Bootstrapping Quasi Likelihood Ratio Tests under Misspecification

We consider quasi likelihood ratio (QLR) tests for restrictions on parameters under potential model misspecification. For convex M-estimation, including quantile regression, we propose a general and simple nonparametric bootstrap procedure that yields asymptotically valid critical values. The method modifies the bootstrap objective function to mimic what happens under the null hypothesis. When testing for an univariate restriction, we show how the test statistic can be made asymptotically pivotal. Our bootstrap can then provide asymptotic refinements as illustrated for a linear regression model. A Monte-Carlo study and an empirical application illustrate that double bootstrap of the QLR test controls level well and is powerful.