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Small Noise Asymptotics for a Stochastic Growth Model

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  • Noah Williams

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

Paper provided by National Bureau of Economic Research, Inc in its series NBER Working Papers with number 10194.

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Date of creation: Dec 2003
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Publication status: published as Williams, Noah. "Small Noise Asymptotics For A Stochastic Growth Model," Journal of Economic Theory, 2004, v119(2,Dec), 2710298.
Handle: RePEc:nbr:nberwo:10194

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  1. 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, Elsevier, vol. 21(2-3), pages 195-232.
  2. Gaspar, Jess & L. Judd, Kenneth, 1997. "Solving Large-Scale Rational-Expectations Models," Macroeconomic Dynamics, Cambridge University Press, Cambridge University Press, vol. 1(01), pages 45-75, January.
  3. Kenneth Kasa, 2004. "Learning, Large Deviations, And Recurrent Currency Crises," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 45(1), pages 141-173, 02.
  4. Jinill Kim and Sunghyun Henry Kim, 2001. "Spurious Welfare Reversals in International Business Cycle Models," Computing in Economics and Finance 2001, Society for Computational Economics 3, Society for Computational Economics.
  5. Blume, Lawrence & Easley, David & O'Hara, Maureen, 1982. "Characterization of optimal plans for stochastic dynamic programs," Journal of Economic Theory, Elsevier, Elsevier, vol. 28(2), pages 221-234, December.
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  9. Manuel S. Santos & Jesus Vigo-Aguiar, 1998. "Analysis of a Numerical Dynamic Programming Algorithm Applied to Economic Models," Econometrica, Econometric Society, Econometric Society, vol. 66(2), pages 409-426, March.
  10. King, Robert G. & Plosser, Charles I. & Rebelo, Sergio T., 1988. "Production, growth and business cycles : II. New directions," Journal of Monetary Economics, Elsevier, Elsevier, vol. 21(2-3), pages 309-341.
  11. Judd, Kenneth L. & Guu, Sy-Ming, 1997. "Asymptotic methods for aggregate growth models," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 21(6), pages 1025-1042, June.
  12. Campbell, John Y., 1994. "Inspecting the mechanism: An analytical approach to the stochastic growth model," Journal of Monetary Economics, Elsevier, Elsevier, vol. 33(3), pages 463-506, June.
  13. Araujo, A, 1991. "The Once but Not Twice Differentiability of the Policy Function," Econometrica, Econometric Society, Econometric Society, vol. 59(5), pages 1383-93, September.
  14. Santos, Manuel S, 1991. "Smoothness of the Policy Function in Discrete Time Economic Models," Econometrica, Econometric Society, Econometric Society, vol. 59(5), pages 1365-82, September.
  15. Klebaner, F. C. & Liptser, R., 1999. "Moderate deviations for randomly perturbed dynamical systems," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 80(2), pages 157-176, April.
  16. Brock, William A. & Mirman, Leonard J., 1972. "Optimal economic growth and uncertainty: The discounted case," Journal of Economic Theory, Elsevier, Elsevier, vol. 4(3), pages 479-513, June.
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Cited by:
  1. Dmitri Kolyuzhnov & Anna Bogomolova, 2004. "Escape Dynamics: A Continuous Time Approximation," Econometric Society 2004 Latin American Meetings, Econometric Society 27, Econometric Society.
  2. Martin Ellison & Liam Graham & Jouko Vilmunen, 2005. "Strong contagion with weak spillovers," Computing in Economics and Finance 2005, Society for Computational Economics 30, Society for Computational Economics.
  3. Huyen Pham, 2007. "Some applications and methods of large deviations in finance and insurance," Papers math/0702473, arXiv.org, revised Feb 2007.
  4. Bruce McGough, 2006. "Shocking Escapes," Economic Journal, Royal Economic Society, Royal Economic Society, vol. 116(511), pages 507-528, 04.
  5. S. B. Aruoba & Jesús Fernández-Villaverde & Juan F. Rubio-Ramirez, 2005. "Comparing Solution Methods for Dynamic Equilibrium Economies," Levine's Bibliography 122247000000000855, UCLA Department of Economics.
  6. 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.
  7. Dmitri Kolyuzhnov & Anna Bogomolova, 2004. "Escape Dynamics: A Continuous Time Approximation," Computing in Economics and Finance 2004, Society for Computational Economics 190, Society for Computational Economics.
  8. Laura Veldkamp, 2003. "Learning Asymmetries in Real Business Cycles," Working Papers, New York University, Leonard N. Stern School of Business, Department of Economics 03-21, New York University, Leonard N. Stern School of Business, Department of Economics.
  9. Olson, Lars J. & Roy, Santanu, 2005. "Theory of Stochastic Optimal Economic Growth," Working Papers, University of Maryland, Department of Agricultural and Resource Economics 28601, University of Maryland, Department of Agricultural and Resource Economics.
  10. Dmitri Kolyuzhnov & Anna Bogomolova, 2004. "Escape Dynamics: A Continuous Time Approximation," Econometric Society 2004 Far Eastern Meetings, Econometric Society 557, Econometric Society.
  11. Robert Feicht & Wolfgang Stummer, 2010. "Complete Closed-form Solution to a Stochastic Growth Model and Corresponding Speed of Economic Recovery preliminary," DEGIT Conference Papers, DEGIT, Dynamics, Economic Growth, and International Trade c015_041, DEGIT, Dynamics, Economic Growth, and International Trade.
  12. 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, Elsevier, vol. 36(4), pages 468-500.
  13. Stutzer, Michael, 2013. "Optimal hedging via large deviation," Physica A: Statistical Mechanics and its Applications, Elsevier, Elsevier, vol. 392(15), pages 3177-3182.
  14. Takanobu Kosugi, 2010. "Assessments of ‘Greenhouse Insurance’: A Methodological Review," Asia-Pacific Financial Markets, Springer, Springer, vol. 17(4), pages 345-363, December.
  15. Sebastian Sienknecht, 2010. "On the Informational Loss Inherent in Approximation Procedures: Welfare Implications and Impulse Responses," Jena Economic Research Papers, Friedrich-Schiller-University Jena, Max-Planck-Institute of Economics 2010-005, Friedrich-Schiller-University Jena, Max-Planck-Institute of Economics.

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