A Principal Component-based Goodness-of-Fit Test for Conditional Distributions

This paper introduces a novel goodness-of-fit test technique for parametric conditional distributions. The proposed tests are based on a residual marked empirical process, for which we develop a conditional Principal Component Analysis. The obtained components provide a basis for various types of new tests in addition to the omnibus one. Component tests that based on each component serve as experts in detecting certain directions. Smooth tests that assemble a few components are also of great use in practice. To further improve testing performance, we introduce a component selection approach, aiming to identify the most contributory components. The finite sample performance of the proposed tests is illustrated through Monte Carlo experiments. Applied to violent crime rates and household food expenditure, our tests show that commonly used log-normal and gamma specifications of the conditional distribution are misspecified and, unlike omnibus tests, precisely identify the distributional features driving the rejections. These diagnostic insights support more reliable economic inference on the determinants of crime and on the shape of Engel curves.