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.

Abstract. A Hidden Markov Model (HMM) is used to classify an out of sample observation vector into either of two regimes. This leads to a procedure for making probability forecasts for changes of regimes in a time series, i.e. for turning points. Instead ofmaximizing a likelihood, the model is estimated with respect to known past regimes. This makes it possible to perform feature extraction and estimation for different forecasting horizons. The inference aspect is emphasized by including a penalty for a wrong decision in the cost function. The method is tested by forecasting turning points in the Swedish and US economies, using leading data. Clear and early turning point signals are obtained, contrasting favourable with earlier HMM studies. Some theoretical arguments for this are given. Keywords. Business Cycle, Feature Extraction, Hidden Markov Switching-Regime

Computing prediction intervals (P.I.s) is an important part of the forecasting process intended to indicate the likely uncertainty in point forecasts. The commonest method of calculating P.I. s is to use theoretical formulae conditional on a best-fitting model. If a normality assumption is used, it needs to be checked. Alternative computational procedures that are not so dependent on a fitted model include the use of empirically based and resampling methods. Some so-called approximate formulae should be avoided. P.I.s tend to be too narrow because out-of-sample forecast accuracy is often poorer than would be expected from within-sample fit, particularly for P.I.s calculated conditional on a model fitted to past data. Reasons for this include uncertainty about the model and a changing environment. Ways of overcoming these problems include using a mixture of models with a Bayesian approach and using a forecasting method that is designed to be robust to changes in the underlying model.

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