Small Noise Asymptotics for a Stochastic Growth Model / Noah Williams.

By:

Williams, Noah

Contributor(s):

National Bureau of Economic Research

Material type: Text

TextSeries: Working Paper Series (National Bureau of Economic Research) ; no. w10194.Publication details: Cambridge, Mass. National Bureau of Economic Research 2003.Description: 1 online resource: illustrations (black and white)Subject(s):

E32 - Business Fluctuations • Cycles
C32 - Time-Series Models • Dynamic Quantile Regressions • Dynamic Treatment Effect Models • Diffusion Processes • State Space Models

Online resources:

Available additional physical forms:

Hardcopy version available to institutional subscribers

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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Previous	nber w10191 The Rise of the Skilled City /	nber w10192 Internal and External Labor Markets: A Personnel Economics Approach /	nber w10193 Varieties of Currency Crises /	nber w10194 Small Noise Asymptotics for a Stochastic Growth Model /	nber w10195 Escaping from a Liquidity Trap and Deflation: The Foolproof Way and Others /	nber w10196 Optimal Policy with Low-Probability Extreme Events /	nber w10197 Should Exact Index Numbers Have Standard Errors? Theory and Application to Asian Growth /	Next

December 2003.

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.