Long-Run Taylor Principle

Definition

The long-run Taylor principle (LRTP) is a determinacy condition for Markov-switching rational expectations (MSRE) models proposed by Davig and Leeper (2007). For an MSRE model $G amm a_{s_{t}} y_{t} = E (y_{t + 1} ∣ O m e g a_{t}) + P s i_{s_{t}} u_{t}$ with h regimes and transition matrix P, the LRTP states that all generalized eigenvalues of $(d ia g (G amm a_{1}, ..., G amm a_{h}), P k ro n I_{n})$ are greater than one in absolute value. Equivalently, if P is invertible, all eigenvalues of $(P^{- 1} k ro n I_{n}) d ia g (G amm a_{1}, ..., G amm a_{h})$ exceed unity.

Intuition

The standard Taylor principle says a central bank must respond more than one-for-one to inflation ( $a lp ha > 1$ ) for determinacy. The LRTP generalizes this to switching environments: even if one regime has $a lp ha < 1$ (passive), determinacy can hold if the other regime is sufficiently active and occurs frequently enough. The LRTP is a “weighted average” eigenvalue condition where the transition matrix P determines the weights. The intuition is that agents, knowing the active regime will return, coordinate expectations to prevent indeterminacy even during passive episodes.

Formal notation

For 2 regimes with transition matrix $P = [p_{ij}]$ , stack the MSRE into the DL representation:

diag(Gamma_1, Gamma_2) y_tilde_t = (P kron I_n) E(y_tilde_{t+1} | Omega_t^{-s}) + diag(Psi_1, Psi_2) u_t

where y_tilde_t = (y_{1t}, y_{2t})' is the 2n-dimensional augmented vector. If P is invertible, define $G amm a_{b} a r = (P^{- 1} k ro n I_{n}) d ia g (G amm a_{1}, G amm a_{2})$ . The LRTP requires: all eigenvalues of $G amm a_{b} a r$ exceed 1 in absolute value (or equivalently, all generalized eigenvalues of $(d ia g (G amm a_{1}, ..., G amm a_{h}), P k ro n I_{n})$ have modulus > 1).

Under the LRTP, the unique non-explosive solution to the DL representation is the regime-by-regime MSV solution $y_{t} = G amm a_{s_{t}}^{- 1} P s i_{s_{t}} u_{t}$ .

Variants

Standard Taylor principle (no switching): When h = 1, the LRTP reduces to “all eigenvalues of Gamma exceed 1,” the classical Blanchard-Kahn condition.
Two-regime case: With P invertible, the LRTP is equivalent to $∣ e i g ((P^{- 1} k ro n I_{n}) d ia g (G amm a_{1}, G amm a_{2})) ∣ > 1$ . The eigenvalues depend on the structural matrices and transition probabilities jointly.
Generalized eigenvalue formulation: For singular P, use the generalized eigenvalues of $(d ia g (G amm a_{1}, ..., G amm a_{h}), P k ro n I_{n})$ via QZ decomposition (Klein 2000, Sims 2001).

Comparison

Versus FWZ (2009) mean-square stability condition: The LRTP is only a necessary condition for determinacy (Cho and Moreno 2016; Farmer et al. 2010 provide an explicit counterexample). The FWZ condition ( $r_{s} i g ma (M_{1} (P h i_{i, j})) >= 1$ for all valid sunspot operators) is necessary AND sufficient. The gap exists because the LRTP checks first-moment stability, while MSS determinacy requires second-moment stability.

Versus MOD method (mod-solution-msre-determinacy): The MOD method provides necessary and sufficient conditions via two spectral conditions. The LRTP corresponds to checking a weaker condition. The set of parameters satisfying the LRTP strictly contains the FWZ/MOD determinacy region.

Versus Hayashi (2017) refinement: The LRTP becomes sufficient (not just necessary) if solutions are restricted to not depend on the previous regime. This is because the DL representation is valid only under this restriction.

When to use

As a quick, necessary screening condition: if the LRTP fails, determinacy definitely fails.
When the economic model implies solutions should not depend on lagged regimes (Hayashi 2017), the LRTP is sufficient.
As a pedagogical tool for understanding how transition probabilities interact with per-regime policy aggressiveness.

Known limitations

Not sufficient in general: The LRTP can hold while the FWZ/MOD condition fails, so the model can be indeterminate despite satisfying the LRTP. Farmer et al. (2010) provide an explicit NK model counterexample.
DL representation validity: The stacked system is only valid when conditional expectations do not depend on the current regime, which fails for general MSRE solutions (Hayashi 2017).
First-moment vs. second-moment: The LRTP ensures first-moment convergence but not second-moment convergence (MSS). Cho (2020) shows this gap is strictly non-empty for MSRE.

Open problems

Quantifying the size of the gap between the LRTP region and the MSS determinacy region for empirically calibrated models.
Whether the regime-independence restriction (Hayashi 2017) is economically reasonable in specific applications.
Extension of the LRTP concept to models with endogenous switching probabilities.

Key papers

long-run-taylor-principle-revisited — Hayashi (2017): identifies the restriction under which LRTP is sufficient.
understanding-markov-switching-rational-expectations-models — FWZ (2009): shows LRTP is not sufficient; provides the correct N&S conditions.
determinacy-classification-markov-switching-rational-expectations — Cho (2020): formalizes the gap as first-moment vs. second-moment stability.
monetary-policy-regime-switches-macroeconomic-dynamics — Foerster (2016): explores LRTP in a calibrated NK model with multiple switching dimensions.

My understanding

The LRTP is an important conceptual result but is not the right determinacy check for production use. For the CRE asset pricing project, the forward-method solver and absorbing-NBC proxy implement the FWZ/Cho conditions, which are strictly stronger. The LRTP would give false positives (pass parameters that are actually indeterminate). The Hayashi (2017) clarification is useful for understanding why the LRTP fails: solutions in the MSRE generically depend on the previous regime through the expectations mechanism, violating the DL representation’s validity condition.

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