| This paper presents analytical and numerical evidence concerning out of sample tests of causality. The relevant environment is one in which the comparative predictive ability of two nested parametric regression models is of interest. This environment occurs naturally when one is interested in testing for causality (Ashley, Granger and Schmalensee, 1980), market efficiency (Pesaran and Timmermann, 1995) and long horizon predictability (Mark, 1995). Existing tests for comparative predictive ability are only applicable when the two models are non-nested and hence the present paper extends the literature on the use of out of sample methods. Results are provided for three statistics: a regression-based statistic suggested by Granger and Newbold (1977), a t-type statistic comparable to those suggested by Diebold and Mariano (1995) and West (1996), and an F-type statistic akin to Theil's U (1966). Since the limiting distributions under the null are nonstandard, tables of asymptotically valid critical values are provided. The null limiting distributions indicate that overfit models should predict poorly and that the Principle of Parsimony should be applied judiciously. Power calculations under a local alternative provide some guidance on the choice of test statistic and the percentage of the sample withheld for predictive evaluation. |