A general approach to conditional moment specification testing with projections

This article develops a general approach for model specification analysis within the conditional moment specification testing framework. The new methodology removes the non-negligible estimation effect of test statistic via a projection-based transformation, exploiting the nature of conditional moment specification testing. That is, the conditional moment restrictions, which are implicitly defined in conditional moment testing framework, not only imply the unconditional moment restrictions we are testing, but also many other unconditional moment restrictions. This approach is robust to departures from the distributional assumptions that are not being tested; moreover, only a preliminary \begin{equation}{\sqrt {T}}\end{equation} T-consistent estimator is needed, and the transformation is asymptotically distribution free. Furthermore, the transformed statistic reaches asymptotic efficiency in the sense of generalized method of moments (GMM) estimation. In some specific alternatives, we establish the optimal tests. We apply the methodology to test the adequacy and nonlinearity of the generalized autoregressive conditional heteroskedasticity (GARCH) models. Finally, an application to the S&P 500 daily data highlights the merits of our approach.