Homoscedasticity, or homogeneity of variance, is a central assumption to most standard statistical models, including linear regression. Violations of homoscedasticity occur when error terms vary unequally across observations (heteroscedasticity). Unfortunately, in practice, datasets often display such heterogeneous patterns, which can compromise the reliability of results and lead to biased conclusions. For these reasons, verifying the structure of the data through homoscedasticity tests is essential. While frequentist tests like Breusch-Pagan and White are widely used, few adaptations of these tests exist within a Bayesian framework. 

Bayesian analysis has been gaining increasing popularity as they address many of the shortcomings of the frequentist methods. Specifically, the Bayes Factor is a powerful measure of evidence, which can be explicitly quantified in favour of both null and alternative hypotheses.

We propose a Bayes Factor test for homoscedasticity in linear regression. This test directly assesses whether the data can be better explained by a homoscedastic or heteroscedastic model. We provide a theoretical rationale for constructing the model, including its application to both linear and non-linear heteroscedasticity and considerations for prior selection. Additionally, we implement the test practically in R. To evaluate its effectiveness, we conduct simulation studies to examine the behaviour of the Bayes Factor under various conditions. Finally, we apply the test to real-life examples where the Breusch-Pagan and White test detected the presence (or not) of homoscedasticity, to validate its accuracy.