Three 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.

The rank test statistic under iid error models is given by

where

and is a score function. The following three score functions are available in the QUANTREG procedure:

Wilcoxon scores:

normal scores:

, where is the normal distribution function

sign scores:

An important feature of the rank test statistic is that, unlike Wald tests or likelihood ratio tests, no estimation of the sparsity function is required.

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