A Consistent Test for Nonlinear Out of Sample Predictive Accuracy

In this paper, we draw on both the consistent specification testing and the predictive ability testing literatures and propose an integrated conditional moment type predictive accuracy test that is similar in spirit to that developed in Bierens (1982,1990) and Bierens and Ploberger (1997). The test is consistent against generic nonlinear alternatives, and is designed for comparing nested models. One important feature of our approach is that the same loss function is used for in-sample estimation and out-of-sample prediction. In this way, we rule out the possibility that the null model can outperform the nesting generic alternative model. It turns out that the limiting distribution of the ICM type test statistic that we propose is a functional of a Gaussian process with a covariance kernel that reflects both the time series structure of the data as well as the contribution of parameter estimation error. As a consequence, critical values are data dependent and cannot be directly tabulated. One approach in this case is to obtain critical value upper bounds using the approach of Bierens and Ploberger (1997). Here, we establish the validity of a conditional p-value method for constructing critical values. The method is similar in spirit to that proposed by Hansen (1996) and Inoue (2001), although we additionally account for parameter estimation error. In a series of Monte Carlo experiments, the finite sample properties of three variants of the predictive accuracy test are examined. Ou