Forward-looking Taylor rule with regime switching

Definition

A monetary policy reaction function in which the central bank sets the short rate as a forward-looking function of expected future inflation and current output, where the coefficients of the rule (response to expected inflation, response to output, intercept, and policy-inertia coefficient) switch with a discrete-state Markov chain. The leading example is

$r_t=m_r(s_t^m)+(1-rho(s_t^m))[alpha(s_t^m)E_{t}p{i}_{t+1}+beta(s_t^m)g_t]+rho(s_t^m)r_{t-1}+sigm{a}_r(s_t^d)ep{s}_t^r$

where $s_t^m$ is the systematic-policy regime variable and $s_t^d$ may optionally govern the volatility of the policy shock independently.

Intuition

Single-regime estimates of the Taylor rule (Clarida-Gali-Gertler 2000) split the U.S. sample into pre- and post-Volcker subsamples and find very different inflation responses ($alpha<1$ pre-Volcker, $alpha>1$ post-Volcker). Treating these as two distinct sub-samples implicitly assumes that agents never expected the regime to change. The regime-switching Taylor rule formalizes a different story: the same Federal Reserve actually switches between an “active” regime ($alpha>1$, satisfies the Taylor principle) and a “passive” regime ($alpha<1$), and rational agents anticipate the switches when forming expectations. This eliminates the sunspot-equilibrium pathology that would arise from a permanent passive regime, because the expected return to the active regime pins down inflation expectations.

Formal notation

In the BC formulation the systematic-policy regime $s_t^{m}in1,2$ governs $(m_r(s_t^m),rho(s_t^m),alpha(s_t^m),beta(s_t^m))$, and the discretion regime $s_t^{d}in1,2$ governs the policy-shock volatility $sigm{a}_r(s_t^d)$. Together with a third regime $s_t^v$ for private-sector shock volatilities, the compound state is $S_t=(s_t^v,s_t^m,s_t^d)$ with $2x2x2=8$ values. The rational-expectations solution of the system is a regime-switching VAR $x_t=mu(S_t)+Phi(S_t)x_{t-1}+Sigma(S_t)ep{s}_t$ in which $mu,Phi,Sigma$ are nonlinear functions of all the underlying structural parameters, including the regime-dependent Taylor-rule coefficients.

Variants

• Two-regime active/passive (BC 2013): the canonical specification, with $alpha(active)>1$ and $alpha(passive)<1$.
• Three or more policy regimes: in principle possible but not yet empirically motivated; identification of the third regime would be very weak.
• Drifting Taylor rule (Ang-Boivin-Dong-Loo-Kung 2011, Cogley-Sargent 2005, Primiceri 2005): instead of discrete regimes, the coefficients drift continuously. Loses the discrete interpretability but is easier to estimate on macro-only data. Bikbov-Chernov argue the discrete formulation is more natural and is identified better with bond yields.
• Inertia-only switching: holds $alpha,beta$ constant and lets only $rho$ switch. Special case of the BC form.
• Compound chains decoupled from volatility regimes: as in BC, the policy-coefficient regime $s_t^m$ is independent of the shock-volatility regime $s_t^v$ and the policy-shock-volatility regime $s_t^d$. This is the cleanest decomposition for identification.

Comparison

• vs single-regime Taylor rule (Clarida-Gali-Gertler 2000, Taylor 1993): the single-regime rule cannot accommodate sub-sample heterogeneity in $alpha$ without an exogenous sub-sample split. Regime-switching makes the switch endogenous and lets agents anticipate it.
• vs drifting-coefficient Taylor rule (Cogley-Sargent 2005, Primiceri 2005): the drift specification is more flexible but harder to interpret (“how active is policy today?” is a continuous question with no sharp answer). Regime-switching gives a discrete answer at the cost of imposing a finite mode count.
• vs DSGE with monetary regime (Amisano-Tristani 2009, Liu-Waggoner-Zha 2011): DSGE alternatives discipline the rule with cross-equation restrictions from a microfoundation. The BC empirical specification trades microfoundation for flexibility in the risk-premium block.

When to use

• Estimating monetary-policy regimes jointly with macro shocks and yield-curve data, where the goal is to date which regime was active when.
• Counterfactual policy analysis: holding shocks fixed at sample paths and asking “what if the active regime had prevailed throughout?”
• Building a structural no-arbitrage term-structure model that admits a regime-switching VAR reduced form (this concept is the structural input that produces the regime-switching-no-arbitrage-term-structure reduced form).

Known limitations

• Identification of $alpha,beta$ from macro data alone is weak — the Bikbov-Chernov SRM estimates have confidence intervals roughly 20x wider than the TSM estimates. Yield-curve information is essentially mandatory for precise inference on the policy regimes.
• Determinacy is not automatic. The compound system needs a generalised Cho-Moreno determinacy condition to rule out non-fundamental equilibria; this is a nontrivial check that has to be re-run at every parameter evaluation during likelihood maximization.
• Sample dependence. The active/passive dating depends on the sample endpoint (BC end in 2008, before the zero lower bound). The interpretation of $alpha<1$ may break down in a binding-ZLB sample.

Open problems

• Endogenizing the regime transition probabilities as functions of macro conditions (e.g., the Fed switches to an active regime conditional on inflation breaching a threshold). State-dependent transitions break tractability of the standard regime-switching affine pricing recursions.
• Disentangling persistent monetary shocks from regime switches in $rho$. Both can produce serial correlation in the short rate.
• Joint estimation with a binding zero lower bound or shadow short rate.

Key papers

• bikbov-chernov-monetary-policy-regimes-term-structure — introduces the active/passive specification with three independent regime chains and estimates it jointly with no-arbitrage bond prices
• regime-switches-agents-beliefs-post-world — estimates Hawk/Dove Taylor rule in a medium-scale MS-DSGE; shows beliefs about regime transitions matter for equilibrium outcomes
• testing-indeterminacy-application-monetary-policy — estimates the Taylor rule inflation response pre/post-Volcker allowing for indeterminacy; establishes the passive (psi_1 < 1) finding that motivates regime switching

My understanding

This is the structural input on the macro side of any structural-NK + regime-switching + no-arbitrage TSM pipeline. The cre-asset-pricing-model project carries an essentially identical Taylor rule (the $m_r,rho,alpha,beta$ parameters are all in the 55-parameter vector), with the difference being that the CRE compound chain is $2x2=4$ states (monetary policy x wage rigidity) rather than BC’s $2x2x2=8$. The CRE pipeline’s construct_structural_matrices_macro step is essentially BC’s structural form, and the transform_struct_to_rf step is BC’s reduced- form $(mu(S_t),Phi(S_t),Sigma(S_t))$ map. The Cho-Moreno forward method generalised to the regime-switching case (BC Online Appendix A) is what the CRE codebase implements as get_forward_solution_msre and check_determinancy_fmsre.