In financial time series analysis, symmetric and asymmetric GARCH models have become essential models for measuring the characteristics of economic volatility. In this article, we propose the consistency and asymptotic normality properties of the self-weighted quasi-maximum likelihood estimation without assuming the existence of the second moment for the moving average model with a class of GARCH error. Numerical simulation shows that the parameter estimation performs well; empirical analysis shows that the self-weighted quasi-maximum likelihood estimation of the moving average model with a class of GARCH error can improve the data fitting effect and prediction ability.