Frequency Domain Calculation of Seasonal VARMA Autocovariances

Abstract

In applications of the fitting and forecasting of Seasonal Vector AutoRegressive Moving Average (SVARMA) models, it is important to quickly and accurately compute the autocovariances. Recursive time domain approaches rely on expressing the seasonal AutoRegressive matrix polynomials as a high order non-seasonal AutoRegressive matrix polynomial; initialization of the recursions is therefore costly, because a large-dimensional matrix must be inverted. However, the resulting high-order polynomial has many zeroes that are not intelligently exploited, indicating that this time domain approach is not optimal. It is proposed to compute the autocovariances via integrating the spectral density, taking advantage of analytical expressions for the inverse AutoRegressive matrix polynomials in terms of determinant and adjoint. The R and RCPP code can be hundreds of times faster than the time domain methods when the dimension and seasonal period are large.