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Data-Based Parametrization for Affine GARCH Models Across Multiple Time Scales—Roughness Implications

Author

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  • Marcos Escobar-Anel

    (Department of Statistical and Actuarial Sciences, University of Western Ontario, London, ON N6A 5B7, Canada)

  • Sebastian Ferrando

    (Department of Mathematics, Toronto Metropolitan University, Toronto, ON M5B 2K3, Canada)

  • Fuyu Li

    (Department of Economics, University of Victoria, Victoria, BC V8P 5C2, Canada)

  • Ke Xu

    (Department of Economics, University of Victoria, Victoria, BC V8P 5C2, Canada)

Abstract

This paper revisits the topic of time-scale parameterizations of the Heston–Nandi GARCH (1,1) model to create a new, theoretically valid setting compatible with real financial data. We first estimate parameters using three US market indices and six frequencies to let data reveal the correct, data-implied, time-scale parameterizations. We compared the data-implied parametrization to two popular candidates in the literature, demonstrating structurally different continuous-time limits, i.e., the data favor fractional Brownian motion (fBM)—instead of the standard Brownian motion (BM)-based parametrization. We then propose a theoretically flexible time-scale parameterization compatible with this fBM behavior. In this context, a fractional derivative analysis of our empirically based parametrization is performed, confirming an anomalous diffusion in the continuous-time limit. Such a finding is yet another endorsement of the recent and popular stylized fact known as rough volatility.

Suggested Citation

  • Marcos Escobar-Anel & Sebastian Ferrando & Fuyu Li & Ke Xu, 2025. "Data-Based Parametrization for Affine GARCH Models Across Multiple Time Scales—Roughness Implications," Econometrics, MDPI, vol. 13(1), pages 1-17, February.
    • Handle: RePEc:gam:jecnmx:v:13:y:2025:i:1:p:6-:d:1589235
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    References listed on IDEAS

    as
    1. Drost, Feike C & Nijman, Theo E, 1993. "Temporal Aggregation of GARCH Processes," Econometrica, Econometric Society, vol. 61(4), pages 909-927, July.
    2. Fulvio Corsi, 2009. "A Simple Approximate Long-Memory Model of Realized Volatility," Journal of Financial Econometrics, Oxford University Press, vol. 7(2), pages 174-196, Spring.
    3. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous‐time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323, October.
    4. Andersen, Torben G. & Bollerslev, Tim, 1997. "Intraday periodicity and volatility persistence in financial markets," Journal of Empirical Finance, Elsevier, vol. 4(2-3), pages 115-158, June.
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