We consider a new class of semiparametric spatio-temporal models with
unknown and banded autoregressive coefficient matrices. The setting
represents a type of sparse structure in order to include as many panels
as possible. We apply the local linear method and least squares method
for Yule-Walker equation to estimate trend function and spatio-temporal
autoregressive coefficient matrices respectively. We also balance the
over-determined and under-determined phenomena in part by adjusting the
order of extracting sample information. Both the asymptotic normality
and convergence rates of the proposed estimators are established. The
proposed methods are further illustrated using both simulation and case
studies, the results also show that our estimator is stable among
different sample size, and it performs better than the traditional
method with known spatial weight matrices.