Estimating weak GARCH representations

The classical definitions of GARCH-type processes rely on strong assumption s on the first two conditional moments. The common practice in empirical st udies, however, has been to test for GARCH by detecting serial correlations in the squared regression errors. This can be problematic because such aut ocorrelation structures are compatible with severe misspecifications of the standard GARCH, Numerous examples are provided in the paper. In consequenc e? standard (quasi-) maximum likelihood procedures can be inconsistent if t he conditional first two moments an misspecified. To alleviate these proble ms of possible misspecification, we consider weak GARCH representations cha racterized by an ARMA structure for the squared error terms. The weak GARCH representation eliminates the need for correct specification of the first two conditional moments. The parameters of the representation are estimated via two-stage least squares. The estimator is shown to be consistent and a symptotically normal. Forecasting issues are also addressed.