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Parametric estimation of hidden Markov models by least squares type estimation and deconvolution

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Abstract

This paper develops a simple and computationally efficient parametric approach to the estimation of general hidden Markov models (HMMs). For non-Gaussian HMMs, the computation of the maximum likelihood estimator (MLE) involves a high-dimensional integral that has no analytical solution and can be difficult to approach accurately. We develop a new alternative method based on the theory of estimating functions and a deconvolution strategy. Our procedure requires the same assumptions as the MLE and deconvolution estimators. We provide theoretical guarantees about the performance of the resulting estimator; its consistency and asymptotic normality are established. This leads to the construction of confidence intervals. Monte Carlo experiments are investigated and compared with the MLE. Finally, we illustrate our approach using real data for ex-ante interest rate forecasts.

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Notes

  1. We refer the reader to Doukhan (1994) for the proof that if \((X_i)_i\) is an ergodic process then the process \((Y_i)_i\), which is the sum of an ergodic process with an i.i.d. noise, is again stationary ergodic. Moreover, by the definition of an ergodic process, if \((Y_i)_i\) is an ergodic process then the couple \(\mathbf {Y}_i=(Y_i, Y_{i+1})\) inherits the property (see Genon-Catalot et al. (2000)).

  2. We argue that our approach can be applied when we introduce a long mean parameter \(\mu \) in the volatility process.

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Authors and Affiliations

  1. Université de Caen - LMNO, Caen, France

    Christophe Chesneau

  2. CREST - ENSAI, Bruz, France

    Salima El Kolei

  3. Université Paris 1 Panthéon-Sorbonne - SAMM, Paris, France

    Fabien Navarro

Authors
  1. Christophe Chesneau
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  2. Salima El Kolei
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  3. Fabien Navarro
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Correspondence to Fabien Navarro.

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Chesneau, C., El Kolei, S. & Navarro, F. Parametric estimation of hidden Markov models by least squares type estimation and deconvolution. Stat Papers 63, 1615–1648 (2022). https://doi.org/10.1007/s00362-022-01288-x

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