The HPQUANTSELECT Procedure

Linear-in-Parameter Model with Non-iid Settings

The general form of a linear quantile regression model is

upper Q Subscript upper Y Baseline left-parenthesis tau vertical-bar bold x right-parenthesis equals bold x prime bold-italic beta left-parenthesis tau right-parenthesis

where the iid assumption is not necessary. Under some regularity conditions, the asymptotic distribution of the general form of quantile regression estimates is

StartRoot n EndRoot left-parenthesis ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis minus bold-italic beta left-parenthesis tau right-parenthesis right-parenthesis right-arrow upper N left-parenthesis 0 comma tau left-parenthesis 1 minus tau right-parenthesis bold upper H Subscript n Superscript minus Baseline bold upper Omega bold upper H Subscript n Superscript minus Baseline right-parenthesis

where bold upper H Subscript n Baseline equals limit Underscript n right-arrow normal infinity Endscripts n Superscript negative 1 Baseline sigma-summation bold x Subscript i Baseline bold x prime Subscript i Baseline f Subscript i Baseline left-parenthesis upper F Subscript i Superscript negative 1 Baseline left-parenthesis tau right-parenthesis right-parenthesis period

Accordingly, the covariance matrix of ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis can be estimated as

ModifyingAbove normal upper Sigma With caret left-parenthesis tau right-parenthesis equals n Superscript negative 2 Baseline tau left-parenthesis 1 minus tau right-parenthesis ModifyingAbove bold upper H With caret Subscript n Superscript minus Baseline left-parenthesis bold upper X prime bold upper X right-parenthesis ModifyingAbove bold upper H With caret Subscript n Superscript minus

where ModifyingAbove bold upper H With caret Subscript n Baseline equals n Superscript negative 1 Baseline sigma-summation left-parenthesis bold x Subscript i Baseline bold x prime Subscript i slash ModifyingAbove s With caret Subscript i Baseline left-parenthesis tau right-parenthesis right-parenthesis.

The sparsity function of the ith observation, ModifyingAbove s With caret Subscript i Baseline left-parenthesis tau right-parenthesis, can be estimated as

ModifyingAbove s With caret Subscript i Baseline left-parenthesis tau right-parenthesis equals StartFraction ModifyingAbove upper F With caret Subscript i Superscript negative 1 Baseline left-parenthesis tau plus h Subscript n Baseline right-parenthesis minus ModifyingAbove upper F With caret Subscript i Superscript negative 1 Baseline left-parenthesis tau minus h Subscript n Baseline right-parenthesis Over 2 h Subscript n Baseline EndFraction

where ModifyingAbove upper F With caret Subscript i Superscript negative 1 Baseline left-parenthesis tau plus-or-minus h Subscript n Baseline right-parenthesis equals bold x prime Subscript i Baseline ModifyingAbove bold-italic beta With caret left-parenthesis tau plus-or-minus h Subscript n Baseline right-parenthesis are the ith predicted quantile values at quantile levels left-parenthesis tau plus-or-minus h Subscript n Baseline right-parenthesis.

Last updated: December 09, 2022