Consider the series , which follows the GARCH process. The conditional distribution of the series Y for time t is written

where denotes all available information at time . The conditional variance is

where


The GARCH model reduces to the ARCH process when . At least one of the ARCH parameters must be nonzero (). The GARCH regression model can be written



where .
In addition, you can consider the model with disturbances following an autoregressive process and with the GARCH errors. The ARGARCH regression model is denoted




The GARCH model is written in ARCH() form as






where is a backshift operator. Therefore, if and . Assume that the roots of the following polynomial equation are inside the unit circle:

where and Z is a complex scalar. and do not share common factors. Under these conditions, , , and these coefficients of the ARCH() process are well defined.
Define . The coefficient is written














where for and for .
Nelson and Cao (1992) proposed the finite inequality constraints for GARCH and GARCH cases. However, it is not straightforward to derive the finite inequality constraints for the general GARCH model.
For the GARCH model, the nonlinear inequality constraints are









For the GARCH model, the nonlinear inequality constraints are















where and are the roots of .
For the GARCH model with , only nonlinear inequality constraints ( for to max()) are imposed, together with the insample positivity constraints of the conditional variance .
The HETERO statement can be combined with the GARCH= option in the MODEL statement to include input variables in the GARCH conditional variance model. For example, the GARCH variance model with two dummy input variables D1 and D2 is






The following statements estimate this GARCH model:
proc autoreg data=one; model y = x z / garch=(p=1,q=1); hetero d1 d2; run;
The parameters for the variables D1 and D2 can be constrained using the COEF= option. For example, the constraints are imposed by the following statements:
proc autoreg data=one; model y = x z / garch=(p=1,q=1); hetero d1 d2 / coef=unit; run;
When you specify both the GARCH= option and the HETERO statement, the GARCH=(TYPE=EXP) option is not valid. The COVEST= option is not applicable to the EGARCH model.
The condition implies that the GARCH process is weakly stationary since the mean, variance, and autocovariance are finite and constant over time. When the GARCH process is stationary, the unconditional variance of is computed as

where and is the GARCH conditional variance.
Sometimes the multistep forecasts of the variance do not approach the unconditional variance when the model is integrated in variance; that is, .
The unconditional variance for the IGARCH model does not exist. However, it is interesting that the IGARCH model can be strongly stationary even though it is not weakly stationary. Refer to Nelson (1990) for details.
The EGARCH model was proposed by Nelson (1991). Nelson and Cao (1992) argue that the nonnegativity constraints in the linear GARCH model are too restrictive. The GARCH model imposes the nonnegative constraints on the parameters, and , while there are no restrictions on these parameters in the EGARCH model. In the EGARCH model, the conditional variance, , is an asymmetric function of lagged disturbances :

where


The coefficient of the second term in is set to be 1 (=1) in our formulation. Note that if . The properties of the EGARCH model are summarized as follows:
The function is linear in with slope coefficient if is positive while is linear in with slope coefficient if is negative.
Suppose that . Large innovations increase the conditional variance if and decrease the conditional variance if .
Suppose that . The innovation in variance, , is positive if the innovations are less than . Therefore, the negative innovations in returns, , cause the innovation to the conditional variance to be positive if is much less than 1.
As shown in many empirical studies, positive and negative innovations have different impacts on future volatility. There is a long list of variations of GARCH models that consider the asymmetricity. Three typical variations are the quadratic GARCH (QGARCH) model (Engle and Ng, 1993), the threshold GARCH (TGARCH) model (Glosten, Jaganathan, and Runkle, 1993; Zakoian, 1994), and the power GARCH (PGARCH) model (Ding, Granger, and Engle, 1993). For more details about the asymmetric GARCH models, see Engle and Ng (1993).
In the QGARCH model, the lagged errors’ centers are shifted from zero to some constant values:

In the TGARCH model, there is an extra slope coefficient for each lagged squared error,

where the indicator function is one if ; otherwise, zero.
The PGARCH model not only considers the asymmetric effect, but also provides another way to model the long memory property in the volatility,

where and .
Note that the implemented TGARCH model is also well known as GJRGARCH (Glosten, Jaganathan, and Runkle, 1993), which is similar to the threshold GARCH model proposed by Zakoian (1994) but not exactly same. In Zakoian’s model, the conditional standard deviation is a linear function of the past values of the white noise. Zakoian’s version can be regarded as a special case of PGARCH model when .
The GARCHM model has the added regressor that is the conditional standard deviation:


where follows the ARCH or GARCH process.
The family of GARCH models are estimated using the maximum likelihood method. The loglikelihood function is computed from the product of all conditional densities of the prediction errors.
When is assumed to have a standard normal distribution (), the loglikelihood function is given by

where and is the conditional variance. When the GARCHM model is estimated, . When there are no regressors, the residuals are denoted as or .
If has the standardized Student’s t distribution, the loglikelihood function for the conditional t distribution is


where is the gamma function and is the degree of freedom (). Under the conditional t distribution, the additional parameter is estimated. The loglikelihood function for the conditional t distribution converges to the loglikelihood function of the conditional normal GARCH model as .
The likelihood function is maximized via either the dual quasiNewton or the trust region algorithm. The default is the dual quasiNewton algorithm. The starting values for the regression parameters are obtained from the OLS estimates. When there are autoregressive parameters in the model, the initial values are obtained from the YuleWalker estimates. The starting value is used for the GARCH process parameters.
The variancecovariance matrix is computed using the Hessian matrix. The dual quasiNewton method approximates the Hessian matrix while the quasiNewton method gets an approximation of the inverse of Hessian. The trust region method uses the Hessian matrix obtained using numerical differentiation. When there are active constraints, that is, , the variancecovariance matrix is given by

where and . Therefore, the variancecovariance matrix without active constraints reduces to .