Stochastic equilibrium: Learning by exponential smoothing

Klaus Pötzelberger, Leopold Sögner

Research output: Contribution to journalArticlepeer-review


This article investigates the question of stability and the impact of learning in a stochastic setup, when only limited information is used to construct estimators. More precisely, we assume that the parameters are estimated by exponential smoothing, where past parameters are down-weighted and the weight of recent observations does not decrease with time. This situation is familiar from applications in finance where price and volatility forecasts are performed by using only some recent data. Even if time series of volatile stocks or bonds are available for a long time, only recent data is used in the analysis. In this situation the prices do not converge and remain a random variable. Already Muth (Econometrica 29 (1961) 315) designed the rational expectations concept to fit to stochastic equilibrium behavior as well. Moreover, e.g. in the work of Stokey and Lucas (Recursive Methods in Economic Dynamics (1989) Harvard University Press, Cambridge, MA), the idea of modeling stochastic equilibrium by the concepts of ergodicity and stationarity has become a familiar concept in economics. In this paper we provide tools to check for ergodicity and apply these tools to a capital market model where the agents learn by means of exponential smoothing. Furthermore, we check whether the implied law is consistent with the data observed by means of an econometric analysis. By applying advanced methods from the theory of stochastic dynamical systems this paper provides a useful example how modern mathematical tools can be used to investigate equilibrium behavior of stochastic models in economics.

Original languageEnglish (US)
Pages (from-to)1743-1770
Number of pages28
JournalJournal of Economic Dynamics and Control
Issue number10
StatePublished - Aug 2003


  • Bounded rationality
  • Learning
  • Stochastic dynamical systems

ASJC Scopus subject areas

  • Economics and Econometrics
  • Control and Optimization
  • Applied Mathematics


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