The QUANTSELECT Procedure

Criteria Used in Model Selection Methods

Subsections:

Single Quantile Effect Selection
Quantile Process Effect Selection

PROC QUANTSELECT supports a variety of fit statistics that you can specify as criteria for the CHOOSE= , SELECT= , and STOP= method-options in the MODEL statement.

Single Quantile Effect Selection

The following fit statistics are available for single quantile effect selection:

AIC: applies the Akaike’s information criterion (Akaike, 1981; Darlington, 1968; Judge et al., 1985).
AICC: applies the corrected Akaike’s information criterion (Hurvich and Tsai, 1989).
SBC: applies the Schwarz Bayesian information criterion (Schwarz, 1978; Judge et al., 1985).
SL<(LR1 | LR2)>: specifies the significance level of a statistic used to assess an effect’s contribution to the fit when it is added to or removed from a model. LR1 specifies likelihood ratio Type I, and LR2 specifies the likelihood ratio Type II. By default, the LR1 statistic is applied.
ADJR1: applies the adjusted quantile regression R statistic.
VALIDATE: applies the average check loss for the validation data.

Table 84.10 provides formulas and definitions for these fit statistics.

Table 84.10: Formulas and Definitions for Model Fit Summary Statistics for Single Quantile Effect Selection

Statistic	Definition or Formula
n	Number of observations
p	Number of parameters including the intercept
$r_ i(\tau )$	Residual for the ith observation; $r_{i}(\tau ) = y_ i-\mb{x}_ i\bbeta (\tau )$
$D(\tau )$	Total sum of check losses; $D(\tau )= \sum _{i=1}^ n \rho _\tau (r_{i})$
$D_0(\tau )$	Total sum of check losses for intercept-only model if intercept is a forced-in effect, otherwise for empty-model.
$\mbox{ACL}(\tau )$	Average check loss; ACL $\rule[.25in]{0in}{0cm}\displaystyle (\tau ) ={D(\tau )\over n}$
$\mbox{R1}(\tau )$	Counterpart of linear regression R-square for quantile regression; $\rule[.25in]{0in}{0cm}\displaystyle 1- {D(\tau )\over D_0(\tau )}$
$\mbox{ADJR1}(\tau )$	Adjusted R1; $(\tau )=$ $\rule[.25in]{0in}{0cm}\displaystyle 1-{(n-1)D(\tau )\over (n-p)D_0(\tau )}$
$\mbox{AIC}(\tau )$	$\rule[.25in]{0in}{0cm}\displaystyle 2n\ln \left( \mbox{ACL}(\tau ) \right) + 2p$
$\mbox{AICC}(\tau )$	$\rule[.25in]{0in}{0cm}\displaystyle 2n\ln \left( \mbox{ACL}(\tau ) \right) + {2pn\over n-p-1}$
$\mbox{SBC}(\tau )$	$\displaystyle 2n\ln \left( \mbox{ACL}(\tau ) \right) + p \ln (n)$

Quantile Process Effect Selection

The following statistics are available for quantile process effect selection:

AIC: specifies Akaike’s information criterion (Akaike, 1981; Darlington, 1968; Judge et al., 1985).
AICC: specifies the corrected Akaike’s information criterion (Hurvich and Tsai, 1989).
SBC: specifies Schwarz Bayesian information criterion (Schwarz, 1978; Judge et al., 1985).
ADJR1: specifies the adjusted quantile regression R statistic.
VALIDATE: specifies average check loss for the validation data.

Table 84.11 provides formulas and definitions for the fit statistics.

Table 84.11: Formulas and Definitions for Model Fit Summary Statistics for Quantile Process Effect Selection

Statistic	Definition or Formula
D	Integral of total sum of check losses; $D=\int _0^1D(\tau ) d\tau$
$D_0$	Integral of total sum of check losses for intercept-only model or empty-model if the NOINT option is used; $D_0=\int _0^1D_0(\tau ) d\tau$
$\mbox{ACL}$	Integral of average check loss; ACL $\rule[.25in]{0in}{0cm}\displaystyle ={D\over n}$
$\mbox{R1}$	$\rule[.25in]{0in}{0cm}\displaystyle 1- {D\over D_0}$
$\mbox{ADJR1}$	Adjusted R1; $\rule[.25in]{0in}{0cm}\displaystyle 1-{(n-1)D\over (n-p)D_0}$
$\mbox{AIC}$	$\rule[.25in]{0in}{0cm}\displaystyle \int _0^1\mr{AIC}(\tau ) d\tau$
$\mbox{AICC}$	$\rule[.25in]{0in}{0cm}\displaystyle \int _0^1\mr{AICC}(\tau ) d\tau$
$\mbox{SBC}$	$\displaystyle \int _0^1\mr{SBC}(\tau ) d\tau$