An improved method of combining forecasts based on fourth cumulant

A well known result in statistics and econometrics is that a linear combination of two point forecasts has a smaller Mean Square Error (MSE) than the two competing forecasts themselves. The only case in which no improvements are possible is when one of the single forecasts is already the optimal one in terms of MSE. The kind of combination methods are dierent, ranging from the simple average (SA) to more robust methods as the one based on median or on a Trimmed Average (TA) to other methods based on regression or optimization techniques. Using the regression-based approach, the resulting combined forecast is a linear function of the individual forecasts where the weights are estimated via Ordinary Least Squares (OLS), minimizing the sum of squared errors. A clear advantage of the OLS forecast combination method is that the combined resulting forecast is unbiased even if one of the individual forecasts is biased. However, other alternative methods were developed, implementing the minimization of a dierent loss function, as happen with the least absolute sum of squares. However, these methods may fail to get a realistic result if the forecasts density are havy-tailed as happen in many situation (e.g. nancial time series). Therefore, we propose a forecast combination method based on Lp-norm estimator, where the minimization of residuals is done according to estimated data kurtosis and the selection of more relevant forecast is achieved via a projection pursuit based on fourth cumulant. A simulation study is presented in order to show improvements in forecasting accuracy.