Consider the linear model
where and are p- and q-dimensional unknown parameters and , , are errors with unknown density function . Let , and let and be the parameter estimates for and , respectively at the quantile. The covariance matrix for the parameter estimates is partitioned correspondingly as with ; and
Three tests are available in the QUANTREG procedure for the linear null hypothesis at the quantile:
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 that is based on the asymptotic covariance is
where and is the estimated sparsity function. The estimator that is 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 let . Set
where is the estimated sparsity function.
The rank test statistic is given by
where
and is one of the following score functions:
Wilcoxon scores:
normal scores: , where is the normal distribution function
sign scores:
tau scores: .
The rank test statistic , unlike Wald tests or likelihood ratio tests, requires no estimation of the nuisance parameter under iid error models (Gutenbrunner etÂ al. 1993).
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 q is the dimension of .
After you obtain the parameter estimates for several quantiles specified in the MODEL statement, you can test whether there are significant differences for the estimates for the same covariates across the quantiles. For example, if you want to test whether the parameters are the same across quantiles, the null hypothesis can be written as , where are the quantiles specified in the MODEL statement. See Koenker and Bassett (1982a) for details.