We construct efficient estimators of the identifiable parameters in a regression model when the errors follow a stationary parametric ARCH(P) process. We do not assume a functional form for the conditional density of the errors, but do require that it be symmetric about zero. The estimators of the mean parameters are adaptive in the sense of Bickel . The ARCH parameters are not jointly identifiable with the error density. We consider a reparameterization of the variance process and show that the identifiable parameters of this process are adaptively estimable.