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Asymptotic theory of integrated conditional moment tests. (English) Zbl 0927.62085
Summary: We derive the asymptotic distribution of the test statistic of a generalized version of the integrated conditional moment (ICM) test of H. J. Bierens [J. Econ. 20, 105-134 (1982; Zbl 0549.62076); ibid. 26, 323-353 (1984; Zbl 0563.62064)], under a class of \(\sqrt n\)-local alternatives, where \(n\) is the sample size. The generalized version involved includes neural network tests as a special case, and allows for testing misspecification of dynamic models.
It appears that the ICM test has nontrivial local power. Moreover, for a class of “large” local alternatives the consistent ICM test is more powerful than the parametric \(t\) test in a neighborhood of the parametric alternative involved. Furthermore, under the assumption of normal errors the ICM test is asymptotically admissible, in the sense that there does not exist a test that is uniformly more powerful. The asymptotic size of the test is case-dependent: the critical values of the test depend on the data-generating process. We derive case-independent upper bounds of the critical values.

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62E20 Asymptotic distribution theory in statistics
62P20 Applications of statistics to economics
62J02 General nonlinear regression
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