Generalised spatial and spatiotemporal autoregressive conditional heteroscedasticity

Abstract

In this paper, we introduce a new spatial model that incorporates heteroscedastic variance depending on neighboring locations. The proposed process is regarded as the spatial equivalent to the temporal autoregressive conditional heteroscedasticity (ARCH) model. We show additionally how the introduced spatial ARCH model can be used in spatiotemporal settings. In contrast to the temporal ARCH model, in which the distribution is known given the full information set of the prior periods, the distribution is not straightforward in the spatial and spatiotemporal setting. However, it is possible to estimate the parameters of the model using the maximum-likelihood approach. Via Monte Carlo simulations, we demonstrate the performance of the estimator for a specific spatial weighting matrix. Moreover, we combine the known spatial autoregressive model with the spatial ARCH model assuming heteroscedastic errors. Eventually, the proposed autoregressive process is illustrated using an empirical example. Specifically, we model lung cancer mortality in 3108 U.S. counties and compare the introduced model with two benchmark approaches.

Stochastic properties of spatial and spatiotemporal ARCH models

“…First, we define this vector h in such a way as to be analogous to the definition in Otto et al (2016Otto et al ( , 2018a. For this model, the vector h O is given by…”

“…The first extensions to spatial models attempted were time series models incorporating spatial effects in temporal lags (see Borovkova andLopuhaa 2012 andCaporin andParuolo 2006, for instance). Instantaneous spatial autoregressive dependence in the conditional second moments, i.e., the conditional variance in each spatial location is influenced by the variance nearby, has been introduced by Otto et al (2016Otto et al ( , 2018a. Their models allow for these instantaneous effects but require certain regularity conditions.…”

“…Eventually, we apply all procedures to a real-data example.Matsuda 2017, 2018a). Moreover, all these models can be used in spatiotemporal settings (see Otto et al 2016; Sato and Matsuda 2018b).In addition to the novel spatial exponential ARCH model, this paper demonstrates the use of the R-package spGARCH. From this practical point of view, the simulation of several spatial ARCH-type models as well as the estimation of a variety of spatial models with conditional heteroscedasticity are shown.…”

“…In particular, we distinguish between three different spatial ARCH-type models. In addition to the original definition of Otto et al (2016), we introduce an exponential spatial ARCH model in this paper. For this new model, maximum-likelihood estimators for the parameters are proposed.…”

“…Matsuda 2017, 2018a). Moreover, all these models can be used in spatiotemporal settings (see Otto et al 2016; Sato and Matsuda 2018b).…”