The QLIM Procedure

HETERO Statement

  • HETERO dependent variables ~ exogenous variables </ options > ;

The HETERO statement specifies variables that are related to the heteroscedasticity of the residuals and the way these variables are used to model the error variance. The heteroscedastic regression model supported by PROC QLIM is

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. You can specify the following options after a slash (/):

LINK=EXP | LINEAR

specifies the functional form. You can specify the following values:

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

By default, LINK=EXP.

NOCONST

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

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

estimates the model by using the square of 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 / discrete;
hetero y ~ z1 / link=linear square;

The option SQUARE does not apply to 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.

You can use the HETERO statement within a Bayesian framework, but you should do this carefully because convergence can be slower than in the homoscedastic case. For more information, see the section Priors for Heteroscedastic Models.

Last updated: August 08, 2024