Which tests should be perfomed after quantile regressions have been estimated? I´m performing a quantile regression. Initially I opted for a linear regression, but as I suspected that variations in X had different effects on the outcome variable across the distribution, I switched to a quantile regression.
My question is a simple one (and one important for my graduation at university). After we estimate a linear regression, some tests must be perfomed ; heterocedasticity test, normality of the residuals, multi-colinearity.
How about after my quantile regression´s have been estimated. Must I perform any tests in order to verifiy that my results are valid from a statistical point of view?
Thank you all!
Arthur
 A: The statement "after linear regression, some tests must be performed..." is misguided, because tests are the wrong way to evaluate assumptions. In particular, the model assumptions are all embedded in the null hypotheses, and they are all false, a priori, so it does not make sense to test them. Also, one does not test multicollinearity at all, even if testing was the right way to proceed, because there is no relevant null hypothesis.
That having been said, there are assumptions for quantile regression, just like linear regression, just fewer of them. Rather than linearity, constant variance, normality, and independence that are assumed for optional OLS, quantile regression just assumes linearity (of the chosen quantile as a function of $x$) and the same independence assumptions as for OLS.
These two assumptions are needed to give reasonable estimates and valid (large sample) inferences in quantile regression. However, unlike the full OLS assumptions, these two assumptions are not enough to guarantee that the estimates are optimal among possible estimates. For example, if you are estimating the median function, the further assumption of a double-exponential error distribution is needed to ensure that the quantile estimates coincide with maximum likelihood.
