Bayesian approach for the study of spatiotemporal GARCH models

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Volatility modeling is essential in many fields, including finance, environmental science, and spatial econometrics, where data often show both spatial and temporal dependencies. Classical GARCH models effectively capture temporal volatility but ignore spatial correlations, limiting their applicability to spatially heterogeneous data. This thesis introduces a spatio-temporal GARCH (STGARCH) model that relaxes the common spatial stationarity assumption by allowing parameters to vary smoothly across space. Observations at each fixed location are modeled as a temporal GARCH process, while spatial variability in parameters is handled through nonparametric estimation. Three estimation methods are proposed. First, we develop the localized least squares (LLS) and local linear least squares (LLLS) estimators; the latter incorporates linear trends to reduce bias. Second, we introduce an Iterative Localized Expectation-Maximization (LEM) algorithm which adopts an Empirical Bayesian perspective to reconstruct latent volatility. We study the theoretical properties of the kernel-based estimators under two asymptotic settings: (i) fixed locations with increasing time, and (ii) infill asymptotics where both locations and time increase. We establish asymptotic normality and, in the latter case, consistency. Simulation studies confirm the estimators’ robustness and highlight the superior performance of the LEM algorithm in capturing spatial non-stationarity. An application to real ozone data demonstrates the model’s ability to capture localized volatility patterns and outperform stationary approaches. This work provides a flexible and statistically sound framework for modeling and forecasting volatility in complex spatio-temporal environments.

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