We introduce a lagged nearest‐neighbour, stationary spatio‐temporal generalized autoregressive conditional heteroskedasticity (GARCH) model on an infinite spatial grid that opens for GARCH innovations in a space‐time ARMA model. This is illustrated by a real data application to a classical dataset of sea surface temperature anomalies in the Pacific Ocean. The model and its translation invariant neighbourhood system are wrapped around a torus forming a model with finite spatial domain, which we call circular spatio‐temporal GARCH. Such a model could be seen as an approximation of the infinite one and simulation experiments show that the circular estimator with a straightforward bias correction performs well on such non‐circular data. Since the spatial boundaries are tied together, the well‐known boundary issue in spatial statistical modelling is effectively avoided. We derive stationarity conditions for these circular processes and study the spatio‐temporal correlation structure through an ARMA representation. We also show that the matrices defined by a vectorized version of the model are block circulants. The maximum quasi‐likelihood estimator is presented and we prove its strong consistency and asymptotic normality by generalizing results from univariate GARCH theory.

中文翻译：

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静止时空 GARCH 模型

我们在无限空间网格上引入了滞后最近邻、平稳时空广义自回归条件异方差 (GARCH) 模型，该模型为时空 ARMA 模型中的 GARCH 创新打开了大门。对太平洋海面温度异常经典数据集的实际数据应用说明了这一点。该模型及其平移不变邻域系统围绕一个环面形成一个具有有限空间域的模型，我们称之为圆形时空 GARCH。这种模型可以看作是无限模型的近似，模拟实验表明，具有直接偏差校正的圆形估计器在此类非圆形数据上表现良好。由于空间边界被捆绑在一起，有效避免了空间统计建模中众所周知的边界问题。我们推导出这些循环过程的平稳性条件，并通过 ARMA 表示研究时空相关结构。我们还表明，由模型的矢量化版本定义的矩阵是块循环。提出了最大拟似然估计量，我们通过推广单变量 GARCH 理论的结果证明了它的强一致性和渐近正态性。