Exploring Heterogeneities with Geographically Weighted Quantile Regression: An Enhancement Based on the Bootstrap Approach

Abstract

Geographically weighted quantile regression (GWQR) has been proposed as a spatial analytical technique to simultaneously explore two heterogeneities, one of spatial heterogeneity with respect to data relationships over space and one of response heterogeneity across different locations of the outcome distribution. However, one limitation of GWQR framework is that the existing inference procedures are established based on asymptotic approximation, which may suffer computation difficulties or yield incorrect estimates with finite samples. In this article, we suggest a bootstrap approach to address this limitation. Our bootstrap enhancement is first validated by a simulation experiment and then illustrated with an empirical U.S. mortality data. The results show that the bootstrap approach provides a practical alternative for inference in GWQR and enhances the utilization of GWQR.

A spatial autoregressive geographically weighted quantile regression to explore housing rent determinants in Amsterdam and Warsaw

“…Table 5 presents the median of local parameters of GWQR-SAR for Warsaw and Amsterdam across studied percentiles (minimum and maximum values in Tables S5 and S6). The optimal bandwidths are highest when examining the fifth and 95th percentiles of the rental price distribution, consistent with Chen et al (2020). When analyzing the local parameters obtained for the structural variables, the apartment area variable, in line with other studies (Helbich et al, 2014), is significant in nearly all locations in both markets.…”

“…Local constant and local linear GWQR can also be estimated using a bootstrap approach instead of an asymptotic approximation (Chen et al, 2020). The bootstrap approach provides reliable estimates of model parameters and standard errors based on the bootstrap distribution of β^(ui,vi).…”

Social Vulnerability and COVID-19 Pandemic Outcomes: Evidence from Spatial Quantile Regression

“…For illustration, Table 4 reports key results of the SAR-GWQR models at three conditional quantiles of the dependent variable, namely the 0.25, 0.5 (median), and 0.75 quantiles. Standard errors are obtained with a bootstrap method with 500 replications (Chen et al, 2020). The optimal bandwidth for model calibration varies as much among the three quantiles as different outcomes.…”

“…This step is taken to avoid an endogeneity bias from estimating Wy that is correlated with the error term. Next, the fitted values from the first step are used to estimate the SAR-GWR with the local-constant QR method detailed by Chen et al (2020). Essentially, the SAR-GWQR method yields "local" quantile parameter estimates of the spatial lag and other covariates at each geographic location across the conditional distribution of the dependent variable.…”

Looking Back, Looking Forward: Progress and Prospect for Spatial Demography

“…And, Multiscale GWR (MGWR) removes the restriction that all independent variables must share the same bandwidth (Fotheringham et al 2017), which offers the flexibility to examine if the impact of one independent variable on the dependent variable is more localized than that of another independent variable. Second, quantile regression has been integrated into the GWR framework (Chen et al 2012), which is known as the geographically weighted quantile regression (GWQR), and the bootstrap approach to statistical inference minimizes the concerns about multiple testing and statistical inferences (Chen et al 2020). GWQR is particularly relevant to demographic and health research because the mean value of a distribution may not be of researchers' interest.…”