Doubly fractional models for dynamic heteroscedastic cycles

Strong cyclical persistence is a common phenomenon that has been documented not only in the levels but also in the volatility of many time series, specially in astronomical or business cycle data. The class of doubly fractional models is extended to include the possibility of long memory in cyclical (non-zero) frequencies in both levels and volatility, and a new model, the GARMA–GARMASV (Gegenbauer AutoRegressive Moving Average–Gegenbauer AutoRegressive Moving Average Stochastic Volatility), is introduced. A sequential estimation strategy based on the Whittle approximation to maximum likelihood is proposed and its finite sample performance is evaluated with a Monte Carlo analysis. Finally, a trifactorial in the mean and bifactorial in the volatility version of the model is proved to successfully fit the well-known sunspot index.