A maximum score test for binary response models

We develop hypothesis tests for binary response models estimated by maximum score. The only methods currently available for developing such tests are smoothed maximum score and subsampling. We propose a new method that uses a “discretization’’ argument to circumvent the intractable asymptotic distribution of the maximum score estimator. The resulting tests require fewer assumptions than smoothed maximum score, less computational time than subsampling and can be applied to a wide range of nested and non-nested hypotheses. The tests are based on a certain asymptotic equivalence that is obtained by discretizing the continuous covariates with the discretization becoming finer as the sample size increases. The discretization parameters (analogous to bandwidths) are specified so that discretization vanishes asymptotically which is critical for the test to be consistent. The test statistics reflect differences in the predictive accuracy of the maximum score estimates under the null and alternative hypotheses. The tests are shown to be asymptotically normal under the null, and converge to infinity under the alternative. We also investigate the size and power properties of the test through Monte Carlo simulations.