Calculating confidence intervals for continuous and discontinuous functions of parameters

Applied researchers often need to estimate confidence intervals for functions of parameters, such as the effects of counterfactual policy changes. If the function is continuously differentiable and has non-zero and bounded derivatives, then they can use the delta method. However, if the function is nondifferentiable (as in the case of simulating functions with zero-one outcomes), has zero derivatives, or unbounded derivatives, then researchers usually use the nonparametric bootstrap or sample from the asymptotic distribution of the estimated parameter vector. Researchers also use these bootstrap approaches when the function is well-behaved but complicated. Indeed, these approaches are advocated by two very influential published articles. We first show that both of these bootstrap procedures can produce confidence intervals whose asymptotic coverage is less than advertised, i.e. confidence intervals that are too small. We then propose two procedures that provide correct coverage. In applications, we find that the bootstrap approaches mentioned above produce confidence intervals that are significantly smaller than their consistent counterparts, suggesting that previous empirical work is likely to have been overly optimistic in terms of the precision of estimated counterfactual effects.