Supplement to “ Inference Based on Conditional Moment Inequalities ”
نویسندگان
چکیده
11. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 12. Supplemental Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 12.1. Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 12.2. Proof of Theorem 2(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 13. Supplemental Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 13.1. Kolmogorov–Smirnov and Approximate CvM Tests and CS’s . . . . . . . . . . . . 13 13.2. Instruments and Weight Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Weight Function Q for GB-spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Data-Dependent Q for Gbox dd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Weight Function Q for Gc/d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 13.3. Sufficient Conditions for Assumption GMS2(a) . . . . . . . . . . . . . . . . . . . 22 13.4. Example: Verification of Assumptions LA1–LA3 and LA3′ . . . . . . . . . . . . . 23 13.5. Uniformity Issues With Infinite-Dimensional Nuisance Parameters . . . . . . . . 26 13.6. Problems With Pointwise Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . 29 13.7. Subsampling Critical Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 13.7.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 13.7.2. Asymptotic Coverage Probabilities of Subsampling Confidence Sets . . . 32 14. Supplemental Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 14.1. Proofs of Lemmas 2 and 3 and Theorem 2(b) . . . . . . . . . . . . . . . . . . . . . 34 14.2. Proofs of Results for Fixed Alternatives . . . . . . . . . . . . . . . . . . . . . . . . 40 14.3. Proofs of Results for n−1/2-Local Alternatives . . . . . . . . . . . . . . . . . . . . . 45 14.4. Proofs Concerning the Verification of Assumptions S1–S4 . . . . . . . . . . . . . 46 15. Supplemental Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 15.1. Proofs of Kolmogorov–Smirnov and Approximate Cramér–von Mises Results . . 49 15.2. Proof of Lemma B2 Regarding GB-spline, Gbox dd, and Gc/d . . . . . . . . . . . . . . . 55 15.3. Proofs of Theorems B4 and B5 Regarding Uniformity Issues . . . . . . . . . . . . 59 15.4. Proofs of Subsampling Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 16. Supplemental Appendix E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 16.1. Preliminary Lemmas E1–E3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 16.2. Proof of Lemma A1(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 16.3. Proof of Lemma A1(b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 16.4. Proof of Lemma E1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 16.5. Proof of Lemma E2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 16.6. Proof of Lemma E3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 17. Supplemental Appendix F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 17.1. Quantile Selection Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 17.1.1. Additional Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 79 17.1.2. Conditional Moment Function Figures . . . . . . . . . . . . . . . . . . . . 80 17.1.3. Description of the CLR-Series, CLR-Local Linear, and LSW Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 17.2. Mean Selection Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 17.2.1. The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
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