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The AUTOREG Procedure |

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 syntax of the HETERO statement is

**HETERO***variables / options***;**

The HETERO statement specifies a model for the conditional variance

The keyword XBETA can be used in the

The errors are assumed to be uncorrelated-- the heteroscedasticity models specified by the HETERO statement cannot be combined with an autoregressive model for the errors. Thus, the HETERO statement cannot be used if the NLAG= option is specified in the MODEL statement.

You can specify the following options in the HETERO statement:

**LINK=***value*-
The LINK= option specifies the functional form of the
heteroscedasticity model.
If you want to estimate the GARCH model whose conditional error variance
contains exogenous variables, you do not need to specify the LINK= option.
The default is LINK=EXP.
Values of the LINK= option are
- EXP
- specifies the exponential link function.
The following model is estimated when you specify LINK=EXP:

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

- 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:

**COEF=***value*-
The COEF= option imposes constraints on the
estimated parameters of the heteroscedasticity model.
The values of the COEF= option are
- NONNEG
- specifies that
the estimated heteroscedasticity parameters must be
nonnegative.
When the HETERO statement is used in conjunction with the
GARCH= option, the default is COEF=NONNEG.
- UNIT
- constrains all heteroscedasticity parameters
to equal 1.
- ZERO
- constrains all heteroscedasticity parameters
to equal 0.
- UNREST
- specifies unrestricted estimation of . When the GARCH= option is not specified, the default is COEF=UNREST.

**STD=***value*-
The STD= option imposes constraints on the
estimated standard deviation of the
heteroscedasticity model.
The values of the STD= option are
- NONNEG
- specifies that
the estimated standard deviation parameter
must be nonnegative.
- UNIT
- constrains the standard deviation parameter
to equal 1.
- UNREST
- specifies unrestricted estimation of . This is the default.

**TEST=***LM*-
The TEST=LM option 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**-
The NOCONST option 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:

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