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Small Noise Asymptotics for a Stochastic Growth Model

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Noah Williams

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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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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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Handle: RePEc:nbr:nberwo:10194

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Find related papers by 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

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  3. Santos, Manuel S, 1993. "On High-Order Differentiability of the Policy Function," Economic Theory, Springer, vol. 3(3), pages 565-70, July.
  4. Kim, Jinill & Kim, Sunghyun Henry, 2003. "Spurious welfare reversals in international business cycle models," Journal of International Economics, Elsevier, vol. 60(2), pages 471-500, August. [Downloadable!] (restricted)
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  1. Dmitri Kolyuzhnov & Anna Bogomolova, 2004. "Escape Dynamics: A Continuous Time Approximation," Econometric Society 2004 Latin American Meetings 27, Econometric Society. [Downloadable!]
  2. 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. [Downloadable!]
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  3. Dmitri Kolyuzhnov & Anna Bogomolova, 2004. "Escape Dynamics: A Continuous Time Approximation," Econometric Society 2004 Far Eastern Meetings 557, Econometric Society. [Downloadable!]
  4. Bruce McGough, 2003. "Shocking Escapes," Computing in Economics and Finance 2003 294, Society for Computational Economics. [Downloadable!]
    Other versions:
  5. Martin Ellison & Liam Graham & Jouka Vilmunen, 2005. "Strong Contagion with Weak Spillovers," Money Macro and Finance (MMF) Research Group Conference 2005 91, Money Macro and Finance Research Group. [Downloadable!]
    Other versions:
  6. Dmitri Kolyuzhnov & Anna Bogomolova, 2004. "Escape Dynamics: A Continuous Time Approximation," Computing in Economics and Finance 2004 190, Society for Computational Economics. [Downloadable!]
  7. 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. [Downloadable!]
    Other versions:
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