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Testing Parametric Conditional Distributions Using the Nonpa

Testing Parametric Conditional Distributions Using the...

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Testing Parametric Conditional Distributions Using the Nonparametric Smoothing Method Xu Zheng * School of Economics, Shanghai Jiao Tong University This Version: December 2007 ABSTRACT: This paper proposes a new goodness of fit test for parametric con- ditional probability distributions using the nonparametric smoothing methodology. An asymptotic normal distribution is established for the test statistic under the null hypothesis of correct specification of the parametric distribution. The test is shown to have power against local alternatives converging to the null at certain rates. The test can be applied to testing for possible misspecifications in a wide variety of parametric models, such as the classical linear regression model with normal error and the censored regression model with normal error. Monte Carlo simulations show that the test has good power against some common alternatives. KEYWORDS: Goodness-of-Fit Test, Conditional Distribution, Nonparametric Smoothing Method, Local Alternatives, Monte Carlo Simulations. JEL subject classification: C14, C12. * Correspondence address: School of Economics, Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200052, China. Phone: 86-21-52301135. Fax: 86-21-62932982. Email: [email protected]
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1. Introduction A fundamental assumption underlying the maximum likelihood based estima- tion and testing procedures is that the probability distributions of random vari- ables belong to parametric families of known distributions. Misspecifications of the parametric distributions may lead to inconsistent estimator and invalid inference (White, 1982). Specification tests of parametric probability distributions date back to the Pearson’s chi-square test, the Kolmogorov-Smirnov test, and the Cram´ er-von Mises test (see Darlin (1957) for these tests). All three tests are based on measuring the distance between the underlying model measured by the empirical distribution function and the fitted parametric model. The Kolmogorov-Smirnov test and the Cram´ er-von Mises test share similar asymptotic properties in terms of the null distribution, local power, and consistency. Both tests have very complicated null distributions (Gregory, 1977). The Pearson’s chi-square test has been extended to econometric models by Andrews (1988a, 1988b) and to various statistical models. Andrews (1988b) demonstrated many useful ways to form the groups or cells to apply to different contexts. However, the chi-square test relies on a more or less arbitrary grouping of the data and is not consistent against all possible misspecifi- cations (Darlin, 1957). Andrews (1997) and Bai (2003) extended the Kolmogorov- Smirnov test to conditional distributions. Tests of parametric distribution functions based on nonparametric smooth- ing techniques have been investigated by Bickel and Rosenblatt (1973), Rosenblatt (1975), Eubank, LaRiccia and Rosenstein (1987), Ghosh and Huang (1991), Robin- son (1991), Eubank and LaRiccia (1992), Zheng (2000, 2008), and others (see Eu- bank, Hart, and LaRiccia (1993) and Hart (1997) for comprehensive surveys on the related work). The work in the literature has focused on testing for unconditional
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