Properties of optimal forecasts under asymmetric loss and nonlinearity

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 that standard properties of optimal forecasts can be invalid under asymmetric loss and nonlinear data generating processes and thus may be very misleading as a benchmark for an optimal forecast. We establish instead that a suitable transformation of the forecast error—known as the generalized forecast error—possesses an equivalent set of properties. The paper also provides empirical examples to illustrate the significance in practice of asymmetric loss and nonlinearities and discusses the effect of parameter estimation error on optimal forecasts.