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The Identification Problem for Linear Rational Expectations Models

Majid M. Al-Sadoon, Piotr Zwiernik

TL;DR

This paper analyzes identifiability in linear rational expectations models (LREMs) by linking parameter-to-observable mappings to spectral-density representations through Wiener–Hopf factorization, extending VARMA analysis. It formalizes a comprehensive parameter-space framework with EU, EUI, and EUI0 subsets to capture existence, uniqueness, and invertibility conditions, and characterizes observational equivalence via a finite-dimensional operator kernel, yielding explicit dimension counts for equivalence classes. The core result shows forward dependence is the main source of non-identifiability and provides a structured decomposition of the identification problem, including a pragmatic pathway to assess structural identifiability under nonlinear restrictions and affine restrictions. Overall, the work extends classical identification theory to LREMs, highlighting how endogeneity of expectations creates new challenges and offering analytic tools for diagnosing and potentially resolving them in empirical work.

Abstract

This version corrects a number of mistakes that appeared in the previous draft. In particular, the (EU-LREM) condition is sufficient for existence and uniqueness but not necessary, as we had claimed. We are grateful to P. C. B. Phillips and to three anonymous referees for the substantial improvements to the paper since it first appeared. Any remaining errors are our own responsibility.

The Identification Problem for Linear Rational Expectations Models

TL;DR

This paper analyzes identifiability in linear rational expectations models (LREMs) by linking parameter-to-observable mappings to spectral-density representations through Wiener–Hopf factorization, extending VARMA analysis. It formalizes a comprehensive parameter-space framework with EU, EUI, and EUI0 subsets to capture existence, uniqueness, and invertibility conditions, and characterizes observational equivalence via a finite-dimensional operator kernel, yielding explicit dimension counts for equivalence classes. The core result shows forward dependence is the main source of non-identifiability and provides a structured decomposition of the identification problem, including a pragmatic pathway to assess structural identifiability under nonlinear restrictions and affine restrictions. Overall, the work extends classical identification theory to LREMs, highlighting how endogeneity of expectations creates new challenges and offering analytic tools for diagnosing and potentially resolving them in empirical work.

Abstract

This version corrects a number of mistakes that appeared in the previous draft. In particular, the (EU-LREM) condition is sufficient for existence and uniqueness but not necessary, as we had claimed. We are grateful to P. C. B. Phillips and to three anonymous referees for the substantial improvements to the paper since it first appeared. Any remaining errors are our own responsibility.

Paper Structure

This paper contains 9 sections, 16 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. The Parameter Space
  4. Core of the Identification Problem for LREMs
  5. Observational Equivalence
  6. Structural Identifiability
  7. Conclusion
  8. Topological Aspects of the Parameter Spaces
  9. Proofs

Key Result

Theorem 1

Let $(B,A)\in\Omega_{n,m,\kappa,\lambda}$. Then $(B,A)$ has a unique solution if and only if there exists a unique $C\in \Sigma^{n\times m}$ such that The impulse responses of the linear process eq:ma are the coefficients of the expansion $C(z)=\sum_{i=0}^\infty C_iz^i$ in a neighbourhood of $\overline{\mathbb{D}}$.

Theorems & Definitions (51)

  • Definition 1
  • Example 1
  • Definition 2
  • Definition 3
  • Theorem 1: Onatski's Criterion
  • Definition 4
  • Example 2
  • Definition 5
  • Definition 6
  • Definition 7
  • ...and 41 more