A goodness-of-fit test for parametric and semi-parametric models in multiresponse regression

TL;DR

This paper introduces an empirical likelihood test for assessing the goodness-of-fit of various parametric and semi-parametric multiresponse regression models, capable of handling multivariate responses and covariates, with good power and bootstrap calibration.

Contribution

It develops a versatile goodness-of-fit test applicable to a broad class of models, including those with shape constraints and infinite-dimensional nuisance functions.

Findings

Test statistic is asymptotically normal under mild conditions.

The test can be calibrated using wild bootstrap methods.

It demonstrates strong power against model departures.

Abstract

We propose an empirical likelihood test that is able to test the goodness of fit of a class of parametric and semi-parametric multiresponse regression models. The class includes as special cases fully parametric models; semi-parametric models, like the multiindex and the partially linear models; and models with shape constraints. Another feature of the test is that it allows both the response variable and the covariate be multivariate, which means that multiple regression curves can be tested simultaneously. The test also allows the presence of infinite-dimensional nuisance functions in the model to be tested. It is shown that the empirical likelihood test statistic is asymptotically normally distributed under certain mild conditions and permits a wild bootstrap calibration. Despite the large size of the class of models to be considered, the empirical likelihood test enjoys good power properties against departures from a hypothesized model within the class.