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Small noise asymptotics for a stochastic growth model

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  • Williams, Noah

Abstract

We develop analytic asymptotic methods to characterize time series properties of nonlinear dynamic stochastic models. We focus on a stochastic growth model which is representative of the models underlying much of modern macroeconomics. Taking limits as the stochastic shocks become small, we derive a functional central limit theorem, a large deviation principle, and a moderate deviation principle. These allow us to calculate analytically the asymptotic distribution of the capital stock, and to obtain bounds on the probability that the log of the capital stock will differ from its deterministic steady state level by a given amount. This latter result can be applied to characterize the probability and frequency of large business cycles. We then illustrate our theoretical results through some simulations. We find that our results do a good job of characterizing the model economy, both in terms of its average behavior and its occasional large cyclical fluctuations.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Economic Theory.

Volume (Year): 119 (2004)
Issue (Month): 2 (December)
Pages: 271-298

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Handle: RePEc:eee:jetheo:v:119:y:2004:i:2:p:271-298

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Web page: http://www.elsevier.com/locate/inca/622869

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References

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  1. Santos, Manuel S, 1991. "Smoothness of the Policy Function in Discrete Time Economic Models," Econometrica, Econometric Society, vol. 59(5), pages 1365-82, September.
  2. Jinill Kim and Sunghyun Henry Kim, 2001. "Spurious Welfare Reversals in International Business Cycle Models," Computing in Economics and Finance 2001 3, Society for Computational Economics.
  3. Campbell, John, 1994. "Inspecting the Mechanism: An Analytical Approach to the Stochastic Growth Model," Scholarly Articles 3196342, Harvard University Department of Economics.
  4. Judd, Kenneth L. & Guu, Sy-Ming, 1997. "Asymptotic methods for aggregate growth models," Journal of Economic Dynamics and Control, Elsevier, vol. 21(6), pages 1025-1042, June.
  5. Brock, William A. & Mirman, Leonard J., 1972. "Optimal economic growth and uncertainty: The discounted case," Journal of Economic Theory, Elsevier, vol. 4(3), pages 479-513, June.
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  7. Amir, Rabah, 1997. "A new look at optimal growth under uncertainty," Journal of Economic Dynamics and Control, Elsevier, vol. 22(1), pages 67-86, November.
  8. Santos, Manuel S, 1993. "On High-Order Differentiability of the Policy Function," Economic Theory, Springer, vol. 3(3), pages 565-70, July.
  9. Klebaner, F. C. & Liptser, R., 1999. "Moderate deviations for randomly perturbed dynamical systems," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 157-176, April.
  10. Gaspar, Jess & L. Judd, Kenneth, 1997. "Solving Large-Scale Rational-Expectations Models," Macroeconomic Dynamics, Cambridge University Press, vol. 1(01), pages 45-75, January.
  11. King, Robert G. & Plosser, Charles I. & Rebelo, Sergio T., 1988. "Production, growth and business cycles : II. New directions," Journal of Monetary Economics, Elsevier, vol. 21(2-3), pages 309-341.
  12. James H. Stock & Mark W. Watson, 1998. "Business Cycle Fluctuations in U.S. Macroeconomic Time Series," NBER Working Papers 6528, National Bureau of Economic Research, Inc.
  13. Kenneth Kasa, 2000. "Learning, large deviations, and recurrent currency crises," Working Paper Series 2000-10, Federal Reserve Bank of San Francisco.
  14. Kydland, Finn E & Prescott, Edward C, 1982. "Time to Build and Aggregate Fluctuations," Econometrica, Econometric Society, vol. 50(6), pages 1345-70, November.
  15. Magill, Michael J. P., 1977. "A local analysis of N-sector capital accumulation under uncertainty," Journal of Economic Theory, Elsevier, vol. 15(1), pages 211-219, June.
  16. Klebaner, F. C. & Nerman, O., 1994. "Autoregressive approximation in branching processes with a threshold," Stochastic Processes and their Applications, Elsevier, vol. 51(1), pages 1-7, June.
  17. Araujo, A, 1991. "The Once but Not Twice Differentiability of the Policy Function," Econometrica, Econometric Society, vol. 59(5), pages 1383-93, September.
  18. King, Robert G. & Plosser, Charles I. & Rebelo, Sergio T., 1988. "Production, growth and business cycles : I. The basic neoclassical model," Journal of Monetary Economics, Elsevier, vol. 21(2-3), pages 195-232.
  19. Manuel S. Santos & Jesus Vigo-Aguiar, 1998. "Analysis of a Numerical Dynamic Programming Algorithm Applied to Economic Models," Econometrica, Econometric Society, vol. 66(2), pages 409-426, March.
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Citations

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Cited by:
  1. Dmitri Kolyuzhnov & Anna Bogomolova & Sergey Slobodyan, 2006. "Escape Dynamics: A Continuous—Time Approximation," CERGE-EI Working Papers wp285, The Center for Economic Research and Graduate Education - Economic Institute, Prague.
  2. Martin Ellison & Liam Graham & Jouko Vilmunen, 2006. "Strong Contagion with Weak Spillovers," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 9(2), pages 263-283, April.
  3. Dmitri Kolyuzhnov & Anna Bogomolova, 2004. "Escape Dynamics: A Continuous Time Approximation," Computing in Economics and Finance 2004 190, Society for Computational Economics.
  4. Huyen Pham, 2007. "Some applications and methods of large deviations in finance and insurance," Papers math/0702473, arXiv.org, revised Feb 2007.
  5. Anderson, Evan W. & Hansen, Lars Peter & Sargent, Thomas J., 2012. "Small noise methods for risk-sensitive/robust economies," Journal of Economic Dynamics and Control, Elsevier, vol. 36(4), pages 468-500.
  6. Robert Feicht & Wolfgang Stummer, 2010. "Complete Closed-form Solution to a Stochastic Growth Model and Corresponding Speed of Economic Recovery preliminary," DEGIT Conference Papers c015_041, DEGIT, Dynamics, Economic Growth, and International Trade.
  7. Takanobu Kosugi, 2010. "Assessments of ‘Greenhouse Insurance’: A Methodological Review," Asia-Pacific Financial Markets, Springer, vol. 17(4), pages 345-363, December.
  8. S. Boragan Aruoba & Jesus Fernandez-Villaverde & Juan F. Rubio-Ramirez, 2003. "Comparing Solution Methods for Dynamic Equilibrium Economies," PIER Working Paper Archive 04-003, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
  9. Dmitri Kolyuzhnov & Anna Bogomolova, 2004. "Escape Dynamics: A Continuous Time Approximation," Econometric Society 2004 Far Eastern Meetings 557, Econometric Society.
  10. Laura Veldkamp, 2003. "Learning Asymmetries in Real Business Cycles," Working Papers 03-21, New York University, Leonard N. Stern School of Business, Department of Economics.
  11. Sebastian Sienknecht, 2010. "On the Informational Loss Inherent in Approximation Procedures: Welfare Implications and Impulse Responses," Jena Economic Research Papers 2010-005, Friedrich-Schiller-University Jena, Max-Planck-Institute of Economics.
  12. Bruce McGough, 2006. "Shocking Escapes," Economic Journal, Royal Economic Society, vol. 116(511), pages 507-528, 04.
  13. Stutzer, Michael, 2013. "Optimal hedging via large deviation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(15), pages 3177-3182.
  14. Olson, Lars J. & Roy, Santanu, 2005. "Theory of Stochastic Optimal Economic Growth," Working Papers 28601, University of Maryland, Department of Agricultural and Resource Economics.
  15. Dmitri Kolyuzhnov & Anna Bogomolova, 2004. "Escape Dynamics: A Continuous Time Approximation," Econometric Society 2004 Latin American Meetings 27, Econometric Society.

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