This paper introduces a multivariate spatiotemporal autoregressive conditional heteroscedasticity (ARCH) model based on a vec-representation. The model includes instantaneous spatial autoregressive spill-over effects, as they are usually present in geo-referenced data. Furthermore, spatial and temporal cross-variable effects in the conditional variance are explicitly modelled. We transform the model to a multivariate spatiotemporal autoregressive model using a log-squared transformation and derive a consistent quasi-maximum-likelihood estimator (QMLE). For finite samples and different error distributions, the performance of the QMLE is analysed in a series of Monte-Carlo simulations. In addition, we illustrate the practical usage of the new model with a real-world example. We analyse the monthly real-estate price returns for three different property types in Berlin from 2002 to 2014. We find weak (instantaneous) spatial interactions, while the temporal autoregressive structure in the market risks is of higher importance. Interactions between the different property types only occur in the temporally lagged variables. Thus, we see mainly temporal volatility clusters and weak spatial volatility spillovers.

中文翻译：

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多元时空 ARCH 模型

本文介绍了一种基于向量表示的多元时空自回归条件异方差（ARCH）模型。该模型包括瞬时空间自回归溢出效应，因为它们通常存在于地理参考数据中。此外，条件方差中的空间和时间交叉变量效应被明确建模。我们使用对数平方变换将模型转换为多元时空自回归模型，并推导出一致的准最大似然估计量（QMLE）。对于有限样本和不同的误差分布，通过一系列蒙特卡罗模拟分析了 QMLE 的性能。此外，我们还通过实际示例说明了新模型的实际用法。我们分析了 2002 年至 2014 年柏林三种不同房产类型的每月房地产价格回报。我们发现弱（瞬时）空间相互作用，而市场风险中的时间自回归结构更为重要。不同属性类型之间的相互作用仅发生在时间滞后变量中。因此，我们主要看到时间波动集群和微弱的空间波动溢出。