This paper develops a parametric family of models of generalized autoregressive heteroskedasticity (GARCH). The family nests the most popular symmetric and asymmetric GARCH models, thereby highlighting the relation between the models and their treatment of asymmetry. Furthermore, the structure permits nested tests of different types of asymmetry and functional forms. Daily U.S. stock return data reject all standard GARCH models in favor of a model in which, roughly speaking, the conditional standard deviation depends on the shifted absolute value of the shocks raised to the power three halves and past standard deviations.

It seems plausible that an increase in stock market volatility raises required stock returns, and thus lowers stock prices. We develop a formal model of this volatility feedback effect using a simple model of changing variance (a quadratic generalized autoregressive conditionally heteroskedastic, or QGARCH, model). Our model is asymmetric and helps to explain the negative skewness and excess kurtosis of U.S. monthly and daily stock returns over the period 1926–1988. We find that volatility feedback normally has little effect on returns, but it can be important during periods of high volatility.