Spatial GARCH Models

This study proposes a spatial extension of time series generalized autoregressive conditional heteroscedasticity (GARCH) models. We call the spatial extended GARCH models as spatial GARCH (S-GARCH) models. S-GARCH models specify conditional variances given simultaneous observations, which constitutes a good contrast with time series GARCH models that specify conditional variances given past observations. The S-GARCH model are transformed into a spatial autoregressive moving-average (SARMA) model and the parameters of the S-GARCH model are estimated by a two step procedure. First step estimation is the quasi maximum likelihood (QML) estimation method and consistency and asymptotic normality of the proposed QML estimators are given. Second step is estimation of an intercept term by the estimator derived from another QML to avoid bias in first step and consistency of the estimator is shown. We demonstrate empirical properties of the model by simulation studies and real data analyses of land price data in Tokyo areas. We find the estimators have small bias regardless of distributions of error terms from simulation studies and real data analyses show that spatial volatility in land price has global spillover and volatility clustering, namely units with higher spatial volatility are clustered in some specific districts like time series financial data.