Because the ARMA–GARCH model can generate data with some important properties such as skewness, heavy tails, and volatility persistence, it has become a benchmark model in analyzing financial and economic data. The commonly employed quasi maximum likelihood estimation (QMLE) requires a finite fourth moment for both errors and the sequence itself to ensure a normal limit. The self-weighted quasi maximum exponential likelihood estimation (SWQMELE) reduces the moment constraints by assuming that the errors and their absolute values have median zero and mean one, respectively. Therefore, it is necessary to test zero median of errors before applying the SWQMELE, as changing zero mean to zero median destroys the ARMA–GARCH structure. This paper develops an efficient empirical likelihood test without estimating the GARCH model but using the GARCH structure to reduce the moment effect. A simulation study confirms the effectiveness of the proposed test. The data analysis shows that some financial returns do not have zero median of errors, which cautions the use of the SWQMELE.

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Fitting an ARMA‐GARCH model has become a common practice in financial econometrics. Because the asymptotic normality of the quasi maximum likelihood estimation (QMLE) requires finite fourth moment for both errors and the sequence itself, self‐weighted quasi maximum exponential likelihood estimation (SWQMELE) has been proposed to reduce the moment constraints but requires the errors to have zero median instead of zero mean. Because changing zero mean to zero median destroys the ARMA‐GARCH structure and has a serious effect on skewed data, this article proposes an efficient empirical likelihood test for zero mean of errors in the application of SWQMELE to ensure that the model still concerns conditional mean. A simulation study confirms the good finite sample performance before applying the test to the US housing price indexes and financial returns for the study of comovement.