Functional Spatial Autoregressive Models

This study introduces a novel spatial autoregressive model in which the dependent variable is a function that may exhibit functional autocorrelation with the outcome functions of nearby units. This model can be characterized as a simultaneous integral equation system, which, in general, does not necessarily have a unique solution. For this issue, we provide a simple condition on the magnitude of the spatial interaction to ensure the uniqueness in data realization. For estimation, to account for the endogeneity caused by the spatial interaction, we propose a regularized two-stage least squares estimator based on a basis approximation for the functional parameter. The asymptotic properties of the estimator including the consistency and asymptotic normality are investigated under certain conditions. Additionally, we propose a simple Wald-type test for detecting the presence of spatial effects. As an empirical illustration, we apply the proposed model and method to analyze age distributions in Japanese cities.