HETERO Statement

The AUTOREG Procedure

HETERO Statement

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 heteroscedastic regression model supported by the HETERO statement is

$y_{t} = x_{t}{\beta} + {\epsilon}_{t}$

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

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

$h_{t} = l( z^{'}_{t}{\eta} )$

The HETERO statement specifies a model for the conditional variance h_t. The vector z_t is composed of the variables listed on the HETERO statement, ${\eta}$ is a parameter vector, and l(·) is a link function that depends on the value of the LINK= option.

The keyword XBETA can be used in the variables list to refer to the model predicted value ${ x^{'}_{t}{\beta}}$ .

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, 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:

$h_{t} = {\exp}( z^{'}_{t}{\eta})$

SQUARE

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

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

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:

$h_{t} = (1+ z^{'}_{t}{\eta})$

COEF= value

The COEF= option imposes constraints on the estimated parameters ${\eta}$ of the heteroscedasticity model. The values of the COEF= option are

NONNEG: specifies that the estimated heteroscedasticity parameters ${\eta}$ 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 ${\eta}$ to equal 1.
ZERO: constrains all heteroscedasticity parameters ${\eta}$ to equal 0.
UNREST: specifies unrestricted estimation of ${\eta}$ . When the GARCH= option is not specified, the default is COEF=UNREST.

STD= value

The STD= option imposes constraints on the estimated standard deviation ${{\sigma}}$ of the heteroscedasticity model. The values of the STD= option are

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 ${{\sigma}}$ . 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:

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

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