Abstract
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.
Highlights
Model instabilities are deeply rooted in real-life forecasting challenges because models are uncertain and mutable
In this paper 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 through some procedures proposed in literature
If the errors follow a Gaussian distribution, the forecast obtained with a linear combination of forecasts using Lp-norm parameters (L2 norm in this case) gives satisfying results, the mean square error is slightly higher than the Mean Square Error (MSE) using the Ordinary Least Squares (OLS) parameters but lower than the other methods
Summary
Model instabilities are deeply rooted in real-life forecasting challenges because models are uncertain and mutable. We propose an alternative and more general approach for combining models and/or forecasts, based on Lp-norm estimators [40]. In this paper we propose to study the effectiveness of Lp-norm estimators in combining volatility forecast. The main motive of this paper is to measure the performance of GARCH techniques for forecasting combination by using different distribution model. The different GARCH distribution models present in the paper are the t-student, the Gaussian, the GED jointly considered with some ARMA models. 3, we introduce the main recent methods for better achieve the forecast combination and we show the advantages and the disadvantages of the different combination approaches. 4, we bring up for discussion the Generalized Error Distributions (G.E.D.) explaining in detail our approach based on Lp-norm estimators by proposing a new algorithm to combine the forecasts. In the appendix A and B we show the commented R studio scripts that we built and used to estimate the forecast and the combination in the simulated and the empirical study, whilst in appendix C some graphical findings of the simulation study are reported
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