The AUTOREG Procedure

The heteroscedasticity consistent covariance matrix estimator (HCCME), also well-known as the sandwich (or robust or empirical) covariance matrix estimator, has been popular in recent years since it gives the consistent estimation of the covariance matrix of the parameter estimates even when the heteroscedasticity structure might be unknown or misspecified. White (1980) first proposes the concept of HCCME, known as HC0. However, the small-sample performance of HC0 is not good in some cases. Davidson and MacKinnon (1993) introduce more improvements of HC0, namely HC1, HC2 and HC3, with the degrees of freedom or leverage adjustment. Cribari-Neto (2004) proposes HC4 to deal with cases that have points of high leverage.

HCCME can be expressed in the following general “sandwich" form:

where , which stands for “bread", is the Hessian matrix and , which stands for “meat", is the outer product of gradient (OPG) with or without adjustment. For HC0, is the OPG without adjustment; that is,

where is the sample size and is the gradient vector of th observation. For HC1, is the OPG with the degrees of freedom correction; that is,

where is the number of parameters. For HC2, HC3, and HC4, the adjustment is related to leverage, namely,

The leverage is defined as , where is defined as follows:

• For an OLS model, is the th observed regressors in column vector form.

• For an AR error model, is the derivative vector of the th residual with respect to the parameters.

• For a GARCH or heteroscedasticity model, is the gradient of the th observation (that is, ).