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A conditional Kolmogorov test. (English) Zbl 0928.62019
This paper introduces a specification test for parametric models for independent observations. The null hypothesis of interest is that the parametric model is correctly specified. The alternative hypothesis is that the parametric model is incorrectly specified. The parametric model we consider is one that specifies the conditional distribution of a vector \(Y_i\in R^V\) of response variables given a vector \(X_i\in R^K\) of covariates (regressors). The distribution of the covariates is not specified by the parametric model. Many models used in microeconometric and biometric applications are of this type.
The test we consider is a generalization of the Kolmogorov (K) test (sometimes called the Kolmogorov-Smirnov test) of goodness-of-fit. We call the test a conditional Kolmogorov (CK) test, because it is designed for parametric models for the conditional distribution of \(Y_i\) given \(X_i\). The CK test has the following attributes: it is (i) consistent against all alternatives to the null hypothesis \(H_0\), (ii) powerful against \(1/\sqrt{n}\) local alternatives to \(H_0\), and (iii) not dependent on any smoothing parameters.

62F03 Parametric hypothesis testing
62P20 Applications of statistics to economics
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