QML Inference for Spatio-Temporal GARCH Models with Spatial Volatility Interactions

We propose a new class of spatio-temporal GARCH models designed to capture volatility dynamics that propagate jointly across time and space. Existing spatio-temporal GARCH formulations typically account for either lagged spatial spillovers or contemporaneous interactions separately, and therefore fail to capture the combined effect of instantaneous spatial volatility feedback and its propagation over time. To address this gap, we introduce a unified framework that incorporates both contemporaneous and lagged spatial volatility interactions within a single coherent model. At each time point, conditional variances evolve according to a temporal GARCH recursion combined with both contemporaneous and lagged spatial volatility interactions defined on a lattice. This structure allows volatility shocks to diffuse instantaneously across neighboring locations and persist over time through spatially structured feedback mechanisms, extending existing spatial and spatio-temporal GARCH formulations. We establish sufficient conditions for the existence of a unique strictly stationary and ergodic solution based on contraction properties of a combined spatial–temporal operator. Statistical inference is conducted via Gaussian quasi-maximum likelihood estimation (QMLE). We derive consistency and asymptotic normality of the QMLE under two asymptotic regimes: (i) increasing temporal domain with fixed spatial size, and (ii) joint asymptotics where both the number of time periods and spatial locations diverge. In both cases, the asymptotic covariance matrix admits a standard sandwich form and can be consistently estimated. An extensive Monte Carlo study confirms the theoretical results. The simulations show that the QMLE performs well even under strong spatial and temporal persistence and remains robust to heavy-tailed innovations. In particular, increasing the spatial domain substantially improves estimation accuracy, highlighting the efficiency gains induced by spatial information. The proposed model provides a flexible and tractable framework for analyzing volatility processes evolving jointly in time and space.