Predictive ability with cointegrated variables

In this paper we outline conditions under which the Diebold and Mariano (DM) (J. Bus. Econom. Statist. 13 (1995) 253) test for predictive ability can be extended to the case of two forecasting models, each of which may include cointegrating relations, when allowing for parameter estimation error. We show that in the cases where either the loss function is quadratic or the length of the prediction period, P, grows at a slower rate than the length of the regression period, R, the standard DM test can be used. On the other hand, in the case of a generic loss function, if P/R → π as T →; ∞, 0 < π < ∞, then the asymptotic normality result of West (Econometrica 64 (1996) 1067) no longer holds. We also extend the "data snooping" technique of White (Econometrica 68 (2000) 1097) for comparing the predictive ability of multiple forecasting models to the case of cointegrated variables. In a series of Monte Carlo experiments, we examine the impact of both short run and cointegrating vector parameter estimation error on DM, data snooping, and related tests. Our results suggest that size is reasonable for R and P greater than 50, and power improves with P, as expected. Furthermore, the additional cost, in terms of size distortion, due to the estimation of the cointegrating relations is not substantial. We illustrate the use of the tests in a nonnested cointegration framework by forming prediction models for industrial production which include two interest rate variables, prices, and either M1, M2, or M3.