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Title page 1

Contents 1

Abstract 2

1. Introduction 3

2. Methodology 7

2.1. Identification of VAR impulse response functions via external instruments 8

2.2. Introducing time-varying coefficients 11

2.3. Joint inference for the reduced form parameters 15

2.4. Inference for impulse response functions 19

2.5. Bandwidth selection 21

3. Monte Carlo Simulations 22

3.1. Data Generating Process 23

3.2. Empirical coverage 24

4. The time-varying effects of oil supply news on US industrial production 29

4.1. Data and identification strategy 31

4.2. Results 33

5. Conclusion 36

References 38

Appendix A. Proofs 43

Appendix B. Inference for structural impulse response functions 47

Appendix C. Supplementary Monte Carlo Results 52

Appendix D. Supplementary empirical results 53

Appendix E. A comparison to alternative time-varying IRF estimators 61

Tables 32

Table 1. Granger causality test results computed at different points of time for the null hypothesis that zt does not predict yt in a VAR for yt = [zt, y't ]' 32

Table 2. Portmanteau test results computed at different points of time for the null hypothesis that Rhq,t = (R₁,t, ... ,Rhq,t) = 0 54

Figures 24

Figure 1. True impulse response functions λh,i,t for horizons h = 0, h = 10 and h = 20 24

Figure 2. Estimated empirical coverage at 95% confidence level obtained for λh,i,t (red) and λh,i,t (blue) at t = 1/2T = 189 (first row) and t = 3/4T = 283 (second row).... 25

Figure 3. Estimated empirical coverage at 95% confidence level obtained for λh,i,t (red) and λh,i,t (blue) at t = 1/2T = 189 (first row) and t = 3/4T = 283 (second row).... 26

Figure 4. Estimated empirical coverage at 95% confidence level obtained for λh,i,t (red) and λh,i,t (blue) at t = 1/2T = 189 (first row) and t = 3/4T = 283 (second row).... 27

Figure 5. Estimated empirical coverage at 95% confidence level obtained for λh,i,t (red) and λh,i,t (blue) at t = 1/2T = 189 (first row) and t = 3/4T = 283 (second row).... 28

Figure 6. US petroleum consumption, production, and net imports (1950-2023) 29

Figure 7. Time-varying impulse response functions to an oil-supply shock of unit variance 33

Figure 8. Estimated impulse response functions to an oil-supply shock of unit variance for selected industries at three-(manufacturing) and four- (mining)... 35

Figure C.9. True impulse response functions λh,i,t 52

Figure C.10. Estimated empirical coverage at 95% confidence level obtained for λh,i,t (red) and λh,i,t (blue) at t = 1/2T = 5655 (first row) and t = 3/4T = 8483... 53

Figure D.11. Estimate of αt obtained in the IV-SVAR 54

Figure D.12. Time-varying impulse response functions of US manufacturing- and mining output to an oil-supply shock. The top two rows contrast... 56

Figure D.13. Time-varying relative impulse response functions λh,i,t,tb to an oil-supply shock. Estimates are obtained using the internal instrument VAR,... 57

Figure D.14. Time-varying impulse response functions to an oil-supply shock of unit variance. Shaded areas indicate 65% and 90% confidence sets,... 58

Figure D.15. Time-varying impulse response functions to an oil-supply shock of unit variance. Shaded areas indicate 65% and 90% confidence sets,... 59

Figure D.16. Time-varying impulse response functions to an oil-supply shock of unit variance. Shaded areas indicate 65% and 90% confidence sets,... 59

Figure D.17. Time-varying impulse response functions to an oil-supply shock of unit variance. Shaded areas indicate 65% and 90% confidence sets,... 60

Figure D.18. Time-varying impulse response functions to an oil-supply shock of unit variance. Shaded areas indicate 65% and 90% confidence sets,... 60

Figure E.19. Kernel based estimator: Point estimates of IRFs alongside 90% confidence intervals at t = [1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8]T for y₁ (first column)... 63

Figure E.20. Path estimator: Point estimates of IRFs alongside 90% confidence intervals at t = [1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8]T for y₁ (first column)... 64

Figure E.21. Bayesian VAR-X: Posterior median estimates of IRFs alongside 90% posterior credible intervals at t = [1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8]T for y₁... 65