Optimal forecasting accuracy using Lp-norm combination

A well-known result in statistics is that a linear combination of two-point forecasts has a smaller Mean Square Error (MSE) than the two competing forecasts themselves (Bates and Granger in J Oper Res Soc 20(4):451–468, 1969). 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 kinds of combination methods are various, ranging from the simple average (SA) to more robust methods such as the one based on median or Trimmed Average (TA) or Least Absolute Deviations or optimization techniques (Stock and Watson in J Forecast 23(6):405–430, 2004). Standard regression-based combination approaches may fail to get a realistic result if the forecasts show high collinearity in several situations or the data distribution is not Gaussian. Therefore, we propose a forecast combination method based on Lp-norm estimators. These estimators are based on the Generalized Error Distribution, which is a generalization of the Gaussian distribution, and they can be used to solve the cases of multicollinearity and non-Gaussianity. In order to demonstrate the potential of Lp-norms, we conducted a simulated and an empirical study, comparing its performance with other standard-regression combination approaches. We carried out the simulation study with different values of the autoregressive parameter, by alternating heteroskedasticity and homoskedasticity. On the other hand, the real data application is based on the daily Bitfinex historical series of bitcoins (2014–2020) and the 25 historical series relating to companies included in the Dow Jonson, were subsequently considered. We showed that, by combining different GARCH and the ARIMA models, assuming both Gaussian and non-Gaussian distributions, the Lp-norm scheme improves the forecasting accuracy with respect to other regression-based combination procedures.