Generalized (Cross) Spectral Tests for Optimal Forecasts and Conditional Predictive Ability Under Generalized Loss Functions

Under the squared error loss, the optimal forecast is the conditional mean, and the one-step forecast error is a martingale difference (MD). The one-step forecast error forms the conditional moment condition obtained from the loss derivative with respect to the forecast. Similarly, under a generalized loss function, the derivative of the loss with respect to the forecast is an MD. Given a loss function, the forecast optimality may be checked by testing for the MD property of the loss derivative. In this paper, we show that the generalized (cross) spectral test of Hong (1999) may be used to evaluate the forecast optimality and that its asymptotic distribution is not affected by the parameter estimation uncertainty, provided that the training sample grows suitably faster than the validation sample and that the parameters are estimated at root-n rate. We also use the generalized (cross) spectral test to compare the conditional predictive ability of competing forecasting models by testing the MD property of their loss differential