  The majority of stylized facts of financial time series and several
Value-at-Risk measures are modeled via univariate or multivariate GARCH
processes. It is not rare that advanced GARCH models fail to converge for
computational reasons, and a usual parsimonious approach is the GJR-GARCH
model. There is a disagreement in the literature and the specialized
econometric software, on which constraints should be used for the parameters,
introducing indirectly the distinction between asymmetry and leverage. We show
that the approach used by various software packages is not consistent with the
Nelson-Cao inequality constraints. Implementing Monte Carlo simulations,
despite of the results being empirically correct, the estimated parameters are
not theoretically coherent with the Nelson-Cao constraints for ensuring
positivity of conditional variances. On the other hand ruling out the leverage
hypothesis, the asymmetry term in the GJR model can take negative values when
typical constraints like the condition for the existence of the second and
fourth moments, are imposed.
