Bayesian Quantile Regression with Multiple Proxy Variables

Data integration has become more challenging with the emerging availability of multiple data sources. This paper considers Bayesian quantile regression estimation when the key covariate is not directly observed, but the unobserved covariate has multiple proxies. In a unified estimation procedure, the proposed method incorporates these multiple proxies, which have various relationships with the unobserved covariate. The proposed approach allows the inference of both the quantile function and unobserved covariate. Moreover, it requires no linearity of the quantile function or parametric assumptions on the regression error distribution and simultaneously accommodates both linear and nonlinear proxies. The simulation studies show that this methodology successfully integrates multiple proxies and reveals the quantile relationship for a wide range of nonlinear data. The proposed method is applied to the administrative data obtained from the Survey of Household Finances and Living Conditions provided by Statistics Korea. The proposed Bayesian quantile regression is implemented to specify the relationship between assets and salary income in the presence of multiple income records.