Identification under regime switching

Definition

The set of conditions under which the structural parameters of a regime- switching model — including the per-regime parameter values and the transition probabilities of the underlying Markov chain — can be uniquely recovered from the joint distribution of observables. In contrast to standard DSGE / VAR identification (where the question is “are the structural shocks identified given the reduced form?”), regime-switching identification has the extra question “are the per-regime parameters identified given that the regime is latent?“.

Intuition

A regime-switching VAR can be cast as a constrained reduced-form VAR with a larger parameter space (one parameter per regime). The structural parameters are identified locally if perturbing them produces a measurable change in the observable joint density. Cochrane (2011) makes the general point that structural restrictions need not entail identification of all structural parameters — they may identify a lower-dimensional subspace. For regime- switching models the question is delicate because the latent regime introduces an additional source of unidentifiability (label-switching of the regimes themselves) on top of the standard Cochrane-style concern.

Bikbov-Chernov’s contribution is twofold: (i) they verify numerically that the structural parameters of their NK + regime-switching Taylor rule + no-arbitrage TSM are locally identified at the estimated parameter point, even when the data consist of macro variables and the short rate alone (no long bond yields); and (ii) they show empirically that adding long yields, while not necessary for local identification in principle, massively reduces the finite-sample bias and variance of the estimated monetary-policy regime parameters — by a factor of 20 in their simulation study.

Formal notation

Let $theta$ be the structural parameter vector and $p(y_{1:T}|theta)$ the likelihood of the observed panel. Local identification holds at $thet{a}_0$ if the Fisher information matrix I(theta_0) = -E[partial^2 log p / partial theta partial theta'] |_{theta_0} is positive definite, equivalently if the score partial log p / partial theta has full column rank in a neighborhood of $thet{a}_0$. For regime-switching models, evaluating $I(theta)$ requires marginalizing over regime histories — typically via the Hamilton filter for Gaussian observation models or a particle filter for nonlinear / non-Gaussian observation models. BC verify positive-definiteness of the numerical Fisher matrix at the MLE using a Sobol-based perturbation strategy.

The distinction between theoretical local identification (the Fisher matrix is full-rank) and finite-sample identification quality (the Fisher matrix is full-rank but extremely ill-conditioned, so confidence intervals are huge) is sharp in this setting. BC’s TSM-vs-SRM comparison is a quantification of finite-sample identification quality: both models are locally identified but the SRM (no long yields) has confidence intervals ~20x wider than the TSM.

Variants

• Theoretical local identification (Cochrane 2011, Komunjer-Ng 2011 for DSGE; BC 2013 Online Appendix for the regime-switching NK + Taylor rule case): rank of the Fisher information at the true parameter.
• Empirical / finite-sample identification quality: width of the finite-sample confidence interval for each parameter; conditioning of the Fisher matrix in a neighborhood of the MLE. BC report this via parametric bootstrap.
• Identification through observation menu enrichment: adding observables (long bond yields, in BC’s case) tightens identification quality without changing the theoretical identification status. This is the practically-relevant version of the question for macro-finance modeling.
• Identification of latent regime probabilities: separate from parameter identification — even if the parameters are identified, the smoothed regime probabilities $P(S_t|y_{1:T})$ may be diffuse if the observation menu is uninformative about which regime is active.

Comparison

• vs DSGE identification (Iskrev 2010, Komunjer-Ng 2011): BC-style identification adds the regime-history marginalization layer. The Fisher matrix is no longer available in closed form and must be evaluated numerically by perturbing the structural parameters and re-running the filter.
• vs reduced-form VAR identification (Sims 1980, Christiano-Eichenbaum- Evans 1999): the regime-switching reduced form has more parameters per regime, so the structural-to-reduced-form mapping is more constraining and identification is generically easier if the regime is observed. When the regime is latent, the practical situation is harder because the filter has to infer it.
• vs identification by sign restrictions / external instruments: the regime-switching framework gives a different angle on identification — it uses the time-series structure of regime switches rather than cross- equation sign restrictions or instrument exogeneity.

When to use

• Diagnosing whether a regime-switching macro model can in principle be estimated from the data on hand, before running an expensive global optimization.
• Justifying the inclusion of additional observation series (long yields, cap rates, credit spreads) by quantifying the resulting identification improvement.
• Designing simulation studies to measure the bias and variance of regime- parameter estimates as a function of the observation menu.

Known limitations

• Local identification is necessary but not sufficient — a model may be locally identified at the true parameter but globally non-identified (the likelihood may have multiple equally-good modes corresponding to relabeled regimes or alternative parameter configurations).
• Numerical Fisher evaluation is expensive. Perturbing each parameter and re-running the filter scales linearly in the parameter count and must be re-done at every candidate point during global optimization.
• Conditioning is regime-state-dependent. A Fisher matrix that is well-conditioned at the MLE may be ill-conditioned elsewhere, which matters for the inner-loop curvature seen by quasi-Newton optimizers. (Cf. the cre-asset-pricing-model Exp 11 finding that the forward-FD diagonal stencil is gradient-contaminated at non-critical points and the spec-level FD bug in rotation_scaling.json.)

Open problems

• General identifiability theorems for regime-switching VARs with structural cross-equation restrictions are not yet available; researchers rely on numerical verification per model.
• Identification of regime transition probabilities versus regime parameters — how informative is the data about persistence vs about per-regime levels?
• Identification under approximate pricing schemes: when bond prices are computed by a non-closed-form approximation (BC’s approximation, the Leather-Sagi Riccati / quadratic-pricing-factor scheme), does the identification depend on the approximation accuracy?

Key papers

• bikbov-chernov-monetary-policy-regimes-term-structure — provides the numerical local-identification verification for the NK + regime-switching Taylor rule + no-arbitrage TSM and quantifies the finite-sample identification gain from adding long yields (factor of 20 bias reduction)
• testing-indeterminacy-application-monetary-policy — discusses identification issues at the determinacy/indeterminacy boundary; shows that the parameter marking the boundary (psi_1) is identifiable under indeterminacy but NOT under determinacy
• dsge-model-based-estimation-new-keynesian — documents how NKPC identification depends on the observation menu and the rest of the model; illustrates the general identification challenges in macro estimation

My understanding

This is the load-bearing concept that justifies the architecture of any macro-finance regime-switching pipeline that uses asset prices to identify macro regimes. BC’s TSM-vs-SRM comparison is the closest published analog to the cre-asset-pricing-model project’s “Hamilton filter (no cap rates) vs RBPF (with cap rates)” comparison. In both cases, the asset-price observation menu does not change the theoretical identification status of the macro parameters (they are locally identified from macro data alone) but it changes the finite-sample identification quality dramatically. The ~1572-nat Hamilton-vs-RBPF gap in the CRE project is the same phenomenon as BC’s 20x bias reduction: long yields / cap rates encode forward-looking information about which macro regime is active, and the filter recovers that information by conditioning on them. Future CRE-project work that quantifies the bias reduction from cap rates explicitly (BC-style simulation study) would be a clean way to publish this finding.