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Uniform inference for value functions

  • KanKan
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Document pages: 44 pages

Abstract: We propose a method to conduct uniform inference for the optimal valuefunction, that is, the function that results from optimizing an objectivefunction marginally over one of its arguments. Marginal optimization is notcompactly differentiable as a map between the spaces of objective and valuefunctions, which is problematic because standard inference methods fornonlinear maps usually rely on compact differentiability. However, we show thatthe map from objective function to uniform test statistics applied to the valuefunction -- specifically, Kolmogorov-Smirnov or Cramér-von Mises statistics-- are directionally differentiable. We establish consistency and weakconvergence of nonparametric plug-in estimates of the test statistics. Forpractical inference, we develop detailed resampling techniques that combine abootstrap procedure with estimates of the directional derivatives. In addition,we establish local size control of tests which use the resampling procedure.Monte Carlo simulations assess the finite-sample properties of the proposedmethods and show accurate empirical size of the procedures. Finally, we applyour methods to the evaluation of a job training program using bounds for thedistribution function of treatment effects.

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