Orthogonal polynomials can be used to modify the moments of the distribution of a random variable. In this paper, polynomially adjusted distributions are employed to model the skewness and kurtosis of the conditional distributions of GARCH models. To flexibly capture the skewness and kurtosis of data, the distributions of the innovations that are polynomially reshaped include, besides the Gaussian, also leptokurtic laws such as the logistic and the hyperbolic secant. Modeling GARCH innovations with polynomially adjusted distributions can effectively improve the precision of the forecasts. This strategy is analyzed in GARCH models with different specifications for the conditional variance, such as the APARCH, the EGARCH, the Realized GARCH, and APARCH with time-varying skewness and kurtosis. An empirical application on different types of asset returns shows the good performance of these models in providing accurate forecasts according to several criteria based on density forecasting, downside risk, and volatility prediction.

正交多项式可用于修改随机变量分布的矩。在本文中，多项式调整分布被用来模拟 GARCH 模型条件分布的偏度和峰度。为了灵活地捕捉数据的偏度和峰度，多项式重塑的创新分布除了高斯分布之外，还包括诸如逻辑和双曲正割等瘦峰定律。使用多项式调整分布对 GARCH 创新进行建模可以有效提高预测的精度。该策略在具有不同条件方差规范的 GARCH 模型中进行分析，例如 APARCH、EGARCH、Realized GARCH 和具有时变偏度和峰度的 APARCH。