Spatial and Spatiotemporal GARCH Models – A Unified Approach
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In time-series analyses and particularly in finance, generalised autoregressive conditional heteroscedasticity (GARCH) models are widely applied statistical tools for modelling volatility clusters (i.e. periods of increased or decreased risks). In contrast, the spatial dependence in conditional second moments of spatial and spatiotemporal processes has been considered rather uncritical until now. Only a few models have been proposed for modelling local clusters of increased risks. In this paper, we introduce a unified spatial and spatiotemporal GARCH-type model, which covers all previously proposed spatial autoregressive conditional heteroscedasticity (ARCH) models but also introduces novel spatial GARCH (spGARCH) and E-GARCH processes. For this common modelling framework, maximum-likelihood estimators are derived. In addition to the theoretical contributions, we suggest a model selection strategy verified by a series of Monte Carlo simulation studies. Eventually, the use of the unified model is demonstrated by an empirical example. In particular, we focus on real-estate prices from 1995 to 2014 in all Berlin ZIP-code areas. For these data, a spatial autoregressive model has been applied, which shows locally varying model uncertainties captured by the spatial GARCH-type models.
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