This paper provides an exact algorithm for efficient computation of the time series of conditional variances, and hence the likelihood function, of models that have an ARCH($\infty$) 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 \cite{JensenNielsen2014}. It takes advantage of the fast Fourier transform (FFT) to achieve an order of magnitude improvement in computational speed. The efficiency of the algorithm allows estimation (and simulation/bootstrapping) of ARCH($\infty$) models, even with very large data sets and without the truncation of the filter commonly applied in the literature. We also show that the elimination of the truncation of the filter substantially reduces the bias of the quasi-maximum-likelihood estimators. Our results are illustrated in two empirical examples.

中文翻译：

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至无穷大：ARCH (∞) 模型的高效计算*

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时间序列以及似然函数。这类模型包括例如分数积分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。它利用快速傅里叶变换 (FFT) 来实现计算速度的数量级改进。该算法的效率允许估计（和模拟/引导）ARCH($\infty$) 模型，即使是非常大的数据集并且没有在文献中常用的过滤器的截断。我们还表明，消除滤波器的截断大大减少了准最大似然估计量的偏差。我们的结果在两个实证例子中得到了说明。