HPQLIM Procedure

HETERO Statement

  • HETERO dependent-variables tilde exogenous-variables </ options > ;

The HETERO statement specifies variables that are related to the heteroscedasticity of the residuals and the way that these variables are used to model the error variance. PROC HPQLIM supports the following heteroscedastic regression model:

y Subscript i Baseline equals bold x prime Subscript i Baseline bold-italic beta plus epsilon Subscript i
epsilon Subscript i Baseline tilde normal upper N left-parenthesis 0 comma sigma Subscript i Superscript 2 Baseline right-parenthesis

For more information about the specification of functional forms, see the section Heteroscedasticity. The following options specify the functional forms of heteroscedasticity:

LINK=EXP | LINEAR

specifies the functional form.

EXP

specifies the exponential link function:

StartLayout 1st Row 1st Column sigma Subscript i Superscript 2 2nd Column equals 3rd Column sigma squared left-parenthesis 1 plus exp left-parenthesis bold z Subscript i Superscript prime Baseline bold-italic gamma right-parenthesis right-parenthesis EndLayout
LINEAR

specifies the linear link function:

StartLayout 1st Row 1st Column sigma Subscript i Superscript 2 2nd Column equals 3rd Column sigma squared left-parenthesis 1 plus bold z Subscript i Superscript prime Baseline bold-italic gamma right-parenthesis EndLayout

The default is LINK=EXP.

NOCONST

specifies that there be no constant in the exponential heteroscedasticity model:

This option is ignored if you do not specify the LINK= option.

SQUARE

estimates the model by using the square of the linear heteroscedasticity function. For example, you can specify the following heteroscedasticity function:

sigma Subscript i Superscript 2 Baseline equals sigma squared left-parenthesis 1 plus left-parenthesis bold z Subscript i Superscript prime Baseline bold-italic gamma right-parenthesis squared right-parenthesis
model y = x1 x2 / censored(lb=0);
hetero y ~ z1 / link=linear square;

The SQUARE option does not apply to the exponential heteroscedasticity function because the square of an exponential function of bold z Subscript i Superscript prime Baseline bold-italic gamma is the same as the exponential of 2 bold z Subscript i Superscript prime Baseline bold-italic gamma. Hence, the only difference is that all bold-italic gamma estimates are divided by two.

This option is ignored if you do not specify the LINK= option. You cannot use the HETERO statement within a Bayesian framework.

Last updated: June 19, 2025