Accelerating GARCH and Score-Driven Models: Optimality, Estimation and Forecasting

Abstract

We first consider an extension of the generalized autoregressive conditional heteroskedasticity (GARCH) model that allows for a more flexible weighting of financial squared-returns for the filtering of volatility. The parameter for the squared-return in the GARCH model is time-varying with an updating function similar to GARCH but with the squared-return replaced by the product of the volatility innovation and its lagged value. This local estimate of the first order autocorrelation of volatility innovations acts as an indicator of the importance of the squared-return for volatility updating. When recent volatility innovations have the same sign (positive autocorrelation), the current volatility estimate needs to adjust more quickly than in a period where recent volatility innovations have mixed signs (negative autocorrelation). The empirical relevance of the accelerated GARCH updating is illustrated by forecasting daily volatility in return series of all individual stocks present in the Standard & Poor’s 500 index. Major improvements are reported for those stock return series that exhibit high kurtosis. The local adjustment in weighting new observational information is generalised to score-driven time-varying parameter models of which GARCH is a special case. It is within this general framework that we provide the theoretical foundations of accelerated updating. We show that acceleration in updating is more optimal in terms of reducing Kullback-Leibler divergence and in comparison to fixed updating. The robustness of our proposed extension is highlighted in a simulation study within a misspecified modelling framework. The score-driven acceleration is also empirically illustrated with the forecasting of US inflation using a model with time-varying mean and variance; we report significant improvements in the forecasting accuracy at a yearly horizon.