Nonparametric Bootstrap Tests for Independence of Generalized Errors

In this paper, we develop a general method of testing for independence when unobservable generalized errors are involved. Our method can be applied to testing for serial independence of generalized errors, and testing for independence between the generalized errors and observ- able covariates. The former can serve as a uni?ed approach to testing adequacy of time series models, as model adequacy often implies that the generalized errors obtained after a suitable transformation are independent and identically distributed. The latter is a key identi?cation assumption in many nonlinear economic models. Our tests are based on a classical sample dependence measure, the Hoe¤ding-Blum-Kiefer-Rosenblat-type empirical process applied to generalized residuals. We establish a uniform expansion of the process, thereby deriving an ex- plicit expression for the parameter estimation e¤ect, which causes our tests not to be nuisance parameter-free. To circumvent this problem, we propose a multiplier-type bootstrap to approx- imate the limit distribution. Our bootstrap procedure is computationally very simple as it does not require a reestimation of the parameters in each bootstrap replication. In a simulation study, we apply our method to test the adequacy of ARMA-GARCH and Hansen (1994) skewed t models, and document a good ?nite sample performance of our test. Finally, an empirical application to some daily exchange rate data highlights the merits of our approach.