Long Memory of Financial Time Series and Hidden Markov Models with Time-Varying Parameters

Peter Nystrup, Henrik Madsen, Erik Lindström

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Hidden Markov models are often used to model daily returns and to infer the hidden state of financial markets. Previous studies have found that the estimated models change over time, but the implications of the time-varying behavior have not been thoroughly examined. This paper presents an adaptive estimation approach that allows for the parameters of the estimated models to be time varying. It is shown that a two-state Gaussian hidden Markov model with time-varying parameters is able to reproduce the long memory of squared daily returns that was previously believed to be the most difficult fact to reproduce with a hidden Markov model. Capturing the time-varying behavior of the parameters also leads to improved one-step density forecasts. Finally, it is shown that the forecasting performance of the estimated models can be further improved using local smoothing to forecast the parameter variations.
Original languageEnglish
JournalJournal of Forecasting
Issue number8
Pages (from-to)989–1002
Publication statusPublished - 2016


  • Hidden Markov models
  • Daily returns
  • Long memory
  • Adaptive estimation
  • Time-varying parameters


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