The QUANTSELECT Procedure

Criteria Used in Model Selection Methods

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 $