  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.
