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.
The data suggest that only slight deacidification of the substratum causes the breakdown of the L. conizaeoides populations. Neither competitors nor parasites of L. conizaeoides that may have profited from reduced SO(2) concentrations are likely causes of the rapid dieback of the species.
This paper investigates the effect of daily wind direction and speed on the spatio-temporal distribution of particulate matter, PM 2.5 . Interdependencies between the PM 2.5 values of different monitoring sites are characterized by incorporating time-varying anisotropic spatial weighting matrices. These weights are parameterized with respect to wind direction, speed and a range that marks the bandwidth of admissible deviations between wind direction and bearing. The empirical analysis is based on daily PM 2.5 values recorded by monitoring sites located across the eastern United States in 2015 as well as several meteorological regressors. More precisely, we propose a space-time dynamic panel data model with different spatial autoregressive, temporal and exogenous dependencies. All model parameters are estimated by the quasi-maximum likelihood approach. The estimation procedure, including the identification of the range and spatial parameters, is verified by Monte Carlo simulations. We show that part of the spatial dependency of PM 2.5 values is explained by wind direction.Estimation of Anisotropic, Time-Varying Spatial Spillovers 255 spillover effects of PM 2.5 (particulate matter with a diameter less than or equal to 2.5 μm). Wind directions induce spatial dependencies that are not uniform in all directions, so the time-varying spatial weights are also considered to be anisotropic. While the influence of meteorological regressors, which contribute considerably to the distribution and even deposition of PM 2.5 , has been extensively investigated (e.g., DeGaetano and Doherty 2004;Fassò 2013;Tai, Mickley, and Jacob 2010), fewer studies have considered the effect of wind direction. Tai, Mickley, and Jacob (2010) investigated which wind directions were strongly associated with high concentrations of major PM 2.5 components. Sanchez-Reyna et al. (2005) also found that particular wind directions were frequently accompanied by PM 10 increases in London. Similarly, Guerra et al. (2006) reported that wind directions from major industrial areas led to higher PM 2.5 and PM 10 concentrations in Kansas. However, in other respects, time-varying anisotropic spatially autoregressive dependencies due to wind direction have been widely ignored when analyzing or interpolating distributions of particulate matter. We therefore investigate the contribution of these dependencies by employing time-varying anisotropic spatial weighting matrices.The specification of a weighting matrix that reflects the spatial structure and interrelations of cross-sectional units of a sample is one of the key issues in spatial regression models. When working with georeferenced data, those weights are typically predetermined by geographical characteristics such as common borders, nearest-neighbour dependencies or distance decays. In these cases, the typical assumption of non-stochastic weights is not violated. In addition, those characteristics are often permanent or at least slowly evolving over time such that it has become conclusive to draw on constant ...
In this paper, a general overview on spatial and spatiotemporal ARCH models is provided. 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. In addition, we consider a new complex-valued definition of the spatial ARCH process. From a practical point of view, the use of the R-package spGARCH is demonstrated. To be precise, we show how the proposed spatial ARCH models can be simulated and summarize the variety of spatial models, which can be estimated by the estimation functions provided in the package. 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. There are several packages implementing geostatistical models, kriging approaches, and other spatial models (cf. Cressie 1993; Cressie and Wikle 2011). One of the most powerful packages used to deal with models of spatial dependence is spdep, written by Bivand and Piras (2015). It implements most spatial models in a user-friendly way, such as spatial autoregressive models, spatial lag models, and so forth (see, also, Elhorst 2010 for an overview). These models are typically called spatial econometrics models, although they are not tied to applications in economics. In contrast, the package gstat provides functions for geostatistical models, variogram estimation, and various kriging approaches (see Pebesma 2004 for details). For dealing with big geospatial data, the Stem package uses an expectationmaximization (EM) algorithm for fitting hierarchical spatiotemporal models (see Cameletti 2015 for details). For a distributed computing environment, the MATLAB software D-STEM from Finazzi and Fasso (2014) also provides powerful tools for dealing with heterogeneous spatial supports, large multivariate data sets, and heterogeneous spatial sampling networks. Additionally, these fitted models are suitable for spatial imputation. Contrary to these EM approaches, Bayesian methods for modeling spatial data are implemented in the R-INLA package (see Rue et al. 2009 for technical details of the integrated nested Laplace approximations and Martins et al. 2013 for recently implemented features). Along with this package, the R-INLA project provides several functions for diverse spatial models incorporating integrated nested Laplace approximations.In contrast to the above mentioned software for spatial models, the prevalent R-package for time series GARCH-type models is rugarch from Ghalanos (2018). Since spGARCH h...
In this paper, we propose a test procedure to detect change points of multidimensional autoregressive processes. The considered process differs from typical applied spatial autoregressive processes in that it is assumed to evolve from a predefined center into every dimension. Additionally, structural breaks in the process can occur at a certain distance from the predefined center. The main aim of this paper is to detect such spatial changes. In particular, we focus on shifts in the mean and the autoregressive parameter. The proposed test procedure is based on the likelihood-ratio approach. Eventually, the goodness-of-fit values of the estimators are compared for different shifts. Moreover, the empirical distribution of the test statistic of the likelihood-ratio test is obtained via Monte Carlo simulations. We show that the generalized Gumbel distribution seems to be a suitable limiting distribution of the proposed test statistic. Finally, we discuss the detection of lung cancer in computed tomography scans and illustrate the proposed test procedure.
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