Sergio Firpo, Antonio F. Galvao, Thomas Parker
arXiv 22 Nov 2019 · Econometrics
arXiv:1911.10215 · PDF · Extracted main text
We propose a method to conduct uniform inference for the (optimal) value function, that is, the function that results from optimizing an objective function marginally over one of its arguments. Marginal optimization is not Hadamard differentiable (that is, compactly differentiable) as a map between the spaces of objective and value functions, which is problematic because standard inference methods for nonlinear maps usually rely on Hadamard differentiability. However, we show that the map from objective function to an $L_p$ functional of a value function, for $1 \leq p \leq \infty$, are Hadamard directionally differentiable. As a result, we establish consistency and weak convergence of nonparametric plug-in estimates of Cram\'er-von Mises and Kolmogorov-Smirnov test statistics applied to value functions. For practical inference, we develop detailed resampling techniques that combine a bootstrap 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 proposed methods and show accurate empirical size and nontrivial power of the procedures. Finally, we apply our methods to the evaluation of a job training program using bounds for the distribution function of treatment effects.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Fan and Park (2010) Sharp Bounds on the Distribution of Treatment Effects and Their Statistical Inference | 1.000 | 5 | 3 | 100% |
| 2 | van der Vaart and Wellner (1996) Weak Convergence and Empirical Processes | 0.899 | 11 | 5 | 73% |
| 3 | Linton, Song, and Whang (2010) An Improved Bootstrap Test of Stochastic Dominance | 0.888 | 10 | 4 | 70% |
| CarcamoCuevasRodriguez19 | unmatched citation key CarcamoCuevasRodriguez19 | 0.811 | 4 | 2 | 100% |
| 5 | Dehejia and Wahba (1999) Causal Effects in Nonexperimental Studies: Reevaluating the Evaluation of Training Programs | 0.811 | 4 | 2 | 100% |
| 6 | Williamson and Downs (1990) Probabilistic Arithmetic I. Numerical Methods for Calculating Convolutions and Dependency Bounds | 0.811 | 4 | 2 | 100% |
| 7 | Shapiro (1990) On Concepts of Directional Differentiability | 0.737 | 3 | 3 | 67% |
| 8 | Firpo and Ridder (2019) Partial Identification of the Treatment Effect Distribution and Its Functionals | 0.737 | 3 | 2 | 100% |
| 9 | LaLonde (1986) Evaluating the Econometric Evaluations of Training Programs with Experimental Data | 0.737 | 3 | 2 | 100% |
| 10 | Fang and Santos (2019) Inference on Directionally Differentiable Functions | 0.705 | 20 | 4 | 35% |
Showing the top 10 of 58 scored citations. 1 of these could not be matched to a bibliography entry, so only the citation key is shown.
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| Citing paper | Intensity | Mentions | Sections | |
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| 1 | Partial Identification and Inference for Conditional Distributions of Treatment Effects | 0.794 | 20 | 8 |
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| 5 | Policy Learning with $$-Expected Welfare | 0.000 | 1 | 1 |