Forecast combinations have frequently been found in empirical studies to produce better forecasts on average than methods based on the ex-ante best individual forecasting model. Moreover, simple combinations that ignore correlations between forecast errors often dominate more refined combination schemes aimed at estimating the theoretically optimal combination weights. In this chapter we analyze theoretically the factors that determine the advantages from combining forecasts (for example, the degree of correlation between forecast errors and the relative size of the individual models’ forecast error variances). Although the reasons for the success of simple combination schemes are poorly understood, we discuss several possibilities related to model misspecification, instability (non-stationarities) and estimation error in situations where thenumbersofmodelsislargerelativetothe available sample size. We discuss the role of combinations under asymmetric loss and consider combinations of point, interval and probability forecasts. Key words: Forecast combinations; pooling and trimming; shrinkage methods; model misspecification, diversification gains

In this paper we describe and analyze a sequential procedure for aggregating linear combinations of a finite family of regression estimates, with particular attention to linear combinations having coefficients in the generalized simplex. The procedure is based on exponential weighting, and has a computationally tractable approximation. Analysis of the procedure is based in part on techniques from the sequential prediction of non-random sequences. Here these techniques are applied in a stochastic setting to obtain cumulative loss bounds for the aggregation procedure. From the cumulative loss bounds we derive an oracle inequality for the aggregate estimator for an unbounded response having a suitable moment generating function. The inequality shows that the risk of the aggregate estimator is less than the risk of the best candidate linear combination in the generalized simplex, plus a complexity term that depends on the size of the coefficient set. The inequality readily yields convergence rates for aggregation over the unit simplex that are within logarithmic factors of known minimax bounds. Some preliminary results on model selection are also presented.

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Statistical models (e.g., ARIMA models) have been commonly used in time series data analysis and forecasting. Typically one model is selected based on a selection criterion (e.g., AIC), hypothesis testing, and/or graphical inspections. The selected model is then used to forecast future values. However, model selection is often unstable and may cause an unnecessarily high variability in the final estimation/prediction. In this work, we propose the use of an algorithm AFTER to convexly combine the models for a better performance of prediction. The weights are sequentially updated after each additional observation. Simulations and real data examples are used to compare performance of our approach with model selection methods. The results show advantage of combining by AFTER over selection in term of forecasting accuracy at several settings.

This paper studies a procedure to combine individual forecasts that achieve theoretical optimal performance. The results apply to a wide variety of loss functions and no conditions are imposed on the data sequences and the individual forecasts apart from a tail condition. The theoretical results show that the bounds are also valid in the case of time varying combination weights, under specific conditions on the nature of time variation. Some experimental evidence to confirm the results is provided.

We would like to thank Yongmiao Hong and Tae-Hwy Lee for sharing their computer program, Qi Li and particularly two anonymous referees for numerous helpful comments and suggestions.

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journal homepage: www.elsevier.com/locate/caie A collaborative demand forecasting process with event-based fuzzy judgements

The distribution of model averaging estimators and an impossibility result regarding its estimation