An Improved Generalized Spectral Test For Conditional Mean Models In Time Series With Conditional Heteroskedasticity Of Unknown Form

Dynamic economic theories usually have implications on and only on the conditional mean dynamics of economic processes. Using a generalized spectral derivative approach, Hong and Lee (2005, Review of Economic Studies 72, 499–541) recently proposed a new class of omnibus nonparametric specification tests for linear and nonlinear time series conditional mean models, where the dimension of the conditioning information set may be infinite. The tests can detect a wide range of model misspecifications in mean while being robust to conditional heteroskedasticity and time-varying higher order moments of unknown form. They enjoy an asymptotic “nuisance parameter–free” property in the sense that parameter estimation uncertainty has no impact on the asymptotic N(0,1) distribution of the test statistics. As a result, only the estimated residuals from the null parametric model are needed to implement the tests, and no specific estimation is required.Although parameter estimation uncertainty has no impact on the asymptotic distribution of the tests, it may have significant impact on the finite-sample distribution, and such an impact may become more substantial as the number of estimated parameters increases. In this paper, we adopt the Wooldridge (1990, Econometric Theory 6, 17–43) device for parametric m-tests to the Hong and Lee (2005) nonparametric tests to reduce the impact of parameter estimation uncertainty. Asymptotic size and power properties of the modified tests are investigated, and simulation studies show that the modified tests generally have better sizes in finite samples and are robust to parameter estimation uncertainty. In the meantime, the size improvement does not cause loss of power against a wide range of alternatives when using the empirical critical values for the tests. These results suggest that the modified generalized spectral derivative tests can be a useful tool in time series conditional mean modeling.We thank Bruce Hansen (the co-editor) and three referees for insightful and helpful comments and suggestions. We also are grateful for financial support from the National Science Foundation Grant SES–0111769 and the Cheung Kong Visiting Scholarship by the Chinese Ministry of Education and Xiamen University, China.