A Consistent ICM-based $χ^2$ Specification Test

Abstract: In spite of the omnibus property of Integrated Conditional Moment (ICM) specification tests, they are not commonly used in empirical practice owing to, e.g., the non-pivotality of the test and the high computational cost of available bootstrap schemes especially in large samples. This paper proposes specification and mean independence tests based on a class of ICM metrics termed the generalized martingale difference divergence (GMDD). The proposed tests exhibit consistency, asymptotic $\chi^2$-distribution under the null hypothesis, and computational efficiency. Moreover, they demonstrate robustness to heteroskedasticity of unknown form and can be adapted to enhance power towards specific alternatives. A power comparison with classical bootstrap-based ICM tests using Bahadur slopes is also provided. Monte Carlo simulations are conducted to showcase the proposed tests' excellent size control and competitive power.