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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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Suggested Citation

  • Noah Williams, 2003. "Small Noise Asymptotics for a Stochastic Growth Model," Computing in Economics and Finance 2003 262, Society for Computational Economics.
    • Handle: RePEc:sce:scecf3:262
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    Cited by:

    1. Albeverio, Sergio & Mastrogiacomo, Elisa, 2022. "Large deviation principle for spatial economic growth model on networks," Journal of Mathematical Economics, Elsevier, vol. 103(C).
    2. Aruoba, S. Boragan & Fernandez-Villaverde, Jesus & Rubio-Ramirez, Juan F., 2006. "Comparing solution methods for dynamic equilibrium economies," Journal of Economic Dynamics and Control, Elsevier, vol. 30(12), pages 2477-2508, December.
    3. Stutzer, Michael, 2020. "Persistence of averages in financial Markov Switching models: A large deviations approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 553(C).
    4. 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.
    5. Dmitri Kolyuzhnov & Anna Bogomolova, 2004. "Escape Dynamics: A Continuous Time Approximation," Econometric Society 2004 Latin American Meetings 27, Econometric Society.
    6. Bruce McGough, 2006. "Shocking Escapes," Economic Journal, Royal Economic Society, vol. 116(511), pages 507-528, April.
    7. Van Nieuwerburgh, Stijn & Veldkamp, Laura, 2006. "Learning asymmetries in real business cycles," Journal of Monetary Economics, Elsevier, vol. 53(4), pages 753-772, May.
    8. 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.
    9. Stutzer, Michael, 2013. "Optimal hedging via large deviation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(15), pages 3177-3182.
    10. Kolyuzhnov, Dmitri & Bogomolova, Anna & Slobodyan, Sergey, 2014. "Escape dynamics: A continuous-time approximation," Journal of Economic Dynamics and Control, Elsevier, vol. 38(C), pages 161-183.
    11. Dmitri Kolyuzhnov & Anna Bogomolova, 2004. "Escape Dynamics: A Continuous Time Approximation," Econometric Society 2004 Far Eastern Meetings 557, Econometric Society.
    12. Huyen Pham, 2007. "Some applications and methods of large deviations in finance and insurance," Papers math/0702473, arXiv.org, revised Feb 2007.
    13. John Stachurski, 2009. "Economic Dynamics: Theory and Computation," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262012774, December.
    14. 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.
    15. Takanobu Kosugi, 2010. "Assessments of ‘Greenhouse Insurance’: A Methodological Review," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 17(4), pages 345-363, December.
    16. Lars J. Olson & Santanu Roy, 2006. "Theory of Stochastic Optimal Economic Growth," Springer Books, in: Rose-Anne Dana & Cuong Le Van & Tapan Mitra & Kazuo Nishimura (ed.), Handbook on Optimal Growth 1, chapter 11, pages 297-335, Springer.
    17. Dmitri Kolyuzhnov & Anna Bogomolova, 2004. "Escape Dynamics: A Continuous Time Approximation," Computing in Economics and Finance 2004 190, Society for Computational Economics.
    18. Sebastian Sienknecht, 2010. "On the Informational Loss Inherent in Approximation Procedures: Welfare Implications and Impulse Responses," Jena Economics Research Papers 2010-005, Friedrich-Schiller-University Jena.

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    Keywords

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    JEL classification:

    • E32 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Business Fluctuations; Cycles
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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