GARCH Estimation with Nelson-Cao Inequality Constraints

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 in-sample positivity constraints of the conditional variance .

Using the HETERO Statement with GARCH Models

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;

Limitations of GARCH and Heteroscedasticity Specifications

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.

IGARCH and Stationary GARCH Model

The condition implies that the GARCH process is weakly stationary since the mean, variance, and autocovariance are finite and constant over time. However, this condition is not sufficient for weak stationarity in the presence of autocorrelation. For example, the stationarity condition for an AR(1)-GARCH process is

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.

EGARCH Model

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.

GARCH-in-Mean

The GARCH-M model has the added regressor that is the conditional standard deviation:

where follows the ARCH or GARCH process.

Maximum Likelihood Estimation

The family of GARCH models are estimated using the maximum likelihood method. The log-likelihood function is computed from the product of all conditional densities of the prediction errors.

When is assumed to have a standard normal distribution (), the log-likelihood function is given by

where and is the conditional variance. When the GARCH-M model is estimated, . When there are no regressors, the residuals are denoted as or .

If has the standardized Student’s t distribution, the log-likelihood 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 log-likelihood function for the conditional t distribution converges to the log-likelihood function of the conditional normal GARCH model as .

The likelihood function is maximized via either the dual quasi-Newton or trust region algorithm. The default is the dual quasi-Newton 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 Yule-Walker estimates. The starting value is used for the GARCH process parameters.

The variance-covariance matrix is computed using the Hessian matrix. The dual quasi-Newton method approximates the Hessian matrix while the quasi-Newton 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 variance-covariance matrix is given by

where and . Therefore, the variance-covariance matrix without active constraints reduces to .