Numerical Bootstrap and Constrained Estimation

Stanford University, Stanford CA

Investigators

Abstract

Many economic models used for policy evaluation are highly nonlinear, and are typically subject to nonlinear constraints on the parameters. The computational challenge in estimating these models has posed a significant obstacle for utilizing these models in effective policy making. This project assists in economic policy making by developing a new method of statistical inference that can be used to provide valid evaluation of the statistical uncertainty associated with policy-related functions of estimated parameters of economic models. This new method combines one-sided numerical differentiation with bootstrap resampling techniques to allow for non-differentiability in the policy function, and is both computationally simple and easy to implement. It can be applied to analyze oligopolistic competition in industrial organization, the effect of education policy such as the impact of smaller class sizes, and many other areas of applied economic analysis. These tools contribute to the welfare of the society by enabling models to evaluate the effectiveness of economic policies. This project studies a numerical Delta method for inference on a directionally differentiable function of regular parameters. This method is computationally efficient, does not require analytic knowledge of the structure of the function of interest, and provides uniformly valid inference for testing a one-sided hypothesis of a convex function of the parameters. In situations where the first order Delta method limiting distribution is degenerate, the second (or higher) order Delta method may provide the necessary nondegenerate large sample approximation. The investigator further generalizes the numerical Delta method to a new resampling technique called the numerical bootstrap that can consistently estimate the limit distribution in many cases -- where the conventional bootstrap is not valid and subsampling has been the most commonly used inference approach, and where the parameters are not known to be directionally differentiable. Applications include constrained and unconstrained M-estimators converging at both regular and nonstandard rates such as the maximum score model, partially identified models, misspecified simulated GMM models, and many sample size dependent statistics.