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
.
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