AUTOREG Procedure

HETERO Statement

  • HETERO 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 of the regression.

The heteroscedastic regression model supported by the HETERO statement is

y Subscript t Baseline equals bold x Subscript t Baseline beta plus epsilon Subscript t
epsilon Subscript t Baseline tilde normal upper N left-parenthesis 0 comma sigma Subscript t Superscript 2 Baseline right-parenthesis
sigma Subscript t Superscript 2 Baseline equals sigma squared h Subscript t
h Subscript t Baseline equals l left-parenthesis bold z prime Subscript t Baseline eta right-parenthesis

The HETERO statement specifies a model for the conditional variance h Subscript t. The vector bold z Subscript t is composed of the variables listed in the HETERO statement, eta is a parameter vector, and l left-parenthesis dot right-parenthesis is a link function that depends on the value of the LINK= option. In the printed output, upper H upper E upper T Baseline 0 represents the estimate of sigma, while upper H upper E upper T Baseline 1 - upper H upper E upper T n are the estimates of parameters in the eta vector.

The keyword XBETA can be used in the variables list to refer to the model predicted value bold x prime Subscript t Baseline beta. If XBETA is specified in the variables list, other variables in the HETERO statement will be ignored. In addition, XBETA cannot be specified in the GARCH process.

For heteroscedastic regression models without GARCH effects, the errors epsilon Subscript t are assumed to be uncorrelated—the heteroscedasticity models specified by the HETERO statement cannot be combined with an autoregressive model for the errors. Thus, when a HETERO statement is used, the NLAG= option cannot be specified unless the GARCH= option is also specified.

You can specify the following options in the HETERO statement.

COEF=value

imposes constraints on the estimated parameters eta of the heteroscedasticity model. You can specify the following values:

NONNEG

specifies that the estimated heteroscedasticity parameters eta must be nonnegative.

UNIT

constrains all heteroscedasticity parameters eta to equal 1.

UNREST

specifies unrestricted estimation of eta.

ZERO

constrains all heteroscedasticity parameters eta to equal 0.

If you specify the GARCH= option in the MODEL statement, then by default COEF=NONNEG. If you do not specify the GARCH= option in the MODEL statement, then by default COEF=UNREST.

LINK=value

specifies the functional form of the heteroscedasticity model. By default, LINK=EXP. If you specify a GARCH model in the HETERO statement, the model is estimated using LINK=LINEAR only. For more information, see the section Using the HETERO Statement with GARCH Models. You can specify the following values:

EXP

specifies the exponential link function. The following model is estimated when you specify LINK=EXP:

h Subscript t Baseline equals exp left-parenthesis bold z prime Subscript t Baseline eta right-parenthesis
LINEAR

specifies the linear function; that is, the HETERO statement variables predict the error variance linearly. The following model is estimated when you specify LINK=LINEAR:

h Subscript t Baseline equals left-parenthesis 1 plus bold z prime Subscript t Baseline eta right-parenthesis
SQUARE

specifies the square link function. The following model is estimated when you specify LINK=SQUARE:

h Subscript t Baseline equals left-parenthesis 1 plus bold z prime Subscript t Baseline eta right-parenthesis squared
NOCONST

specifies that the heteroscedasticity model does not include the unit term for the LINK=SQUARE and LINK=LINEAR options. For example, the following model is estimated when you specify the options LINK=SQUARE NOCONST:

h Subscript t Baseline equals left-parenthesis bold z prime Subscript t Baseline eta right-parenthesis squared
STD=value

imposes constraints on the estimated standard deviation sigma of the heteroscedasticity model. You can specify the following values:

NONNEG

specifies that the estimated standard deviation parameter sigma must be nonnegative.

UNIT

constrains the standard deviation parameter sigma to equal 1.

UNREST

specifies unrestricted estimation of sigma.

By default, STD=UNREST.

TEST=LM

produces a Lagrange multiplier test for heteroscedasticity. The null hypothesis is homoscedasticity; the alternative hypothesis is heteroscedasticity of the form specified by the HETERO statement. The power of the test depends on the variables specified in the HETERO statement.

The test may give different results depending on the functional form specified by the LINK= option. However, in many cases the test does not depend on the LINK= option. The test is invariant to the form of h Subscript t when h Subscript t Baseline left-parenthesis 0 right-parenthesis equals 1 and h prime Subscript t Baseline left-parenthesis 0 right-parenthesis not-equals 0. (The condition h Subscript t Baseline left-parenthesis 0 right-parenthesis equals 1 is satisfied except when the NOCONST option is specified with LINK=SQUARE or LINK=LINEAR.)

Last updated: June 19, 2025