A regression test of semiparametric index model specifications

Abstract

This paper presents a straightforward regression test of parametric and semiparametric index models against more general semiparametric and nonparametric alternative models. The test is based on the regression coefficient of the restricted model residuals on the fitted values of the more general model. A goodness-of-fit interpretation is shown for the regression coefficient, and the test is based on the squared "t-statistic" of the coefficient estimate, where the variance of the coefficient has been adjusted for the use of nonparametric estimators. An asymptotic theory is given for the situation where kernel estimators are used to estimate unknown regression functions, and the variance adjustment terms are given for this case. The methods are applied to the empirical problem of characterizing environmental effects on housing prices in the Boston Housing data, where a partial index model is found to be preferable to a standard log-linear equation, yet not rejected against general nonparametric regression. Various issues in the asymptotic theory and other features of the test are discussed.