This paper examines the asymptotic and finite-sample properties of out-of-sample tests for equal accuracy and encompassing as applied to nested models. With nested models, these tests can be viewed as Granger causality tests. Applied to nested models, however, the standard asymptotic critical values for many tests of equal accuracy and encompassing are invalid. Statistics such as those proposed by Diebold and Mariano (1995) and Harvey, et. al. (1998) fail to converge to the standard normal distribution when the models are nested rather than nonnested. Building on McCracken's (1999) results for equal accuracy tests, this paper derives the asymptotic distributions for a set of standard encompassing tests and one new encompassing test. Numerical simulations are used to generate the appropriate asymptotic critical values. Monte Carlo simulations are then used to evaluate the size and power of a battery of equal forecast accuracy and encompassing tests, as well as standard F-tests of causality. In these experiments, forecasts from an estimated VAR model are compared to those from a null estimated AR model. The simulation results indicate that McCracken's out-of-sample F-type test of equal accuracy and the encompassing test proposed in this paper can be more powerful than standard F-tests of causality. The Monte Carlo simulations also show that using invalid asymptotic critical values can produce misleading inferences.

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This paper describes a large, monthly frequency, macroeconomic database with the goal of establishing a convenient starting point for empirical analysis that requires "big data." The dataset mimics the coverage of those already used in the literature but has three appealing features. First, it is designed to be updated monthly using the FRED database. Second, it will be publicly accessible, facilitating comparison of related research and replication of empirical work. Third, it will relieve researchers from having to manage data changes and revisions. We show that factors extracted from our dataset share the same predictive content as those based on various vintages of the so-called Stock-Watson dataset. In addition, we suggest that diffusion indexes constructed as the partial sum of the factor estimates can potentially be useful for the study of business cycle chronology.JEL Classification: C30, C33, G11, G12.

This paper examines the asymptotic and finite-sample properties of out-of-sample tests for equal accuracy and encompassing as applied to nested models. With nested models, these tests can be viewed as Granger causality tests. Applied to nested models, however, the standard asymptotic critical values for many tests of equal accuracy and encompassing are invalid. Statistics such as those proposed by Diebold and Mariano (1995) and Harvey, et. al. (1998) fail to converge to the standard normal distribution when the models are nested rather than nonnested. Building on McCracken's (1999) results for equal accuracy tests, this paper derives the asymptotic distributions for a set of standard encompassing tests and one new encompassing test. Numerical simulations are used to generate the appropriate asymptotic critical values. Monte Carlo simulations are then used to evaluate the size and power of a battery of equal forecast accuracy and encompassing tests, as well as standard F-tests of causality. In these experiments, forecasts from an estimated VAR model are compared to those from a null estimated AR model. The simulation results indicate that McCracken's out-of-sample F-type test of equal accuracy and the encompassing test proposed in this paper can be more powerful than standard F-tests of causality. The Monte Carlo simulations also show that using invalid asymptotic critical values can produce misleading inferences.