Equivalence between out-of-sample forecast comparisons and Wald statistics

Abstract

We demonstrate the asymptotic equivalence between commonly used test statistics for out-of-sample forecasting performance and conventional Wald statistics. This equivalence greatly simplifies the computational burden of calculating recursive out-of-sample test statistics and their critical values. For the case with nested models, we show that the limit distribution, which has previously been expressed through stochastic integrals, has a simple representation in terms of ?2-distributed random variables and we derive its density. We also generalize the limit theory to cover local alternatives and characterize the power properties of the test.