Weighted Maximum Likelihood for Risk Prediction

Most time series models used in econometrics and empirical finance are estimated withmaximum likelihood methods, in particular when interest centers on density and Value{at{Risk (VaR) prediction. The standard maximum likelihood principle implicitly placesequal weight on each of the observations in the sample, but depending on the extent towhich the model and the true data generating process deviate this can be improved upon.For example, in the context of modeling financial time series, weighting schemes whichplace relatively more weight on observations in the recent past result in improvement ofout{of{sample density forecasts, compared to the default of equal weights. Also, if insteadof accurate forecasting of the entire density, interest is restricted to just downside risk,placing more weight on the negative observations in the sample improves results further.In this paper, a third and quite general strategy of shifting more weight towards certainobservations of the sample is proposed. Weights are derived from external variables thatconvey additional information about the true DGP, like trading volume, news arrivals oreven investor sentiment. As such, those observations are down weighted that bear a highprobability of being destructive outliers with no benefit of using them when fitting themodel. Considerable improvements in forecast accuracy for a variety of data sets anddifferent time series models can be realized.[...]