A consistent conditional moment test of functional form.

*(English)*Zbl 0737.62058It is shown that any conditional moment (CM) test of functional form of nonlinear regression models can be converted into a chi-square test that is consistent against all deviations from the null hypothesis (i.e. the model represents the conditional expectation of the dependent variable relative to the vector of regressors). The consistency of this test does not rely on randomization.

In section 2, the consistent CM test is presented for a parametric family of probability distribution functions. In case of a finite set of moment conditions the test cannot be consistent against all possible alternatives. The use of an infinite set provides a solution of the consistency. In sections 3 and 4, on the base of two lemmas and five theorems, the conversion of a CM test into a consistent CM test is derived. A quick-and-easy procedure is given in theorem 5. In section 5 it is shown by a limited Monte Carlo simulation (500 replications) how the test performs in finite samples in the form of theorem 5. An interpretation of the results follows. Section 6 is concerned with the question what kind of information about the true model is provided by the test if the null hypothesis is rejected. The appendix contains the formal statements of the main assumptions and the proofs of the lemmas.

In section 2, the consistent CM test is presented for a parametric family of probability distribution functions. In case of a finite set of moment conditions the test cannot be consistent against all possible alternatives. The use of an infinite set provides a solution of the consistency. In sections 3 and 4, on the base of two lemmas and five theorems, the conversion of a CM test into a consistent CM test is derived. A quick-and-easy procedure is given in theorem 5. In section 5 it is shown by a limited Monte Carlo simulation (500 replications) how the test performs in finite samples in the form of theorem 5. An interpretation of the results follows. Section 6 is concerned with the question what kind of information about the true model is provided by the test if the null hypothesis is rejected. The appendix contains the formal statements of the main assumptions and the proofs of the lemmas.

Reviewer: R.Fahrion (Heidelberg)