The asymmetric moving average model (asMA) is extended to allow forasymmetric quadratic conditional heteroskedasticity (asQGARCH). Theasymmetric parametrization of the conditional variance encompassesthe quadratic GARCH model of Sentana (1995). We introduce a framework fortesting asymmetries in theconditional mean and the conditional variance, separately or jointly.Some of the new model's moment properties are also derived. Empiricalresults are given for the daily returns of the compositeindex of the New York Stock Exchange. There is strong evidence ofasymmetry in both the conditional mean and conditional variancefunctions. In a genuine out-of-sample forecasting experiment theperformance of the best fitted asMA-asQGARCH model is compared topure asMA and no-change forecasts. This is done both in terms ofconditional mean forecasting as well as in terms of risk forecasting.