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 . 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:

1. Fit a quantile regression model and compute the residuals. Each residual can be viewed as an estimated realization of the corresponding error .

2. 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 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 .

3. Compute residual quantiles and as follows:

1. Set and .

2. Use the equation

where is the ith smallest residual and .

3. 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 .

4. Estimate the sparsity function as