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On global identification in structural vector autoregressions

Emanuele Bacchiocchi, Toru Kitagawa

Abstract

In a landmark contribution to the structural vector autoregression (SVARs) literature, Rubio-Ramirez, Waggoner, and Zha (2010, `Structural Vector Autoregressions: Theory of Identification and Algorithms for Inference,' Review of Economic Studies) shows a necessary and sufficient condition for equality restrictions to globally identify the structural parameters of a SVAR. The simplest form of the necessary and sufficient condition shown in Theorem 7 of Rubio-Ramirez et al (2010) checks the number of zero restrictions and the ranks of particular matrices without requiring knowledge of the true value of the structural or reduced-form parameters. However, this note shows by counterexample that this condition is not sufficient for global identification. Analytical investigation of the counterexample clarifies why their sufficiency claim breaks down. The problem with the rank condition is that it allows for the possibility that restrictions are redundant, in the sense that one or more restrictions may be implied by other restrictions, in which case the implied restriction contains no identifying information. We derive a modified necessary and sufficient condition for SVAR global identification and clarify how it can be assessed in practice.

On global identification in structural vector autoregressions

Abstract

In a landmark contribution to the structural vector autoregression (SVARs) literature, Rubio-Ramirez, Waggoner, and Zha (2010, `Structural Vector Autoregressions: Theory of Identification and Algorithms for Inference,' Review of Economic Studies) shows a necessary and sufficient condition for equality restrictions to globally identify the structural parameters of a SVAR. The simplest form of the necessary and sufficient condition shown in Theorem 7 of Rubio-Ramirez et al (2010) checks the number of zero restrictions and the ranks of particular matrices without requiring knowledge of the true value of the structural or reduced-form parameters. However, this note shows by counterexample that this condition is not sufficient for global identification. Analytical investigation of the counterexample clarifies why their sufficiency claim breaks down. The problem with the rank condition is that it allows for the possibility that restrictions are redundant, in the sense that one or more restrictions may be implied by other restrictions, in which case the implied restriction contains no identifying information. We derive a modified necessary and sufficient condition for SVAR global identification and clarify how it can be assessed in practice.

Paper Structure

This paper contains 9 sections, 1 theorem, 29 equations.

Table of Contents

  1. Introduction
  2. Model
  3. An illustrative example
  4. An example
  5. Analytical investigation
  6. Detecting the failure of global identification
  7. Other examples of transformation
  8. Extending the necessary and sufficient condition for exact identification
  9. Conclusion

Key Result

Theorem 1

Consider an SVAR with admissible restrictions represented by $R$. The SVAR is exactly identified at the point $(A_0,A_+)\in R$ if and only if $q_j=n-j$ for $j=1,\ldots,n$ and the restrictions are non-redundant at $(A_0,A_+)$.

Theorems & Definitions (7)

  • Definition 1: Normalization rule
  • Definition 2: Exact identification
  • Example 1: Short-run zero restrictions on a monetary policy shock
  • Example 2: Short-run, long-run, cumulated long-run restrictions
  • Definition 3: Non-redundant restrictions
  • Theorem 1: A necessary and sufficient condition for exact identification
  • proof