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Testing for Indeterminacy: An Application to U.S. Monetary Policy

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Problem

Linear rational expectations (LRE) models can have multiple equilibria (indeterminacy) when monetary policy is “passive” — the central bank does not raise the nominal interest rate aggressively enough in response to inflation. Prior to this paper, system-based econometric inference for DSGE models was limited to the determinacy region of the parameter space, meaning that (i) parameter estimates were biased if the true data-generating process was indeterminate, and (ii) there was no systematic way to assess the quantitative importance of sunspot shocks vs fundamental shocks under indeterminacy.

Key idea

Extend Bayesian likelihood-based estimation of DSGE models to the full parameter space, including the indeterminacy region. Under indeterminacy, the propagation of fundamental shocks is not uniquely determined, and sunspot shocks can affect equilibrium allocations. The paper indexes the multiple solutions through additional parameters (characterizing the non-unique transmission of fundamental shocks and the distribution of sunspot shocks) and uses Bayesian posterior probabilities to weight determinacy vs indeterminacy. This is the bayesian-indeterminacy-testing approach.

Method

  • Prototypical three-equation New Keynesian model: IS curve, new-keynesian-phillips-curve, and Taylor rule with parameters psi_1 (inflation response), psi_2 (output response), rho_R (inertia).
  • Cast in canonical LRE form (Sims 2002): Gamma_0 s_t = Gamma_1 s_{t-1} + Psi e_t + Pi eta_t.
  • Under determinacy (psi_1 > 1): forecast errors eta_t = A_1 e_t with A_2 = 0 (no sunspots), unique solution.
  • Under indeterminacy (psi_1 < 1): eta_t = A_1 e_t + A_2 zeta_t where both A_1 and A_2 are free parameters; M (non-unique fundamental transmission) and sunspot variance sigma_zeta must be estimated.
  • Likelihood function constructed over the full parameter space: determinacy and indeterminacy regions have different likelihood expressions. Bayesian posterior weights pi_T(I) and pi_T(D) summarize evidence for indeterminacy vs determinacy.
  • Estimated on U.S. quarterly data: output, inflation, interest rate. Two sub-samples: pre-Volcker (1960:Q1-1979:Q2) and post-1982 (1982:Q4-1997:Q4).

Results

  • Post-1982: posterior probability of determinacy is ~0.95-1.00 across prior specifications. U.S. monetary policy after Volcker is consistent with determinacy (psi_1 > 1, active monetary policy).
  • Pre-Volcker (1960:Q1-1979:Q2): posterior probability of indeterminacy is ~0.80-0.97 depending on prior. Monetary policy was passive (psi_1 < 1), consistent with the Clarida-Gali-Gertler (2000) finding but now established through full-system Bayesian estimation.
  • Two interpretations of pre-Volcker dynamics under indeterminacy:
    • (A) Indeterminacy changed the propagation of fundamental shocks, but sunspot shocks played no role (M != 1, sigma_zeta = 0). Data slightly favor this interpretation.
    • (B) Fundamental shock responses resemble determinacy, but additional sunspot shocks caused substantial inflation and interest rate fluctuations with small output effects (M = 1, sigma_zeta > 0).
  • The parameter psi_1 (marking the determinacy boundary) is identifiable under indeterminacy but NOT under determinacy — a non-trivial identification result.
  • Estimates are robust to comparison with a richer model (habit formation, backward-looking price setters) restricted to determinacy: the data favor the simple model with indeterminacy over the rich model with determinacy.

Limitations

  • Sensitivity to model specification: the association of passive monetary policy with indeterminacy is model-specific. Dupor (2001) shows that in a continuous-time model with endogenous investment, passive policy can be consistent with determinacy.
  • The richer endogenous dynamics under indeterminacy can capture model misspecification, potentially biasing posteriors toward indeterminacy when the true model is more complex.
  • Small, stylized model — only three observables. Larger DSGE models might change the posterior weights.
  • Sub-sample split rather than a regime-switching framework: agents are assumed not to anticipate regime changes. regime-switches-agents-beliefs-post-world addresses this limitation.

Open questions

  • How do indeterminacy results change in medium-scale DSGE models with more shocks and frictions?
  • Can the approach be combined with regime-switching to avoid the sub-sample split?
  • What are the welfare implications of the estimated sunspot shocks?
  • How does indeterminacy interact with the zero lower bound on nominal interest rates?

My take

This is a foundational paper for Bayesian DSGE estimation and for the understanding of U.S. monetary policy history. The contribution is both methodological (extending likelihood-based estimation to the indeterminacy region) and empirical (the pre-Volcker indeterminacy finding). For the CRE project, this paper provides the intellectual foundation for the claim that the pre-Volcker Fed was passive — the same claim that motivates the regime-switching Taylor rule in Bikbov-Chernov and Leather-Sagi. The key limitation (sub-sample split rather than regime switching) was subsequently addressed by regime-switches-agents-beliefs-post-world, which is why the Leather-Sagi model uses a Markov chain rather than a structural break.

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