Jensen's Inequality and the Success of Linear Prediction Pools

Abstract:     
Recent empirical studies have shown that linear density forecast combinations (prediction pools) for macroeconomic variables tend to outperform most or all individual models which enter the combination. The performance measure used by these studies is the log scoring rule; evidence for other scoring rules is rare for (continuous) economic variables. The present paper contributes to the literature by analyzing linear pools under the quadratic and continuous ranked probability scores.

I first review some analytical results which highlight the commonalities and differences of the scoring rules. In particular, Stael von Holstein (1970) has shown that all three scoring rules are concave functions of the forecast density, i.e. the score of the pool generally satisfies a lower bound which is a function of the individual models' scores. I then consider an empirical analysis of monthly US macro data. While linear pools generally perform well, there is evidence that their performance (relative to the individual models) is somewhat better under the log score than under the other two rules. This finding can be explained by the increased dispersion of linear pools, which is beneficial under the log score but not necessarily under the other two rules.

Keywords: Density Forecasting, Forecast Combination, Scoring Rules

JEL Classification: C52

working papers series