On testing the equality of mean and quantile effects

This paper proposes tests for equality of the mean regression (MR) and quantile regression (QR) coefficients. The tests are based on the asymptotic joint distribution of the ordinary least squares and QR estimators. First, we formally derive the asymptotic joint distribution of these estimators. Second, we propose a Wald test for equality of the MR and QR coefficients considering a single fixed quantile, and also describe a more general test using multiple quantiles simultaneously. A very salient feature of these tests is that they produce asymptotically distribution-free nature of inference. In addition, we suggest a sup-type test for equality of the coefficients uniformly over a range of quantiles. For the estimation of the variance-covariance matrix, the use sample counterparts and bootstrap methods. An important attribute of the proposed tests is that they can be used as a heteroskedasticity test. Monte Carlo studies are conducted to evaluate the finite sample properties of the tests in terms of size and power. Finally, we briefly illustrate the implementation of the tests using Engel data.