The QTRSELECT Procedure
Quantile Regression
This section describes the basic concepts and notations for quantile regression and quantile regression model selection.
Let denote a data set of observations, where are responses and are regressors. Koenker and Bassett (1978) define the regression quantile at quantile level as any solution to the minimization problem
where is a check loss function in which and .
If you specify weights , in the WEIGHT statement, then weighted quantile regression is carried out by solving
The QTRSELECT procedure fits a quantile regression model by using a predictor-corrector interior point algorithm, which was originally designed to solve support vector machine classifiers for large data sets (Gertz and Griffin 2005, 2010).
Linear Model with Independent and Identically Distributed Errors
A linear model with independent and identically distributed (iid) errors assumes that the distribution of the response conditional on follows the linear model,
where for are iid in the distribution function F. General linear regression uses this model for deriving its statistical inferences and model selection criteria. Although quantile regression can analyze heterogeneous data without the iid errors assumption, this model is helpful for deriving some quantile regression model selection criteria. For more information about using a linear model with iid errors in deriving the quasi-likelihood information criteria, see the section Criteria Used in Model Selection.
Linear-in-Parameter Model with Non-iid Settings
The more general form of a linear quantile regression model is
where the iid assumption is not necessary.
Let denote the sparsity function of the ith observation. Under some regularity conditions, the asymptotic distribution of the general form of quantile regression estimates is
The sparsity function of the ith observation, , can be estimated as
where are the quantile predictions of the ith observation at quantile levels .
Accordingly, the covariance matrix of can be estimated as
You can specify the bandwidth method for computing by using either the COV=SPARSITY(BF) option for the Bofinger bandwidth method or the COV=SPARSITY(HF) option for the Hall-Sheather bandwidth method:
The Bofinger bandwidth optimizes the mean square 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
More Statistics for Parameter Estimates
Let denote the jth parameter estimate, and let denote the vector of the parameter estimates.
PROC QTRSELECT outputs the standard error, t value, and Pr > probability for each in the parameter estimates table. If you specify the CLB option in the MODEL statement, PROC QTRSELECT also outputs confidence limits in the parameter estimates table. Table 18.8 summarizes these statistics for .
Here is the element of , and denotes the -level student’s t score with 1 degree of freedom.