You can specify the SPARSITY(IID) option in the MODEL
statement to assume that the distribution of conditional on
follows the linear model
where for
are iid in the distribution function F. Let
denote the density function of F. Further assume that
in a neighborhood of
. Then, under some mild conditions, Koenker and Bassett (1982) prove that the asymptotic distribution of the quantile regression estimates is
where and
The reciprocal of the density function,
, is called the sparsity function.
Accordingly, the covariance matrix of can be estimated as
where is the design matrix and
is an estimate of
. Under the iid assumption, the algorithm for computing
is as follows:
Fit a quantile regression model and compute the residuals. Each residual can be viewed as an estimated realization of the corresponding error
.
Compute the quantile level bandwidth . The HPQUANTSELECT procedure provides two bandwidth methods:
The Bofinger bandwidth is an optimizer of mean squared error for standard density estimation:
The Hall-Sheather bandwidth is based on Edgeworth expansions for studentized quantiles,
satisfies
for the construction of
confidence intervals, where T is the cumulative distribution function for the t distribution and
is the residual degrees of freedom.
The quantity
is not sensitive to f and can be estimated by assuming f is Gaussian as
where .
Compute residual quantiles and
as follows:
Set and
.
Use the equation
where is the ith smallest residual and
.
If , find i that satisfies
and
. If such an i exists, reset
so that
. Also find j that satisfies
and
. If such a j exists, reset
so that
.
Estimate the sparsity function as