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Stylized Facts of Financial Time Series and Three Popular Models of Volatility


  • Malmsten, Hans

    () (Dept. of Economic Statistics, Stockholm School of Economics)

  • Teräsvirta, Timo

    () (Dept. of Economic Statistics, Stockholm School of Economics)


Properties of three well-known and frequently applied first-order models for modelling and forecasting volatility in financial series such as stock and exchange rate returns are considered. These are the standard Generalized Autoregressive Conditional Heteroskedasticity (GARCH), the Exponential GARCH and the Autoregressive Stochastic Volatility model. The focus is on finding out how well these models are able to reproduce characteristic features of such series, also called stylized facts. These include high kurtosis and a rather low-starting and slowly decaying autocorrelation function of the squared or absolute-valued observations. Another stylized fact is that the autocorrelations of absolute-valued returns raised to a positive power are maximized when this power equals unity. A number of results for moments of the three models are given as well as the autocorrelation function of squared observations or, when available, the autocorrelation function of the absolute-valued observations raised to a positive power. These results make it possible to consider kurtosis-autocorrelation combinations that can be reproduced with these models and compare them with ones that have been estimated from financial time series. The ability of the models to reproduce the stylized fact that the autocorrelations of powers of absolute-valued observations are maximized when the power equals one is discussed as well. Finally, it is pointed out that none of these basic models can generate realizations with a skewed marginal distribution. Not unexpectedly, a conclusion that emerges from these considerations, largely based on results on the moment structure of these models, is that none of the models dominates the others when it comes to reproducing stylized facts in typical financial time series.

Suggested Citation

  • Malmsten, Hans & Teräsvirta, Timo, 2004. "Stylized Facts of Financial Time Series and Three Popular Models of Volatility," SSE/EFI Working Paper Series in Economics and Finance 563, Stockholm School of Economics, revised 03 Sep 2004.
  • Handle: RePEc:hhs:hastef:0563

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    References listed on IDEAS

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    Cited by:

    1. Asai, Manabu & McAleer, Michael & Medeiros, Marcelo C., 2012. "Modelling and forecasting noisy realized volatility," Computational Statistics & Data Analysis, Elsevier, vol. 56(1), pages 217-230, January.
    2. repec:kap:iaecre:v:15:y:2009:i:1:p:71-87 is not listed on IDEAS
    3. Oleg Korenok & Stanislav Radchenko, 2005. "The smooth transition autoregressive target zone model with the Gaussian stochastic volatility and TGARCH error terms with applications," Econometrics 0508015, EconWPA.
    4. Guido VENIER, 2008. "A New Model For Stock Price Movements," Journal of Applied Economic Sciences, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. 3(3(5)_Fall), pages 329-350.
    5. Kovačić, Zlatko, 2007. "Forecasting volatility: Evidence from the Macedonian stock exchange," MPRA Paper 5319, University Library of Munich, Germany.
    6. Dalla, Violetta, 2015. "Power transformations of absolute returns and long memory estimation," Journal of Empirical Finance, Elsevier, vol. 33(C), pages 1-18.
    7. Haas, Markus, 2009. "Persistence in volatility, conditional kurtosis, and the Taylor property in absolute value GARCH processes," Statistics & Probability Letters, Elsevier, vol. 79(15), pages 1674-1683, August.
    8. repec:fgv:epgrbe:v:66:n:3:a:3 is not listed on IDEAS
    9. Teräsvirta, Timo, 2006. "An introduction to univariate GARCH models," SSE/EFI Working Paper Series in Economics and Finance 646, Stockholm School of Economics.
    10. McAleer, Michael & Medeiros, Marcelo C., 2008. "A multiple regime smooth transition Heterogeneous Autoregressive model for long memory and asymmetries," Journal of Econometrics, Elsevier, vol. 147(1), pages 104-119, November.
    11. Matos, Paulo & Beviláqua, Giovanni & Filho, Jaime, 2012. "Previsão do câmbio real-dólar sob um arcabouço de apreçamento de ativos," Revista Brasileira de Economia - RBE, FGV/EPGE - Escola Brasileira de Economia e Finanças, Getulio Vargas Foundation (Brazil), vol. 66(3), October.
    12. Matei, Marius, 2011. "Non-Linear Volatility Modeling of Economic and Financial Time Series Using High Frequency Data," Journal for Economic Forecasting, Institute for Economic Forecasting, vol. 0(2), pages 116-141, June.
    13. Mora Galán, Alberto & Pérez, Ana & Ruiz, Esther, 2004. "Stochastic volatility models and the Taylor effect," DES - Working Papers. Statistics and Econometrics. WS ws046315, Universidad Carlos III de Madrid. Departamento de Estadística.

    More about this item


    Autoregressive conditional heteroskedasticity; evaluation of volatility models; exponential GARCH; GARCH; modelling return series; stochastic volatility;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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