The QUANTREG Procedure
| Linear Test |
Consider the linear model
 |
where 
and 
are 
and 
dimensional unknown parameters, and 
, 
, are errors with unknown density function 
. Let 
; 
and 
be the parameter estimates for and respecitively at the 
th quantile. The covariance matrix 
for the parameter estimates is partitioned correspondingly as 
with 
; and 
Testing Effects of Covariates
Three tests are available in the QUANTREG procedure for the linear null hypothesis 
at the th 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 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 is given by
 |
where
 |
 |
 |
 |
 |
 |
 |
 |
and 
is a score function.
The following score functions are available in the QUANTREG procedure:
- 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 et 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 .
Testing for Heteroscedasticity
After you obtain the parameter estimates for several quantiles specified in the MODEL statement, you can test whether there are significant difference 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 (1982) for details.