# Tests for validity of the semiparametric heteroskedastic transformation model

## TL;DR

This paper develops a goodness-of-fit test for the validity of a semiparametric heteroskedastic transformation model, using characteristic functions and residual-based methods, with theoretical and empirical validation.

## Contribution

It introduces a new test for the model's validity under a parametric transformation, addressing the challenge of unobserved errors with residual-based estimation and resampling techniques.

## Key findings

- Test accurately detects model validity under various conditions
- Resampling procedure effectively approximates critical values
- Monte Carlo experiments demonstrate good finite-sample performance

## Abstract

There exist a number of tests for assessing the nonparametric heteroscedastic location-scale assumption. Here we consider a goodness-of-fit test for the more general hypothesis of the validity of this model under a parametric functional transformation on the response variable. Specifically we consider testing for independence between the regressors and the errors in a model where the transformed response is just a location/scale shift of the error. Our criteria use the familiar factorization property of the joint characteristic function of the covariates under independence. The difficulty is that the errors are unobserved and hence one needs to employ properly estimated residuals in their place. We study the limit distribution of the test statistics under the null hypothesis as well as under alternatives, and also suggest a resampling procedure in order to approximate the critical values of the tests. This resampling is subsequently employed in a series of Monte Carlo experiments that illustrate the finite-sample properties of the new test. We also investigate the performance of related test statistics for normality and symmetry of errors, and apply our methods on real data sets.