Rubio-Ramírez, J. F. and Fernández-Villaverde, J. (2005) Estimating dynamic equilibrium economies: Linear versus nonlinear likelihood. Journal of Applied Econometrics 20 (7), 891–910.Google Scholar
Stokey, N. L. (2009) The Economics of Inaction. New Jersey: Princeton University Press.Google Scholar
Swanson, E. T. (2012) Risk aversion and the labor margin in dynamic equilibrium models. American Economic Review 102 (4), 1663–1691.Google Scholar
Tapiero, C. S. and Sulem, A. (1994) Computational aspects in applied stochastic control. Computational Economics 7 (2), 109–146.Google Scholar
Taylor, J. B. and Uhlig, H. (1990) Solving nonlinear stochastic growth models: A comparison of alternative solution methods. Journal of Business & Economic Statistics 8 (1), 1–17.Google Scholar
Turnovsky, S. J. and Smith, W. T. (2006) Equilibrium consumption and precautionary savings in a stochastically growing economy. Journal of Economic Dynamics and Control 30 (2), 243–278.Google Scholar
Wachter, J. (2013) Can time-varying risk of rare disasters explain aggregate stock market volatility? Journal of Finance 68 (3), 987–1035.Google Scholar
Wälde, K. (2011) Production technologies in stochastic continuous time models. Journal of Economic Dynamics and Control 35 (4), 616–622.Google Scholar
Xia, Y. (2001) Learning about predictability: The effects of parameter uncertainty on dynamic asset allocation. Journal of Finance 56 (1), 205–246.Google Scholar