Two tests are available in the QUANTREG procedure for the linear null hypothesis . Here denotes a subset of the parameters, where the parameter vector is partitioned as , and the covariance matrix for the parameter estimates is partitioned correspondingly as with ; and

The Wald test statistic, which is based on the estimated coefficients for the unrestricted model, is given by

where is an estimator of the covariance of . The QUANTREG procedure provides two estimators for the covariance, as described in the previous section. The estimator based on the asymptotic covariance is

where and is the estimated sparsity function. The estimator based on the bootstrap covariance is the empirical covariance of the MCMB samples.

The likelihood ratio test is based on the difference between the objective function values in the restricted and unrestricted models. Let and , and set

where is the estimated sparsity function.

Koenker and Machado (1999) prove that these two tests are asymptotically equivalent and that the distributions of the test statistics converge to under the null hypothesis, where is the dimension of .