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On the ARCH model with stationary liquidity

Author

Listed:
  • Marko Voutilainen

    (Aalto University School of Science)

  • Pauliina Ilmonen

    (Aalto University School of Science)

  • Soledad Torres

    (CIMFAV - Facultad de Ingeniería, Universidad de Valparaíso)

  • Ciprian Tudor

    (UFR Mathématiques, Université de Lille 1)

  • Lauri Viitasaari

    (University of Helsinki
    Aalto University School of Business)

Abstract

The classical ARCH model together with its extensions have been widely applied in the modeling of financial time series. We study a variant of the ARCH model that takes account of liquidity given by a positive stationary process. We provide minimal assumptions that ensure the existence and uniqueness of the stationary solution for this model. Moreover, we give necessary and sufficient conditions for the existence of the autocovariance function. After that, we derive an AR(1) characterization for the stationary solution yielding Yule–Walker type quadratic equations for the model parameters. In order to define a proper estimation method for the model, we first show that the autocovariance estimators of the stationary solution are consistent under relatively mild assumptions. Consequently, we prove that the natural estimators arising out of the quadratic equations inherit consistency from the autocovariance estimators. Finally, we illustrate our results with several examples and a simulation study.

Suggested Citation

  • Marko Voutilainen & Pauliina Ilmonen & Soledad Torres & Ciprian Tudor & Lauri Viitasaari, 2021. "On the ARCH model with stationary liquidity," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(2), pages 195-224, February.
    • Handle: RePEc:spr:metrik:v:84:y:2021:i:2:d:10.1007_s00184-020-00779-x
    DOI: 10.1007/s00184-020-00779-x
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    References listed on IDEAS

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    1. Tim Bollerslev, 2008. "Glossary to ARCH (GARCH)," CREATES Research Papers 2008-49, Department of Economics and Business Economics, Aarhus University.
    2. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    3. Bahamonde, Natalia & Torres, Soledad & Tudor, Ciprian A., 2018. "ARCH model and fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 70-78.
    4. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
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