Bayesian nonlinear expectation for time series modelling and its application to Bitcoin

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This paper proposes a two-stage approach to parametric nonlinear time series modelling in discrete time with the objective of incorporating uncertainty or misspecification in the conditional mean and volatility. At the first stage, a reference or approximating time series model is specified and estimated. At the second stage, Bayesian nonlinear expectations are introduced to incorporate model uncertainty or misspecification in prediction via specifying a family of alternative models. The Bayesian nonlinear expectations for prediction are constructed from closed-form Bayesian credible intervals evaluated using conjugate priors and residuals of the estimated approximating model. Using real Bitcoin data including some periods of Covid 19, applications of the proposed method to forecasting and risk evaluation of Bitcoin are discussed via three major parametric nonlinear time series models, namely the self-exciting threshold autoregressive model, the generalized autoregressive conditional heteroscedasticity model and the stochastic volatility model.
Original languageEnglish
Pages (from-to)505–537
Number of pages33
JournalEmpirical Economics
Issue number1
Publication statusPublished - Jan 2023

Bibliographical note

© The Author(s) 2022, corrected publication 2022. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.


  • Parametric time series modelling
  • Nonlinear expectations
  • Bayesian statistics
  • Girsanov’s transform
  • Drift and volatility uncertainties
  • Bitcoin


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