5

Recent Developments in Density

Forecasting

Stephen G. Hall and James Mitchell

Abstract
With the growing recognition that point forecasts, the traditional focus, are better seen as the
central points of ranges of uncertainty, in recent years increased emphasis has been given to den-
sity forecasts. This chapter reviews these recent developments, with a focus on the production
and use of density forecasts in macroeconomics. Particular attention is paid to the evaluation and
combination of density forecasts.

5.1 Introduction 200
5.2 The importance of density forecasts 201
5.2.1 Forecasting under general loss functions 202
5.3 The production of density forecasts 203
5.3.1 Sources of uncertainty 204
5.3.2 Model-based densities 205
5.3.3 Subjective density forecasts 208
5.3.4 Combining model-based and subjective density forecasts 209
5.4 The evaluation of density forecasts 211
5.4.1 Interval forecasts 213
5.4.2 Rolling density forecasts 214
5.4.2.1 Goodness-of-fit tests: in theory 215
5.4.2.2 Goodness-of-fit tests: in practice 215
5.4.2.3 Scoring rules 220
5.4.2.4 Comparing competing density forecasts 220
5.5 The combination of density forecasts 222
5.5.1 Combination methods 223
5.5.2 The linear opinion pool 224
5.5.3 Equal weights 225
5.5.4 KLIC minimizing weights 226
5.5.4.1 Bayesian Model Averaging (BMA) 227
5.5.4.2 Out-of-sample measures of fit 229
5.5.4.3 Empirical applications combining and evaluating density
forecasts 230
5.6 Conclusion 232

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