Introduction

Most hypotheses in limited dependent variable (LDV) models are composite, meaning that the null hypothesis H0 does not completely specify the datagenerating process (DGP). In this case, the null specifies only that the DGP belongs to a given set. As a consequence, the sampling distribution of a test statistic under H0 is unknown except in special cases, because it depends on the true DGP in the set specified by H0. The problem is how to test the null hypothesis in this situation. In this chapter, the bootstrap is used to solve the hypothesis-testing problem.

The LDV models considered in this chapter are the simple binary probit model and the simple censored normal linear regression model. For these models, a typical null hypothesis H0 is that the slope coefficient is zero. This null is composite, the remaining parameters being nuisance parameters. This null is tested using the Lagrange multiplier (LM), likelihood ratio (LR), and Wald test statistics. In our Monte Carlo experiments, we compare the powers of the competing tests when the tests use bootstrap-based critical values. We argue that the powers of the tests with bootstrap-based critical values are empirically relevant because these critical values can be calculated in applications.

There are two basic approaches to obtaining critical values for testing a composite null hypothesis. One approach employs the concept of the size of a test. The size is the supremum of the test's rejection probability over all DGPs contained in H0.