A Stationary Spatio‐Temporal GARCH Model

Abstract

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.