QED Working Paper Number
1425

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.

JEL Codes
Keywords
Circular convolution theorem
Conditional heteroskedasticity
Fast Fourier transform
FIGARCH
Truncation
Working Paper