Weighted maximum likelihood for risk prediction

Abstract

Most time series models used in econometrics and empirical finance are estimated with maximum likelihood methods, in particular when interest centers on density and Value-at-Risk (VaR) prediction. The standard maximum likelihood principle implicitly places equal weight on each of the observations in the sample, but depending on the extent to which 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 which place relatively more weight on observations in the recent past result in improvement of out-of-sample density forecasts, compared to the default of equal weights. Also, if instead of 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 certain observations of the sample is proposed. Weights are derived from external variables that convey additional information about the true DGP, like trading volume, news arrivals or even investor sentiment. As such, those observations are down weighted that bear a high probability of being destructive outliers with no bene¯t of using them when fitting the model. Considerable improvements in forecast accuracy for a variety of data sets and different time series models can be realized.