Forecasting financial market activity using a semiparametric fractionally integrated Log-ACD

This paper considers the modeling and forecasting of long memory and a smooth scale function in different nonnegative financial time series aggregated from high-frequency data based on a fractionally integrated Log-ACD (FI-Log-ACD) and its semiparametric extension (Semi-FI-Log-ACD). Necessary and sufficient conditions for the existence of a stationary solution of the FI-Log-ACD and its properties under the log-normal assumption are studied in detail. An approximately best linear predictor based on the truncated AR(∞) form of the logarithmic process is proposed, and approximate variances of the prediction errors for an individual observation and for the conditional mean are obtained. Forecasting intervals for these quantities in the log-transformed data and in the original process are calculated under the log-normal assumption. Finally, applications to realized volatility, trading volumes and other data sets show that the proposal works very well in practice.