Differentially private methods for managing model uncertainty in linear regression models

In this article, we develop differentially private tools for handling model uncertainty in linear regression models. We introduce hypothesis tests for nested linear models and methods for model averaging and selection. We consider Bayesian approaches based on mixtures of $g$-priors as well as non-Bayesian approaches based on information criteria. The procedures are straightforward to implement with existing software for non-private data and are asymptotically consistent under certain regularity conditions. We address practical issues such as calibrating the tests so that they have adequate type I error rates or quantifying the uncertainty introduced by the privacy mechanisms. Additionally, we provide specific guidelines to maximize the statistical utility of the methods in finite samples.