A Note on Consistent Conditional Moment Tests

In this paper we propose a new consistent conditional moment test, which synergizes Bierens’ approach with the consistent test of overidentifying restrictions. It relies on a transformation-based empirical process combining both approaches. This new empirical process enjoys some advantages. Firstly it is not affected by the uncertainty from the parameter estimation. Moreover this estimation-effect-free property requires much less restrictive rate condition than in the consistent test of overidentifying restrictions alone. Furthermore the integrated conditional moment (ICM) test based on the new empirical process have power against Pitman local alternatives. We prove, under some regularity conditions, the admissibility of the ICM test based on this transformation-based empirical process in the case that there exists heteroskedasticity of unknown form, extending the result in Bierens and Ploberger (1997). The new consistent test also allows us to propose a much simpler bootstrap procedure than the standard ones. A version of Bierens (1990) test based on the new empirical process is also discussed, and its asymptotic properties are analyzed. Monte Carlo simulations show that Bierens (1990) test based on the new empirical process is more powerful for a large number of alternatives when heteroskedasticity of unknown form is presented.