Bayesian indeterminacy testing

Definition

A Bayesian econometric framework for estimating dynamic stochastic general equilibrium (DSGE) models in which the parameter space is partitioned into a determinacy region (unique stable equilibrium) and an indeterminacy region (multiple stable equilibria), with the likelihood function properly defined over both regions. Under indeterminacy, additional parameters characterize the non-unique transmission of fundamental shocks and the distribution of sunspot shocks. The posterior distribution yields probability weights for determinacy vs indeterminacy conditional on the observed data.

Intuition

Standard DSGE estimation restricts parameters to the determinacy region. If the true data-generating process lies in the indeterminacy region, this restriction produces biased estimates. Bayesian indeterminacy testing removes the restriction: the prior covers both regions, and the posterior tells you how much probability mass falls in each. Under determinacy, sunspot parameters are unidentified (the prior is not updated); under indeterminacy, the data inform both the structural parameters and the additional sunspot/transmission parameters.

Formal notation

The canonical LRE form is $G amm a_{0} (t h e t a) s_{t} = G amm a_{1} (t h e t a) s_{t - 1} + P s i (t h e t a) e_{t} + P i (t h e t a) e t a_{t}$ , where $e t a_{t}$ are rational expectations forecast errors. Under indeterminacy, $e t a_{t} = A_{1} (t h e t a, M) e_{t} + A_{2} (t h e t a) ze t a_{t}$ , where $M$ characterizes the non-unique fundamental-shock transmission and $ze t a_{t} N (0, s i g m a_{z} e t a^{2})$ is a sunspot shock.

The full likelihood is: $L (t h e t a, M ∣ Y^{⊤}) = t h e t ain T h e t a^{I} L_{I} (t h e t a, M ∣ Y^{⊤}) + t h e t ain T h e t a^{D} L_{D} (Y^{⊤})$

where the determinacy likelihood $L_{D}$ is flat in $t h e t a$ when the model lacks endogenous propagation, and the indeterminacy likelihood depends on both $t h e t a$ and $M$ . The posterior probability of indeterminacy is $p i_{T} (I) = in t e g r a l_{t h e t ain T h e t a^{I}} p (t h e t a, M ∣ Y^{⊤}) d (t h e t a, M)$ .

Variants

Centered parameterization (Lubik-Schorfheide 2004): $M$ is reparameterized as a deviation from the MSV solution ( $M = 1 + M_{t} i l d e$ ) to ensure continuity at the determinacy boundary.
Posterior model weights (Bayesian model comparison): compute $p i_{T} (D)$ and $p i_{T} (I)$ as posterior weights for model averaging.
Full-information Bayesian (as in the original): uses all equilibrium cross-equation restrictions.
Limited-information analogs: Andrews-Ploberger optimal tests for indeterminacy from single equations (more difficult due to identification).

Comparison

vs sub-sample estimation under determinacy only (CGG 2000): restricting to determinacy biases parameter estimates if the truth is indeterminate. The Bayesian approach avoids this bias.
vs regime-switching models (monetary-policy-regime-switching): indeterminacy testing treats the pre/post-Volcker periods as separate sub-samples. Regime-switching models (regime-switches-agents-beliefs-post-world) treat policy changes as stochastic and recurrent, which eliminates the sub-sample split but introduces different assumptions.
vs reduced-form change-point tests (Bai-Perron): these can detect parameter instability but cannot assess whether the instability implies sunspot equilibria.

When to use

Assessing whether a particular monetary policy rule is consistent with equilibrium determinacy before designing a regime-switching model.
Estimating the quantitative importance of sunspot shocks in historical episodes of macroeconomic instability.
Evaluating the robustness of DSGE model estimates by checking whether the maximum of the posterior lies in the determinacy or indeterminacy region.

Known limitations

Model sensitivity: the indeterminacy/determinacy boundary depends on the specific model. Richer models may shift the boundary.
Sunspot specification: the distribution of sunspot shocks is partially arbitrary (the variance is identified from data, but the interpretation depends on the model).
Sub-sample assumption: the original approach uses a sub-sample split, assuming no anticipation of regime changes.

Open problems

Extending to nonlinear DSGE models where the determinacy/indeterminacy distinction is more complex.
Joint estimation of indeterminacy and regime switching (the model could be indeterminate in one regime and determinate in another).
Computational challenges: the likelihood under indeterminacy requires evaluating solutions for all parameter combinations, which is expensive for large models.

Key papers

testing-indeterminacy-application-monetary-policy — introduces the Bayesian approach and applies it to U.S. monetary policy

My understanding

This concept provides the econometric foundation for why we believe the pre-Volcker Fed was passive. In the CRE project context, the Leather-Sagi model takes this finding as given (monetary policy regimes exist and switch) and embeds it in a regime-switching framework where agents anticipate the switches. The Lubik-Schorfheide approach established the empirical fact; the regime-switching approach (Bianchi 2013, Bikbov-Chernov 2013) provides the structural framework for modeling it. The check_determinancy_fmsre function in the CRE codebase is the operational descendant of the determinacy checking that Lubik-Schorfheide formalized.

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