The Multiple Quantile Regression on Quantile Ranges Model

Abstract

Motivated by the fact that a linear speci fication in a quantile regression setting is unable to describe the non-linear relations among economic variables, as documented in the empirical econometrics literature, we are the first to formulate and analyze a multiple threshold quantile regression model. Generalizing Hansen (2000) framework, we propose an asymptotic framework to analyze the properties of the model parameters together with the unknown threshold values and develop inferential procedures (Wald tests) to identify heterogeneous e ffects of di fferent covariate quantile ranges on diff erent quantiles of the response. We derive the limiting distribution of the threshold values under two asymptotic frameworks: one assuming fixed and another assuming shrinking magnitude of shifts and we discuss the case where more than one quantile of the response is a effected by some regime change. Finally, we propose a double supremum Wald test for zero versus k regime-changes and a likelihood-ratio-type test for l versus l 1 regime-changes in the covariate and derive their limiting distributions. Simulations assess favorably the relevance of our testing procedures. Our asymptotic results complement those of Gonzalo and Pitarakis (2006) and Li and Ling (2011) to the quantile regression setting.