On the Local Stationarity Approximation in Spatio-Temporal GARCH Modeling

• Atika Aouri (Abdelhafid Boussouf University Center, Mila, Algeria)

In this work, we investigate the approximation of spatially nonstationary spatio-temporal GARCH (ST-GARCH) processes by spatially stationary counterparts at fixed locations. This approach enables a localized analysis of complex spatio-temporal volatility structures. Building upon the model's recursive formulation, we establish that the ST-GARCH process can be represented as a sum of random matrix products, allowing us to derive conditions under which the process admits a Lipschitz continuous approximation. We prove that, under mild regularity and continuity assumptions, the nonstationary process \(X^2_t(s)\) can be closely approximated by a spatially stationary process \(X^2_{t,s_0}(s)\) at a fixed point \(s_0\), with a convergence rate governed by the spatial distance \(\|s - s_0\|_\infty\). Furthermore, using a Taylor expansion and a derivative-based construction, we refine this approximation by including the first-order spatial derivative, yielding an improved representation as a linear combination of two spatially stationary processes. Our theoretical findings lay the groundwork for practical localized modeling and inference in real-world applications involving heterogeneous spatio-temporal data.