To “ Estimation and Inference with Weak , Semi - Strong , and Strong Identification ”
نویسندگان
چکیده
7. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8. Supplemental Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 8.1. Description of Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 8.2. Assumption V1 for Vector β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 8.3. Details for the Type 2 Robust CS With NI Critical Values . . . . . . . . . . . . . . 8 8.4. Assumption B3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 8.5. Assumption C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 8.6. Assumption C6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 8.7. Assumptions C1 and D1: Quadratic Expansions for Sample Average Criterion Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 8.8. Bivariate Probit Model With Endogeneity and Reparametrization . . . . . . . . . 18 9. Supplemental Appendix B: Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 9.1. Proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 9.2. Proofs of Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 9.3. Proofs of t Asymptotic Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 30 9.4. Proofs of QLR Asymptotic Distributions and Restricted Estimator Results and Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 9.5. Proofs of Asymptotic Size Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 9.6. Proofs of Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 10. Supplemental Appendix C: Verification of Assumptions for the ARMA(1 1) Example 66 10.1. ARMA Example: Initial Conditions Adjustment . . . . . . . . . . . . . . . . . . . 66 10.2. ARMA Example: Derivation of Formulae for Key Quantities . . . . . . . . . . . 68 10.3. ARMA Example: Verification of Assumptions . . . . . . . . . . . . . . . . . . . . 74 10.4. Proof of the ARMA Initial Conditions Lemma . . . . . . . . . . . . . . . . . . . . 90 11. Supplemental Appendix D: ARMA(1 1) Numerical Results . . . . . . . . . . . . . . . 96 11.1. Table of Constants for Type 2 Robust CI’s With NI Critical Values . . . . . . . . . 96 11.2. Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 11.3. Additional Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 12. Supplemental Appendix E: Nonlinear Regression Example . . . . . . . . . . . . . . . . 118 12.1. Nonlinear Regression Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 12.2. Parameter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 12.3. Criterion Function Limit Assumption . . . . . . . . . . . . . . . . . . . . . . . . . 119 12.4. Close to β= 0 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 12.5. Distant From β= 0 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 12.6. Key Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 12.7. Variance Matrix Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 12.8. Failure of Assumption C of Stock and Wright (2000) . . . . . . . . . . . . . . . . 128 13. Supplemental Appendix F: LIML Example . . . . . . . . . . . . . . . . . . . . . . . . . 130 13.1. Key Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 13.2. Asymptotic Distributions of the Statistics . . . . . . . . . . . . . . . . . . . . . . . 135 13.3. Simplified Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 13.4. Unrestricted ICS Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
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