The 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

$y_{t} = \mb {x} _{t}{\beta } + {\epsilon }_{t}$

${\epsilon }_{t} \sim \mr {N}(0, {\sigma }_{t}^{2})$

${\sigma }^{2}_{t} = {\sigma }^{2}h_{t}$

$h_{t} = l( {\mb {z}’}_{t}{\eta } )$

The HETERO statement specifies a model for the conditional variance ${h_{t}}$ . The vector ${\mb {z} _{t}}$ is composed of the variables listed in the HETERO statement, ${{\eta }}$ is a parameter vector, and ${l({\cdot })}$ is a link function that depends on the value of the LINK= option. In the printed output, represents the estimate of sigma, while - 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 ${ {\mb {x}’}_{t}{\beta }}$ . 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 ${{\epsilon }_{t}}$ 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.

LINK=value

specifies the functional form of the heteroscedasticity model. By default, LINK=EXP. If you specify a GARCH model with the HETERO statement, the model is estimated using LINK= LINEAR only. For details, see the section Using the HETERO Statement with GARCH Models. Values of the LINK= option are as follows:

EXP

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

$h_{t} = {\exp }( {\mb {z}’}_{t}{\eta })$

SQUARE

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

$h_{t} = (1+ {\mb {z}’}_{t}{\eta })^{2}$

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:

$h_{t} = (1+ {\mb {z}’}_{t}{\eta })$

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.
ZERO: constrains all heteroscedasticity parameters ${{\eta }}$ to equal 0.
UNREST: specifies unrestricted estimation of ${{\eta }}$ .

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

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 ${{\sigma }}$ .

The default is 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 ${h_{t}}$ when ${h_{t}(0) = 1}$ and $h’_{t}(0) \ne 0$ . (The condition ${h_{t}(0) = 1}$ is satisfied except when the NOCONST option is specified with LINK=SQUARE or LINK=LINEAR.)

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:

$h_{t} = ( {\mb {z}’}_{t}{\eta })^{2}$