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Asymptotic properties of Bayesian inference for structural changes in multivariate regressions

  • 이재원 (이재원, 대학원 경제학과 경제학전공)

초록/요약

In the first chapter, Under a fairly general set of assumptions where heteroskedasticity and autocorrelation are allowed, we explore the asymptotic properties of Bayesian inference in multivariate regression models with a structural break, where changes can occur in both regression coefficients and the covariance matrix of the errors. We first establish boundedness on the marginal posterior distribution of a breakdate, indicating that the distortion between the true breakdate and posterior distribution is asymptotically neglectable for inference of regression parameters. Moreover, we validate a Bernstein-von Mises-type theorem for regression parameters in the context of multivariate regressions, where the problem of misspecification boils down to the same one as the regular linear regression model, making the problem be solved as in the regular models. Our Monte Carlo analysis confirms the Bernstein-von Mises theorem, supporting our findings. In the second chapter, we explore the asymptotic properties of Bayesian inference in multivariate regression models allowing multiple structural breaks. We establish asymptotic equivalence between the highest posterior density (HPD) region and confidence sets for breakdates, along with boundedness on the joint marginal posterior distribution and a large-sample correspondence between the posterior density ratio and the likelihood ratio. Moreover, we validate a Bernstein-von Mises-type theorem for regression coefficients in the context of multivariate regressions with multiple breaks. The consequences of misspecifying the model are discussed. Our Monte Carlo analysis confirms the Bernstein-von Mises theorem and the similar behavior of HPD regions and inverted likelihood ratio confidence sets, supporting our findings. Finally, we apply the results to real-world data. Keywords: Structural Break, Bayesian Asymptotics, Multivariate Regression, Model Misspecification

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목차

Abstract i
Acknowledgement ii
Contents iii
List of Tables vi
1 Asymptotic properties of Bayesian inference for a structural change under misspecification 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The Model and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 A Bayesian Approach Under Normal Likelihood . . . . . . . . . . . . . . . 8
1.3.1 Marginal Posterior of T1 . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Bernstein-von Mises theorem for θ . . . . . . . . . . . . . . . . . . 10
1.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.1 Univariate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.2 Multivariate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Asymptotic properties of Bayesian inference for structural changes in
multivariate regressions 26
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Model and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 A Bayesian Approach Under Normal Likelihood . . . . . . . . . . . . . . . 34
2.3.1 Marginal Posterior of T˜ . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.2 Bernstein-von Mises theorem for βj . . . . . . . . . . . . . . . . . . 40
2.4 Asymptotic properties of Chib (1998)’s prior for changepoints . . . . . . . 42
2.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5.1 When the Model is Correctly Specified . . . . . . . . . . . . . . . . 46
2.5.2 When the Model is Misspecified . . . . . . . . . . . . . . . . . . . . 56
2.6 Revisiting Garcia and Perron (1996) and Bai and Perron (2003) . . . . . . 58
2.6.1 An Efficient Way to Compute Bayes Factors . . . . . . . . . . . . 59
2.6.2 Estimated Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Bibliography 68
A Appendix of Chapter 1 72
A.1 Proofs of Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
A.1.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 72
A.1.2 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A.2 Proof of Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A.2.1 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . 79
A.2.2 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . 83
A.3 Derivation of QMLE and Asymptotic Covariance Matrix . . . . . . . . . . 88
B Appendix of Chapter 2 91
B.1 Proofs of Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
B.1.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 91
B.1.2 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 97
B.1.3 Proof of Corollary 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 100
B.1.4 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.1.5 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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