This paper continues the study of time series models generated by non-negative innovations which was begun in Feigin and Resnick (1992,1994). We concentrate on moving average processes. Estimators for moving average coefficients are proposed and consistency and asymptotic distributions established for the case of an order one moving average assuming either the right or left tail of the innovation distribution is regularly varying. The rate of convergence can be superior to that of the Yule--Walker or maximum likelihood estimators.