HPQLIM Procedure

Heteroscedasticity

If the variance of regression disturbance, (epsilon Subscript i), is heteroscedastic, the variance can be specified as a function of variables

upper E left-parenthesis epsilon Subscript i Superscript 2 Baseline right-parenthesis equals sigma Subscript i Superscript 2 Baseline equals f left-parenthesis bold z prime Subscript i Baseline bold-italic gamma right-parenthesis

Table 2 shows various functional forms of heteroscedasticity and the corresponding options to request each model.

Table 2: Specification Summary for Modeling Heteroscedasticity

Number Model Options
1 f left-parenthesis bold z prime Subscript i Baseline bold-italic gamma right-parenthesis equals sigma squared left-parenthesis 1 plus exp left-parenthesis bold z prime Subscript i Baseline gamma right-parenthesis right-parenthesis LINK=EXP (default)
2 f left-parenthesis bold z prime Subscript i Baseline bold-italic gamma right-parenthesis equals sigma squared exp left-parenthesis bold z prime Subscript i Baseline gamma right-parenthesis LINK=EXP NOCONST
3 f left-parenthesis bold z prime Subscript i Baseline bold-italic gamma right-parenthesis equals sigma squared left-parenthesis 1 plus sigma-summation Underscript l equals 1 Overscript upper L Endscripts gamma Subscript l Baseline z Subscript l i Baseline right-parenthesis LINK=LINEAR
4 f left-parenthesis bold z prime Subscript i Baseline bold-italic gamma right-parenthesis equals sigma squared left-parenthesis 1 plus left-parenthesis sigma-summation Underscript l equals 1 Overscript upper L Endscripts gamma Subscript l Baseline z Subscript l i Baseline right-parenthesis squared right-parenthesis LINK=LINEAR SQUARE


In model 3, variances of some observations might be negative. Although PROC HPQLIM assigns a large penalty to move the optimization away from such a region, the optimization might not be able to improve the objective function value and might become locked in the region. Signs of such an outcome include extremely small likelihood values or missing standard errors in the estimates. In model 2, variances are guaranteed to be greater than or equal to 0, but variances of some observations might be very close to 0. In these scenarios, standard errors might be missing. Models 1 and 4 do not have such problems. Variances in these models are always positive and never close to 0.

For more information, see the section Heteroscedasticity and Box-Cox Transformation.

Last updated: June 19, 2025