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 10 provides formulas and definitions for these fit statistics.

Table 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 Subscript i Baseline left-parenthesis tau right-parenthesis Residual for the ith observation; r Subscript i Baseline left-parenthesis tau right-parenthesis equals y Subscript i Baseline minus bold x Subscript i Baseline bold-italic beta left-parenthesis tau right-parenthesis
upper D left-parenthesis tau right-parenthesis Total sum of check losses; upper D left-parenthesis tau right-parenthesis equals sigma-summation Underscript i equals 1 Overscript n Endscripts rho Subscript tau Baseline left-parenthesis r Subscript i Baseline right-parenthesis
upper D 0 left-parenthesis tau right-parenthesis Total sum of check losses for intercept-only model if intercept is a forced-in effect, otherwise for empty-model
ACL left-parenthesis tau right-parenthesis Average check loss; ACL left-parenthesis tau right-parenthesis equals StartFraction upper D left-parenthesis tau right-parenthesis Over n EndFraction
upper R 1 left-parenthesis tau right-parenthesis Counterpart of linear regression R-square for quantile regression; 1 minus StartFraction upper D left-parenthesis tau right-parenthesis Over upper D 0 left-parenthesis tau right-parenthesis EndFraction
ADJR 1 left-parenthesis tau right-parenthesis Adjusted R1; ADJR 1 left-parenthesis tau right-parenthesis equals 1 minus StartFraction left-parenthesis n minus 1 right-parenthesis upper D left-parenthesis tau right-parenthesis Over left-parenthesis n minus p right-parenthesis upper D 0 left-parenthesis tau right-parenthesis EndFraction
AIC left-parenthesis tau right-parenthesis Akaike’s information criterion; AIC left-parenthesis tau right-parenthesis equals 2 n ln left-parenthesis ACL left-parenthesis tau right-parenthesis right-parenthesis plus 2 p
AICC left-parenthesis tau right-parenthesis Corrected Akaike’s information criterion; AICC left-parenthesis tau right-parenthesis equals 2 n ln left-parenthesis ACL left-parenthesis tau right-parenthesis right-parenthesis plus StartFraction 2 p n Over n minus p minus 1 EndFraction
SBC left-parenthesis tau right-parenthesis Schwarz Bayesian information criterion; SBC left-parenthesis tau right-parenthesis equals 2 n ln left-parenthesis ACL left-parenthesis tau right-parenthesis right-parenthesis plus p ln left-parenthesis n right-parenthesis


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 11 provides formulas and definitions for the fit statistics.

Table 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; upper D equals integral Subscript 0 Superscript 1 Baseline upper D left-parenthesis tau right-parenthesis d tau
upper D 0 Integral of total sum of check losses for intercept-only model or empty-model if the NOINT option is used; upper D 0 equals integral Subscript 0 Superscript 1 Baseline upper D 0 left-parenthesis tau right-parenthesis d tau
ACL Integral of average check loss; ACL equals StartFraction upper D Over n EndFraction
upper R 1 Counterpart of linear regression R-square for quantile process regression; upper R 1 equals 1 minus StartFraction upper D Over upper D 0 EndFraction
ADJR 1 Adjusted R1; ADJR 1 equals 1 minus StartFraction left-parenthesis n minus 1 right-parenthesis upper D Over left-parenthesis n minus p right-parenthesis upper D 0 EndFraction
AIC Akaike’s information criterion; AIC equals integral Subscript 0 Superscript 1 Baseline AIC left-parenthesis tau right-parenthesis d tau
AICC Corrected Akaike’s information criterion; AICC equals integral Subscript 0 Superscript 1 Baseline AICC left-parenthesis tau right-parenthesis d tau
SBC Schwarz Bayesian information criterion; SBC equals integral Subscript 0 Superscript 1 Baseline SBC left-parenthesis tau right-parenthesis d tau


FQPR Effect Selection

If you use the QUANTILE=FQPR option to perform the fast quantile process regression, the following statistics are available for FQPR 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 12 provides formulas and definitions for the fit statistics.

Table 12: Formulas and Definitions for Model Fit Summary Statistics for FQPR Effect Selection

Statistic Definition or Formula
q Number of quantile levels for the FQPR quantile-level grid
tau Subscript i The ith quantile level of the FQPR quantile-level grid
D Average of total sum of check losses; upper D equals StartFraction 1 Over q EndFraction sigma-summation Underscript i equals 1 Overscript q Endscripts upper D left-parenthesis tau Subscript i Baseline right-parenthesis
upper D 0 Average of total sum of check losses for intercept-only model or empty-model if the NOINT option is used; upper D 0 equals StartFraction 1 Over q EndFraction sigma-summation Underscript i equals 1 Overscript q Endscripts upper D 0 left-parenthesis tau Subscript i Baseline right-parenthesis
ACL Average of average check loss; ACL equals StartFraction upper D Over n EndFraction
upper R 1 Counterpart of linear regression R-square for FQPR; upper R 1 equals 1 minus StartFraction upper D Over upper D 0 EndFraction
ADJR 1 Adjusted R1; ADJR 1 equals 1 minus StartFraction left-parenthesis n minus 1 right-parenthesis upper D Over left-parenthesis n minus p right-parenthesis upper D 0 EndFraction
AIC Akaike’s information criterion; AIC equals StartFraction 1 Over q EndFraction sigma-summation Underscript i equals 1 Overscript q Endscripts AIC left-parenthesis tau Subscript i Baseline right-parenthesis
AICC Corrected Akaike’s information criterion; AICC equals StartFraction 1 Over q EndFraction sigma-summation Underscript i equals 1 Overscript q Endscripts AICC left-parenthesis tau Subscript i Baseline right-parenthesis
SBC Schwarz Bayesian information criterion; SBC equals StartFraction 1 Over q EndFraction sigma-summation Underscript i equals 1 Overscript q Endscripts SBC left-parenthesis tau Subscript i Baseline right-parenthesis


Last updated: December 09, 2022