Interest Rate Pegs and the Reversal Puzzle: On the Role of Anticipation

Layer 1: Overview

This paper revisits the “reversal puzzle” — the counterintuitive result, first documented by Carlstrom, Fuerst and Paustian (CFP, 2015), that in standard New Keynesian models the effect of forward guidance (technically implemented as a perfectly anticipated interest rate peg) can switch from expansionary to contractionary as the duration of the peg increases. The authors’ central claim is that the appearance of the puzzle hinges on agents’ degree of anticipation of the peg, and they examine three polar/intermediate cases: perfect anticipation, no anticipation, and imperfect anticipation.

Model and setup: The laboratory is the medium-scale DSGE model of Carlstrom, Fuerst and Paustian (2017), which features funding constraints and market segmentation (only financial intermediaries can hold long-term public and private bonds, subject to a leverage constraint from a hold-up problem and net-worth adjustment costs; households face a loan-in-advance constraint on investment). These frictions break Wallace neutrality so that QE has real and inflationary effects. The model has standard New Keynesian features: habit consumption, monopolistic competition, Erceg-Henderson-Levin (2000) sticky prices and wages with Christiano-Eichenbaum-Evans (2005) indexation, investment adjustment costs, and a Taylor rule with interest-rate smoothing. It is estimated with Bayesian methods on eight euro-area observables over 1998Q1-2013Q4, with a subset of parameters calibrated to CFP (β=0.99, capital share α=0.33, depreciation δ=0.025, price/wage markup elasticities ε_p=ε_w=5, steady-state leverage 6). The initial impulse in all experiments is the launch of a QE programme, modeled as a single shock to an AR(2) process for the real market value of long-term bonds (purchases last 6 quarters). Without a peg, QE raises inflation (the orthodox result).

Main findings: (1) Perfect anticipation (perfect-foresight solution): reversals are a robust phenomenon. As peg duration P rises, the inflation response first grows and then explodes near a critical value; in the baseline this critical value is eight quarters. For P of 9-14 quarters inflation reverses sign (deflation instead of inflation); for 15-23 quarters the sign flips back to positive; for 24-50 quarters it turns negative again. Thus output and inflation responses oscillate with P. The authors give analytical intuition via the forward solution: complex unstable eigenvalues of matrix J, written in polar form, mean powers of J enter the solution as trigonometric functions of P (de Moivre’s formula), producing the oscillation. (2) No anticipation (extended-path method, agents expect E_t[ε_{t+n}]=0 each period and are “surprised”): the reversal puzzle is absent for all durations 0-50; the initial inflation response is always positive, because powers of J no longer enter the solution. (3) Imperfect anticipation (Markov-switching model solved with Maih’s 2015 RISE toolbox): two regimes — Taylor rule (regime 1) vs. peg (regime 2, where ρ=τ_Π=τ_y=0). Agents know transition probabilities, so the frequency F2 and average duration AD2 of the peg are known; frequency is interpreted as the degree of anticipation. Generalized impulse responses (50,000 draws) for average durations of 4, 11.5, 19, 37, 50 quarters and frequencies of 10%, 15%, 20%, 30%, 40%, 50% show: at the empirically relevant frequency of 10% (post-WWII US ZLB experience, ~7 years in 73) and at 15% and 20%, no reversals occur for any average duration. Reversals appear only at implausibly high frequencies: at 30% only for AD2=4 quarters; at 40% for AD2=4, 11.5, 19 quarters; at 50% for all average durations.

Implications: A Markov-switching treatment of pegs/ZLB delivers more plausible model outcomes than perfect foresight and is a promising tool for policy simulations to avoid the reversal pathology, since under realistic anticipation forward guidance is less powerful and reversals do not arise.

Layer 2: Deep Dive

What exactly is the reversal puzzle and where did it originate?

It is the counterintuitive result that the macroeconomic effect of forward guidance — implemented technically as a perfectly anticipated interest rate peg — can switch from expansionary to contractionary depending on the peg’s duration, producing sizeable deflation instead of inflation. Carlstrom, Fuerst and Paustian (2015) first analyzed and named it. Similar sign reversals are noted in Lindé-Smets-Wouters (2016) and Binning-Maih (2017).

What is the identification/solution strategy for each anticipation case, and what distinguishes them?

Perfect anticipation: perfect-foresight (deterministic) solution where the peg is implemented via binary dummy shocks (ε^TR in {0,1}) set to one for P pre-announced quarters; agents know all future ε_{t+n}, so powers of the eigenvalue matrix J enter the forward solution. No anticipation: the extended-path method, running a deterministic simulation each period with the previous period as initial condition and steady state as terminal condition, imposing E_t(ε_{t+n})=0 — agents are surprised the peg continues, so powers of J drop out. Imperfect anticipation: a Markov-switching framework (Maih 2015) with non-zero transition probabilities between a Taylor-rule regime and a peg regime; the peg is a recurring stochastic event whose frequency and average duration are known to agents.

What is the formal mechanism for the oscillation under perfect foresight?

The forward-looking (explosive) variables solve as w2,t = -E_t{Σ J^{n-1} Ω22^{-1} Q2 Φ ε_{t+n}}. Some diagonal elements of J (the unstable generalized eigenvalues) are complex; in polar form z_jj = r(cos φ + i sin φ), and by de Moivre z_jj^k = r^k(cos kφ + i sin kφ) for k=0,…,P-1. Because nonzero anticipated future shocks bring in powers of J, the solution involves trigonometric functions of the peg length P, so simulations approach an asymptote, switch sign, approach another asymptote, switch again — hence oscillation as P grows.

Why are reversals absent under no anticipation, given the same complex eigenvalues?

Complex eigenvalues are only a necessary, not sufficient, condition. Under no anticipation E_t(ε_{t+n})=0, so the solution for w2,t no longer depends on powers of J; the simulations do not ‘move along’ the trigonometric functions, so the explosive complex eigenvalues cannot induce cyclical/explosive effects. A sufficient degree of anticipation is necessary for reversals to occur.

How are frequency and average duration of the peg pinned down in the Markov-switching model?

p12 is the transition probability from Taylor regime (1) to peg regime (2); p21 from 2 to 1. Average peg duration AD2 = 1/p21. Frequency F2 = AD2/(AD1+AD2) with AD1 = 1/p12. Table 2 maps the (AD2, F2) grid to the implied p12, p21. The authors check the mean-square-stability condition for each calibration before computing generalized impulse responses from 50,000 draws.

What is the empirically relevant peg frequency and how is it justified?

About 10%, based on the post-WWII US zero-lower-bound experience (7 years at the ZLB out of 73 years), the same value used by Dordal-i-Carreras, Coibion, Gorodnichenko and Wieland (2016). The paper stresses that even at double this value (20%) reversals are absent for all average durations considered.

How does the reversal pattern under imperfect anticipation differ from perfect anticipation?

The patterns differ. Under perfect foresight the lowest sub-range of durations (0-8 quarters) shows no reversal, whereas under imperfect anticipation at frequencies of 30% and 40% a reversal occurs for the lowest average duration (4 quarters). Reversals also appear ‘grouped’ across adjacent average durations. The regime-specific IRFs explain this: given the peg regime (regime 2), higher average durations lead to reversals at low frequencies; given the no-peg regime (regime 1), only frequencies of 30%+ permit reversals and there lower average durations reverse. The GIRF blends both regimes, so its resemblance to a regime’s IRF depends on how frequently that regime occurs.

What robustness checks are performed?

An extensive grid search (Appendix D) varies each structural parameter one at a time around benchmark values under perfect foresight. Reducing forward-lookingness (lower β) or raising habit, changing depreciation δ or investment adjustment cost ψi, varying the Calvo price/wage parameters (θp, θw) and indexation (ιp, ιw), and varying Taylor-rule coefficients (ρ, τπ, τy) all only change the peg duration required for the reversal to appear, not its existence. Notably, even shutting down price and wage indexation jointly (ιp=ιw=0) does not eliminate reversals in this medium-scale model, because other endogenous state variables (capital, wages, net worth) generate complex eigenvalues. More aggressive inflation stabilization (higher τπ) or longer Calvo durations (>0.9) require a longer peg before reversal appears.

How does this paper relate to and differ from closely related prior work?

It is complementary to CFP (2015), who showed reversals require complex eigenvalues from endogenous states and that switching from sticky-price to sticky-information removes the puzzle; this paper instead goes beyond perfect foresight to show the degree of anticipation is key. It differs from De Graeve-Ilbas-Wouters (2014), Maliar-Taylor (2019), and Bundick-Smith (2020), who rely on realistic calibration to weaken forward guidance; here the resolution comes from realistic modeling of expectations. Unlike de Groot and Mazelis (2020) — who modify the linearized solution so agents are fully aware of the peg — the Markov-switching approach treats the peg as a recurring stochastic event. Methodologically closest is Chen (2017), who compares perfect-foresight and Markov-switching implementations of the ZLB; consistent with her, the authors find Markov-switching delivers more plausible outcomes.

What are the policy implications and their scope conditions?

Because the ZLB and forward guidance must be accounted for in model simulations, and these are often modeled as interest-rate pegs, policy evaluations risk spurious reversals. The Markov-switching approach circumvents this pathology and yields qualitatively plausible outcomes. Scope conditions: the result holds for empirically relevant peg frequencies (up to ~20%, double the 10% benchmark) across average durations of 4-50 quarters; reversals can still arise but only under extreme, arguably implausible frequencies (30%+). The conclusions are derived within the CFP (2017) segmented-markets model estimated on euro-area data, with QE as the initiating impulse.

How is the QE programme modeled and what is its transmission?

QE is a single shock to a persistent AR(2) process for the real market value of long-term bonds held by the public, generating an inverse hump shape with purchases lasting 6 quarters before gradual return to steady state. Transmission: lower bond supply to FIs raises bond prices and lowers yield-to-maturity and the term premium; FI net worth and leverage fall but net-worth mobility is limited by adjustment costs, so FIs raise demand for (perfect-substitute) investment bonds, raising their price, relaxing households’ loan-in-advance constraint, boosting investment, output, and inflation; monetary policy then raises the policy rate under the Taylor rule.

Are there caveats about the no-anticipation case as a ‘solution’?

Yes. The authors state the no-anticipation case is obviously not a suitable solution to the puzzle — it is an unrealistic polar case (agents are surprised every period). Both polar cases (perfect and no anticipation) are unrealistic, which motivates the imperfect-anticipation Markov-switching analysis as the realistic middle ground.

Key Concepts

Reversal puzzle: The counterintuitive switching of forward guidance’s effect from expansionary to contractionary (deflation rather than inflation) as the duration of a perfectly anticipated interest rate peg increases; in this paper, the inflation response oscillates in sign across peg durations.

Degree of anticipation: The extent to which agents expect a future interest rate peg. The paper’s central organizing concept: in the stochastic case it is operationalized by the frequency of the peg regime, since a higher frequency makes agents consider a peg more likely.

Interest rate peg: A regime in which the central bank abandons the Taylor rule and holds the nominal short-term rate fixed for a period — the technical implementation of forward guidance and the ZLB in this analysis.

Imperfect anticipation (Markov-switching implementation): A scenario where agents attach non-zero transition probabilities to entering and exiting a recurring peg regime, so individual peg episodes are stochastic in occurrence and duration but their frequency and average duration are known.

Frequency of the peg (F2): The long-run share of time the economy spends in the peg regime, F2 = AD2/(AD1+AD2); interpreted as the degree of anticipation, with ~10% taken as the empirically relevant post-WWII US ZLB value.

Complex eigenvalues / forward solution: Unstable generalized eigenvalues of the solution matrix J that are complex-valued; their polar-form powers introduce trigonometric functions of peg length P into the forward solution — a necessary but not sufficient condition for reversals, which require sufficient anticipation to activate.

Wallace neutrality breakdown: The property, induced by FI funding constraints and bond-market segmentation in the CFP (2017) model, that asset purchases (QE) affect real activity and inflation rather than being neutral as in the standard New Keynesian model.