Quantiles as Optimal Point Predictors

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The loss function plays a central role in the theory and practice of forecasting. If the loss is quadratic, the mean of the predictive distribution is the unique optimal point predictor. If the loss is linear, any median is an optimal point forecast. The title of the paper refers to the simple, possibly surprising fact that quantiles arise as optimal point predictors under a general class of economically relevant loss functions, to which we refer as generalized piecewise linear (GPL). The level of the quantile depends on a generic asymmetry parameter that reflects the possibly distinct costs of underprediction and overprediction. A loss function for which quantiles are optimal point predictors is necessarily GPL, similarly to the classical fact that a loss function for which the mean is optimal is necessarily of the Bregman type. We prove general versions of these results that apply on any decision-observation domain and rest on weak assumptions. The empirical relevance of the choices in the transition from the predictive distribution to the point forecast is illustrated on the Bank of England’s density forecasts of United Kingdom inflation rates, and probabilistic predictions of wind energy resources in the Pacific Northwest.

Key words and phrases: asymmetric loss function; Bayes predictor; density forecast; mean; median; mode; optimal point predictor; quantile; statistical decision theory

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