Quantile Regression on Quantile Ranges

Motivated by the fact that a linear specification in a quantile regression setting is unable to describe the non-linear relations among economic variables, well documented in the empirical econometrics literature, we formulate a threshold quantile regression model for one, known and unknown threshold value. Our modeling framework is more general than that of Hansen (2000), Galvao et. al. (2010) in that we consider covariates that do not exhibit threshold effects themselves, but are allowed to be affected by the covariate subject to regime-changes. We are also dealing with different inference problems. We derive the asymptotic properties of the model parameters as well as the threshold value(s) and develop inferencial procedures to identify heterogeneous effects of different covariate quantile ranges on different quantiles of the response. We conduct inference by developing a sup-Wald test that converges to a two-parameter Gaussian process. In addition, we derive the limiting distribution of the threshold value(s) for the single and multiple threshold effects settings under two asymptotic frameworks: one assuming fixed and another assuming shrinking shifts. Furthermore, we construct confidence intervals for the unknown threshold value via a likelihood-ratio-type test statistic and conduct a simulation study to investigate the coverage probability of the confidence intervals for different quantiles. Our asymptotic results complement those found in the existing literature on threshold regression models