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Constistent model specification tests: Omitted variables and semiparametric functional forms. (English) Zbl 0854.62038
Summary: We develop several consistent tests in the context of a nonparametric regression model. These include tests for the significance of a subset of regressors and tests for the specification of the semiparametric functional form of the regression function, where the latter covers tests for a partially linear and a single index specification against a general nonparametric alternative.
One common feature to the construction of all these tests is the use of the central limit theorem for degenerate $$U$$-statistics of order higher than two. As a result, they share the same advantages over most of the corresponding existing tests in the literature: (a) They do not depend on any ad hoc modifications such as sample splitting, random weighting, etc. (b) Under the alternative hypotheses, the test statistics in this paper diverge to positive infinity at a faster rate than those based on ad hoc modifications.

##### MSC:
 62G07 Density estimation 62G10 Nonparametric hypothesis testing 62P20 Applications of statistics to economics 62G20 Asymptotic properties of nonparametric inference
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