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.

由于 ARMA-GARCH 模型可以生成具有偏度、重尾和波动持续性等重要属性的数据，因此它已成为分析金融和经济数据的基准模型。常用的准最大似然估计 (QMLE) 要求误差和序列本身都有一个有限的四阶矩，以确保正常限制。自加权准最大指数似然估计 (SWQMELE) 通过假设误差及其绝对值分别具有中值零和均值 1 来减少矩约束。因此，有必要在应用 SWQMELE 之前测试零中值误差，因为将零均值更改为零中值会破坏 ARMA-GARCH 结构。本文开发了一种有效的经验似然检验，没有估计 GARCH 模型，而是使用 GARCH 结构来减少矩效应。模拟研究证实了所提议测试的有效性。数据分析表明，一些财务回报的误差中位数不为零，这提醒了使用 SWQMELE。