Properties of Optimal Forecasts

Abstract

Evaluation of forecast optimality in economics and finance has almost exclusively been conducted under the assumption of mean squared error loss. Under this loss function optimal forecasts should be unbiased and forecast errors serially uncorrelated at the single-period horizon with increasing variance as the forecast horizon grows. Using analytical results we show in this Paper that all the standard properties of optimal forecasts can be invalid under asymmetric loss and non-linear data-generating processes and thus may be very misleading as a benchmark for an optimal forecast. Our theoretical results suggest that many of the conclusions in the empirical literature concerning sub-optimality of forecasts could be premature. We extend the properties that an optimal forecast should have to a more general setting than previously considered in the literature. We also present new results on forecast error properties that may be tested when the forecaster's loss function is unknown but restrictions can be imposed on the data-generating process, and introduce a change of measure, following which the optimum forecast errors for general loss functions have the same properties as optimum errors under MSE loss.

Keywords: Forecast evaluation, loss function, rationality, efficient markets

JEL Classification: C22, C52, C53