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Gram-Charlier展开(GCE)在动态条件高阶矩GARCH模型中得到了广泛应用。相比于常见的分布,GCE的高阶矩形式更加直观,能更直接地刻画条件高阶矩的动态特征。此展开函数并不非负,在实际应用时需要平方并归一化,但已有文献大多忽略此处理后高阶矩所发生的变化。本文研究了平方处理方法对展开函数高阶矩的影响,推导了正确的高阶矩形式。实证研究表明,用原始参数作为近似高阶矩会产生显著偏误,在VaR预测时会严重低估风险。
Gram-Charlier expansion (GCE) has been widely used in dynamic conditional high-order GARCH models. Compared with the common distribution, the higher order moments of GCE are more intuitive and can describe the dynamic characteristics of higher order moments more directly. This expansion function is not non-negative, it needs to be square and normalized in practical application, but most of the existing literature neglects the changes of higher moments after this processing. In this paper, we study the influence of the square processing method on the higher moments of the expansion function and deduce the correct higher order moments. Empirical studies show that the use of the original parameters as approximate higher-order moments can produce significant bias and seriously underestimate the risk in VaR predictions.