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Stochastic properties of spatial and spatiotemporal ARCH models

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  • Philipp Otto

    (Leibniz University Hannover)

  • Wolfgang Schmid

    (European University Viadrina)

  • Robert Garthoff

    (European University Viadrina)

Abstract

In this paper, we provide some results on the class of spatial autoregressive conditional heteroscedasticity (ARCH) models, which have been introduced in recent literature to model spatial conditional heteroscedasticity. That means that the variance in some locations depends on the variance in neighboring locations. In contrast to the temporal ARCH model, for which the distribution is known, given the full information set for the prior periods, the distribution is not straightforward in the spatial and spatiotemporal settings. Thus, we focus on the probability structure of these models. In particular, we derive the conditional and unconditional moments of the process as well as the distribution of the process, given a known error distribution. Eventually, it is shown that the process is strictly stationary under certain conditions.

Suggested Citation

  • Philipp Otto & Wolfgang Schmid & Robert Garthoff, 2021. "Stochastic properties of spatial and spatiotemporal ARCH models," Statistical Papers, Springer, vol. 62(2), pages 623-638, April.
    • Handle: RePEc:spr:stpapr:v:62:y:2021:i:2:d:10.1007_s00362-019-01106-x
    DOI: 10.1007/s00362-019-01106-x
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    References listed on IDEAS

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    1. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    2. Jonathan R. Stroud & Peter Müller & Bruno Sansó, 2001. "Dynamic models for spatiotemporal data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(4), pages 673-689.
    3. Montserrat Fuentes, 2002. "Spectral methods for nonstationary spatial processes," Biometrika, Biometrika Trust, vol. 89(1), pages 197-210, March.
    4. J. Elhorst, 2010. "Applied Spatial Econometrics: Raising the Bar," Spatial Economic Analysis, Taylor & Francis Journals, vol. 5(1), pages 9-28.
    5. 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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    Cited by:

    1. Raffaele Mattera & Philipp Otto, 2023. "Network log-ARCH models for forecasting stock market volatility," Papers 2303.11064, arXiv.org.
    2. Philipp Otto & Wolfgang Schmid, 2023. "A general framework for spatial GARCH models," Statistical Papers, Springer, vol. 64(5), pages 1721-1747, October.

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