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A mixture autoregressive model based on Student's $t$-distribution

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  • Mika Meitz
  • Daniel Preve
  • Pentti Saikkonen

Abstract

A new mixture autoregressive model based on Student's $t$-distribution is proposed. A key feature of our model is that the conditional $t$-distributions of the component models are based on autoregressions that have multivariate $t$-distributions as their (low-dimensional) stationary distributions. That autoregressions with such stationary distributions exist is not immediate. Our formulation implies that the conditional mean of each component model is a linear function of past observations and the conditional variance is also time varying. Compared to previous mixture autoregressive models our model may therefore be useful in applications where the data exhibits rather strong conditional heteroskedasticity. Our formulation also has the theoretical advantage that conditions for stationarity and ergodicity are always met and these properties are much more straightforward to establish than is common in nonlinear autoregressive models. An empirical example employing a realized kernel series based on S&P 500 high-frequency data shows that the proposed model performs well in volatility forecasting.

Suggested Citation

  • Mika Meitz & Daniel Preve & Pentti Saikkonen, 2018. "A mixture autoregressive model based on Student's $t$-distribution," Papers 1805.04010, arXiv.org.
  • Handle: RePEc:arx:papers:1805.04010
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    1. Hajo Holzmann & Axel Munk & Tilmann Gneiting, 2006. "Identifiability of Finite Mixtures of Elliptical Distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(4), pages 753-763, December.
    2. Leena Kalliovirta & Mika Meitz & Pentti Saikkonen, 2015. "A Gaussian Mixture Autoregressive Model for Univariate Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(2), pages 247-266, March.
    3. Patton, Andrew J., 2011. "Volatility forecast comparison using imperfect volatility proxies," Journal of Econometrics, Elsevier, vol. 160(1), pages 246-256, January.
    4. Kalliovirta, Leena & Meitz, Mika & Saikkonen, Pentti, 2016. "Gaussian mixture vector autoregression," Journal of Econometrics, Elsevier, vol. 192(2), pages 485-498.
    5. Goffe, William L. & Ferrier, Gary D. & Rogers, John, 1994. "Global optimization of statistical functions with simulated annealing," Journal of Econometrics, Elsevier, vol. 60(1-2), pages 65-99.
    6. C. S. Wong & W. S. Chan & P. L. Kam, 2009. "A Student t-mixture autoregressive model with applications to heavy-tailed financial data," Biometrika, Biometrika Trust, vol. 96(3), pages 751-760.
    7. Dorsey, Robert E & Mayer, Walter J, 1995. "Genetic Algorithms for Estimation Problems with Multiple Optima, Nondifferentiability, and Other Irregular Features," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(1), pages 53-66, January.
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    Cited by:

    1. Savi Virolainen, 2021. "Gaussian and Student's $t$ mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area," Papers 2109.13648, arXiv.org, revised Jun 2022.
    2. Patrick Toman & Nalini Ravishanker & Nathan Lally & Sanguthevar Rajasekaran, 2023. "Latent Autoregressive Student- t Prior Process Models to Assess Impact of Interventions in Time Series," Future Internet, MDPI, vol. 16(1), pages 1-17, December.
    3. Henri Karttunen, 2020. "An autoregressive model based on the generalized hyperbolic distribution," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(3), pages 787-816, September.

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