A bootstrap version of the Hausman test to assess the impact of cluster-level endogeneity beyond the random intercept model.

In the random intercept model for clustered data, the random effect is typically assumed to be independent of predictors. Violation of this assumption due to unmeasured cluster-level confounding (endogeneity) induces bias in the estimates of effects of within-cluster predictors. Treating cluster-specific intercepts as fixed rather than random avoids this bias. The Hausman test contrasts the fixed effect estimator with the traditional random effect estimator in the random intercept model to test for the presence of cluster-level endogeneity and has a known asymptotic -distribution under correct model specification. Unmeasured cluster-level heterogeneity may, however, interact with predictors as well, necessitating random slope models. Relying on either cluster or residual resampling in a bootstrap procedure, we propose two extensions of the Hausman test that can easily be used beyond the random intercept model. We compare the original Hausman test and its robust version to the newly proposed bootstrap tests in terms of empirical type I error rate and power. Under additive unmeasured heterogeneity, all methods perform equally well, whereas the original and robust Hausman tests are too liberal or too conservative under additional slope heterogeneity, both bootstrap Hausman tests maintain appropriate performance. Moreover, both bootstrap tests show robustness against misspecification in the presence of unit-level heteroscedasticity and temporal correlation.