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SVARs with breaks: Identification and inference

Emanuele Bacchiocchi, Toru Kitagawa

Abstract

In this paper we propose a class of structural vector autoregressions (SVARs) characterized by structural breaks (SVAR-WB). Together with standard restrictions on the parameters and on functions of them, we also consider constraints across the different regimes. Such constraints can be either (a) in the form of stability restrictions, indicating that not all the parameters or impulse responses are subject to structural changes, or (b) in terms of inequalities regarding particular characteristics of the SVAR-WB across the regimes. We show that all these kinds of restrictions provide benefits in terms of identification. We derive conditions for point and set identification of the structural parameters of the SVAR-WB, mixing equality, sign, rank and stability restrictions, as well as constraints on forecast error variances (FEVs). As point identification, when achieved, holds locally but not globally, there will be a set of isolated structural parameters that are observationally equivalent in the parametric space. In this respect, both common frequentist and Bayesian approaches produce unreliable inference as the former focuses on just one of these observationally equivalent points, while for the latter on a non-vanishing sensitivity to the prior. To overcome these issues, we propose alternative approaches for estimation and inference that account for all admissible observationally equivalent structural parameters. Moreover, we develop a pure Bayesian and a robust Bayesian approach for doing inference in set-identified SVAR-WBs. Both the theory of identification and inference are illustrated through a set of examples and an empirical application on the transmission of US monetary policy over the great inflation and great moderation regimes.

SVARs with breaks: Identification and inference

Abstract

In this paper we propose a class of structural vector autoregressions (SVARs) characterized by structural breaks (SVAR-WB). Together with standard restrictions on the parameters and on functions of them, we also consider constraints across the different regimes. Such constraints can be either (a) in the form of stability restrictions, indicating that not all the parameters or impulse responses are subject to structural changes, or (b) in terms of inequalities regarding particular characteristics of the SVAR-WB across the regimes. We show that all these kinds of restrictions provide benefits in terms of identification. We derive conditions for point and set identification of the structural parameters of the SVAR-WB, mixing equality, sign, rank and stability restrictions, as well as constraints on forecast error variances (FEVs). As point identification, when achieved, holds locally but not globally, there will be a set of isolated structural parameters that are observationally equivalent in the parametric space. In this respect, both common frequentist and Bayesian approaches produce unreliable inference as the former focuses on just one of these observationally equivalent points, while for the latter on a non-vanishing sensitivity to the prior. To overcome these issues, we propose alternative approaches for estimation and inference that account for all admissible observationally equivalent structural parameters. Moreover, we develop a pure Bayesian and a robust Bayesian approach for doing inference in set-identified SVAR-WBs. Both the theory of identification and inference are illustrated through a set of examples and an empirical application on the transmission of US monetary policy over the great inflation and great moderation regimes.
Paper Structure (27 sections, 9 theorems, 116 equations, 6 figures, 1 table)

This paper contains 27 sections, 9 theorems, 116 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related literature
  3. Econometric framework: SVAR with breaks
  4. Theory of identification
  5. Identification
  6. Identifying restrictions
  7. Global vs local identification in SVAR-WBs
  8. Conditions for identification in SVAR-WB
  9. Estimating the structural parameters of locally-identified SVAR-WBs
  10. A general algorithm for the estimation of just-identified SVAR-WBs
  11. Drawing inference in locally- or set-identified SVAR-WBs
  12. Drawing inference in locally-identified SVAR-WBs
  13. Bayesian Inference
  14. Frequentist-valid inference
  15. Drawing inference in set-identified SVAR-WBs
  16. ...and 12 more sections

Key Result

Theorem 1

Consider an SVAR-WB as in Eq. (eq:SVARWB) with $s$ regimes and admissible parameters represented by $\mathcal{A}(\phi)$. A necessary condition for the identification of the parameters is that $f\geq s\,\tilde{n}$, where $\tilde{n}=n(n-1)/2$ .

Figures (6)

  • Figure 1: Impulse response functions of output gap to a monetary policy shock: Model I (locally-identified SVAR-WB: equality restrictions, only).
  • Figure 2: Impulse response functions of output gap to a monetary policy shock: Model II (locally-identified SVAR-WB: equality and sign restrictions).
  • Figure 3: Impulse response functions of output gap to a monetary policy shock: Model III (set-identified SVAR-WB: equality and sign restrictions).
  • Figure 4: Impulse response functions of output gap to a monetary policy shock: Model IV (set-identified SVAR-WB: sign restrictions within regimes, only).
  • Figure 5: Impulse response functions of output gap to a monetary policy shock: Model V (set-identified SVAR-WB: sign restrictions + ranking restrictions + FEV restrictions).
  • ...and 1 more figures

Theorems & Definitions (31)

  • Definition 1: Global identification in SVAR-WB
  • Definition 2: Local identification in SVAR-WB
  • Definition 3: Normalization
  • Definition 4: Admissible infinitesimal rotations
  • Theorem 1
  • proof
  • Theorem 2: Sufficient rank condition for identification
  • proof
  • Corollary 1
  • proof
  • ...and 21 more