Exponential Smoothing, Long Memory and Volatility Prediction

Extracting and forecasting the volatility of financial markets is an important empirical problem. The paper provides a time series characterization of the volatility components arising when the volatility process is fractionally integrated, and proposes a new predictor that can be seen as extension of the very popular and successful forecasting and signal extraction scheme, known as exponential smoothing (ES). First, we derive a generalization of the Beveridge-Nelson result, decomposing the series into the sum of fractional noise processes with decreasing orders of integration. Secondly, we consider three models that are natural extensions of ES: the fractionally integrated first order moving average (FIMA) model, a new integrated moving average model formulated in terms of the fractional lag operator (FLagIMA), and a fractional equal root integrated moving average (FerIMA) model, proposed originally by Hosking. We investigate the properties of the volatility components and the forecasts arising from these specification, which depend uniquely on the memory and the moving average parameters. For statistical inference we show that, under mild regularity conditions, the Whittle pseudo-maximum likelihood estimator is consistent and asymptotically normal. The estimation results show that the log-realized variance series are mean reverting but nonstationary. An out-of-sample rolling forecast exercise illustrates that the three generalized ES predictors improve significantly upon commonly used methods for forecasting realized volatility, and that the estimated model confidence sets include the newly proposed fractional lag predictor in all occurrences.