Delta method and nonparametric percentile bootstrap confidence intervals

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Extension of the delta method for ratios of expectations in the sequence-of-model framework

A lower bound on the length of nonasymptotic confidence intervals

Nonparametric estimation in conditional moment restricted models via Generalized Empirical Likelihood

Table of contents :

Remerciements
1 Introduction/ Résumé substantiel en français
2 Introduction in English
3 Nonparametric estimation in conditional moment restricted models via Generalized Empirical Likelihood
3.1 Introduction
3.2 A general presentation of GEL estimators
3.2.1 From GMCs to GELs
3.2.2 Construction of the estimation procedure
3.3 Results
3.3.1 Consistency
3.3.2 Rate
3.4 Conclusion
3.5 Proofs of the main results
3.5.1 Proof of Theorem 3.1
3.5.2 Proof of Theorem 3.2
3.6 Appendix
3.6.1 Lemmas
3.6.2 Proofs
3.6.2.1 Proof of Lemma 3.2
3.6.2.2 Proof of Lemma 3.3
3.6.2.3 Proof of Lemma 3.4
3.6.2.4 Proof of Lemma 3.5
3.6.2.5 Proof of Lemma 3.6
3.6.2.6 Proof of Lemma 3.7
3.6.2.7 Proof of Lemma 3.8
3.6.2.8 Proof of Lemma 3.9
3.6.2.9 Proof of Lemma 3.10
3.6.2.10 Proof of Lemma 3.11
3.6.2.11 Proof of Lemma 3.12
3.6.2.12 Proof of Lemma 3.13
3.6.2.13 Proof of Lemma 3.14
4 Empirical Process Results for Exchangeable Arrays
4.1 Introduction
4.2 The set up and main results
4.2.1 Set up
4.2.2 Uniform laws of large numbers and central limit theorems
4.2.3 Convergence of the bootstrap process
4.2.4 Application to nonlinear estimators
4.3 Extensions
4.3.1 Heterogeneous number of observations
4.3.2 Separately exchangeable arrays
4.4 Simulations and real data example
4.4.1 Monte Carlo simulations
4.4.2 Application to international trade data
4.5 Conclusion
4.6 Appendix A
4.6.1 Proof of Lemma 2.2
4.6.1.1 A decoupling inequality
4.7 Appendix B
4.7.1 Proofs of the main results
4.7.1.1 Lemma 4.3
4.7.1.2 Theorem 4.1
4.7.1.2.1 Uniform law of large numbers
4.7.1.2.2 Uniform central limit theorem
4.7.1.3 Theorem 4.2
4.7.1.4 Theorem 4.3
4.7.1.5 Theorem 4.4
4.7.2 Proofs of the extensions
4.7.2.1 Theorem 4.5
4.7.2.1.1 Uniform law of large numbers
4.7.2.1.2 Uniform central limit theorem
4.7.2.2 Convergence of the bootstrap process
4.7.2.3 Theorem 4.6
4.7.2.3.1 Uniform law of large numbers
4.7.2.3.2 Uniform central limit theorem
4.7.2.3.3 Convergence of the bootstrap process
4.7.3 Technical lemmas
4.7.3.1 Results related to the symmetrisation lemma
4.7.3.1.1 Proof of Lemma S4.4
4.7.3.1.2 Proof of Lemma S4.5
4.7.3.2 Results related to laws of large numbers
4.7.3.2.1 Proof of Lemma S4.6
4.7.3.2.2 Proof of Lemma S4.7
4.7.3.2.3 Proof of Lemma S4.8
4.7.3.2.4 Proof of Lemma S4.9
4.7.3.3 Covering and entropic integrals
4.7.3.3.1 Proof of Lemma S4.10
4.7.3.3.2 Proof of Lemma S4.11
5 On the construction of confidence intervals for ratios of expectations
5.1 Introduction
5.2 Our framework
5.3 Limitations of the delta method: when are asymptotic confidence intervals valid?
5.3.1 Asymptotic approximation takes time to hold
5.3.2 Asymptotic results may not hold in the sequence-of-model framework
5.3.3 Extension of the delta method for ratios of expectations in the sequence-of-model framework
5.3.4 Validity of the nonparametric bootstrap for sequences of models
5.4 Construction of nonasymptotic confidence intervals for ratios of expectations
5.4.1 An easy case: the support of the denominator is well-separated from 0
5.4.2 General case: no assumption on the support of the denominator
5.5 Nonasymptotic CIs: impossibility results and practical guidelines
5.5.1 An upper bound on testable confidence levels
5.5.2 A lower bound on the length of nonasymptotic confidence intervals
5.5.3 Practical methods and plug-in estimators
5.6 Numerical applications
5.6.1 Simulations
5.6.2 Application to real data
5.7 Conclusion
5.8 General definitions about confidence intervals
5.9 Proofs of the results in Sections 5.3, 5.4 and 5.5
5.9.1 Proof of Theorem 5.1
5.9.2 Proof of Theorem 5.2
5.9.2.1 Proof of Lemma 5.4
5.9.2.2 Proof of Lemma 5.5
5.9.3 Proof of Example 5.3
5.9.4 Proof of Theorem 5.3
5.9.5 Proof of Theorem 5.4
5.9.5.1 Proof of Lemma 5.6
5.9.6 Proof of Theorem 5.5
5.9.6.1 Proof of Lemma 5.7
5.9.7 Proof of Theorem 5.6
5.9.7.1 Proof of Lemma 5.8
5.10 Adapted results for “Hoeffding” framework
5.10.1 Concentration inequality in an easy case: the support of the denominator is well-separated from 0
5.10.2 Concentration inequality in the general case
5.10.3 An upper bound on testable confidence levels
5.10.4 Proof of Theorems 5.8 and 5.9
5.10.5 Proof of Theorem 5.10
5.10.5.1 Proof of Lemma 5.9
5.11 Additional simulations
5.11.1 Gaussian distributions
5.11.2 Student distributions
5.11.3 Exponential distributions
5.11.4 Pareto distributions
5.11.5 Bernoulli distributions
5.11.6 Poisson distributions
5.11.7 Delta method and nonparametric percentile bootstrap confidence intervals
6 Fuzzy Differences-in-Differences with Stata
6.1 Introduction
6.2 Set-up
6.2.1 Parameters of interest, assumptions, and estimands
6.2.2 Estimators
6.3 Extensions
6.3.1 Including covariates
6.3.2 Multiple periods and groups
6.3.3 Other extensions
6.3.3.1 Special cases
6.3.3.2 No “stable” control group
6.3.3.3 Non-binary treatment
6.4 The fuzzydid command
6.4.1 Syntax
6.4.2 Description
6.4.3 Options
6.4.4 Saved results
6.5 Example
6.6 Monte Carlo Simulations
6.7 Conclusion