Simplex quantile regression without crossing

Noncrossing quantile regression with an emphasis on model misspecification is investigated. While many sophisticated methods for noncrossing quantile have been developed under the assumption of correct model specification, model misspecification and extrapolation are two issues rarely considered in the literature. In this paper, a monotonicity representation for quantile regression models is obtained under simplex embedding, which leads to the simplex quantile regression (SQR) method. SQR model advocates the use of the Barycentric coordinate system and is immune to quantile crossing. An innovative concept, the maximum effective region, which defines the allowable region for extrapolation without quantile crossing is introduced. Under model misspecification, the existence of pseudo true quantile regression parameters, the asymptotic property of SQR estimator and asymptotic optimality of cross-validation for model selection and model averaging are established. The advantages of our new approach are also illustrated by numerical results. The proposed method is applied to U.S. financial market data. Our SQR-based strategy of asset pricing analysis allows for head-to-head transparent comparisons of five periods between October 1987 (Black Monday crash period) and February 2020 (COVID-19 pandemic).