Nearly Optimal Tests when a Nuisance Parameter is Present Under the Null Hypothesis

This paper considers nonstandard hypothesis testing problems that involve a nui-sance parameter. We establish a bound on the weighted average power of all valid tests, and develop a numerical algorithm that determines a feasible test with power close to the bound. The approach is illustrated in six applications: inference about a linear regression coefficient when the sign of a control coefficient is known; small sample inference about the difference in means from two independent Gaussian sam-ples from populations with potentially different variances; inference about the break date in structural break models with moderate break magnitude; predictability tests when the regressor is highly persistent; inference about an interval identified parame-ter; and inference about a linear regression coefficient when the necessity of a control is in doubt