  Spatial dependent data frequently occur in many fields such as spatial
econometrics and epidemiology. To deal with the dependence of variables and
estimate quantile-specific effects by covariates, spatial quantile
autoregressive models (SQAR models) are introduced. Conventional quantile
regression only focuses on the fitting models but ignores the examination of
multiple conditional quantile functions, which provides a comprehensive view of
the relationship between the response and covariates. Thus, it is necessary to
study the different regression slopes at different quantiles, especially in
situations where the quantile coefficients share some common feature. However,
traditional Wald multiple tests not only increase the burden of computation but
also bring greater FDR. In this paper, we transform the estimation and
examination problem into a penalization problem, which estimates the parameters
at different quantiles and identifies the interquantile commonality at the same
time. To avoid the endogeneity caused by the spatial lag variables in SQAR
models, we also introduce instrumental variables before estimation and propose
two-stage estimation methods based on fused adaptive LASSO and fused adaptive
sup-norm penalty approaches. The oracle properties of the proposed estimation
methods are established. Through numerical investigations, it is demonstrated
that the proposed methods lead to higher estimation efficiency than the
traditional quantile regression.
