Interquantile Shrinkage in General Spatial Quantile Autoregressive Regression models

To deal with the dependence of variables and evaluate quantile-specific effects by covariates, spatial quantile autoregressive models (SQAR models) were introduced. Conventional quantile regression only focused on the fitting models but ignores the examinations of multiple conditional quantile functions, which provided 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 increased the burden of computation but also brought greater False Discovery Rate (FDR). In this paper, we transformed 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. Based on instrumental variables and shrinkage techniques, we proposed a Two-stage Interquantile Estimation method, including Fused Adaptive LASSO (FAL) and Fused Adaptive Sup-norm (FAS) estimators. The oracle properties of the new estimators were established. Through Monte Carlo simulations and numerical investigations of Crime data, it is demonstrated that the proposed method leads to higher estimation efficiency than the traditional quantile regression.