到无穷远：ARCH（\ infty）模型的高效计算

To infinity and beyond: Efficient computation ofARCH(\infty) models

作者：

莫滕·勒加尔德·尼尔森（Morten lrregaard Nielsen）

安托万·诺尔（Antoine No.l）

This paper provides an exact algorithm for efficient computation of the time series of conditional variances, and hence the likelihood function, of models that havean ARCH(∞) representation. This class of models includes, e.g., the fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) model.Our algorithm is a variation of the fast fractional difference algorithm of Jensen andNielsen (2014). It takes advantage of the fast Fourier transform (FFT) to achieve anorder of magnitude improvement in computational speed. The efficiency of the algorithm allows estimation (and simulation/bootstrapping) of ARCH(∞) models, evenwith very large data sets and without the truncation of the filter commonly applied inthe literature. We also show that the elimination of the truncation of the filter substantially reduces the bias of the quasi-maximum-likelihood estimators. Our resultsare illustrated in two empirical examples.
本文提供了一种精确算法，用于有效计算具有ARCH（$ \ infty $）表示的模型的条件方差的时间序列，从而有效地计算似然函数。这类模型包括，例如，分数积分的广义自回归条件异方差（FIGARCH）模型。我们的算法是\ cite {JensenNielsen2014}的快速分数差分算法的变体。它利用快速傅立叶变换（FFT）来实现计算速度的数量级改进。该算法的效率允许对ARCH（$ \ infty $）模型进行估计（和仿真/自举），即使具有非常大的数据集且没有删减文献中常用的滤波器也是如此。我们还表明，消除了滤波器的截断现象，大大降低了准最大似然估计量的偏差。我们的结果在两个经验示例中得到说明。