A Unified Framework for the Estimation and Inference in Linear Quantile Regression: A Local Polynomial Approach

For linear quantile regressions with smooth coefficient functions, we establish an asymptotically valid inference procedure for quantile coefficients at all quantile levels including 0 and 1. This is made possible by employing the local polynomial estimator of the linear quantile coefficient proposed in Gimenes and Guerre (2015). In contrast to the Bassett and Koenker (1978) estimator, we show that under the assumptions adopted in this paper, the local polynomial estimator is unique (with probability approaching 1) and asymptotically normally distributed for all quantile levels. Asymptotic normality at all quantile levels and existence of the Hessian matrix allow us to construct a self-normalized statistic based on which we develop a uni…fied inference for linear quantile regressions. A small simulation study is carried out to compare the new procedure with existing ones. An empirical application for extreme quantiles is also provided.