The HPQUANTSELECT Procedure
Linear Model with iid Errors
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. 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 