Modeling inflation expectations in forward-looking interest rate and money growth rules

The paper asks whether rational expectations can be built directly into a small structural VAR — without first mapping the system to a fully specified DSGE model — and then uses that framework as a testbed to compare two candidate monetary policy indicators: the Wu and Xia (2016) shadow federal funds rate versus Divisia M4 nominal money growth. The motivation is a documented empirical problem with the federal funds rate (and its shadow-rate extension through the effective lower bound) in low-dimensional recursive VARs: it tends to produce “puzzling” impulse responses (e.g., inflation rising after a contractionary shock), a pattern the authors say is “amply documented in Ramey (2016) and references therein.” Standard fixes add more information variables to the VAR, but commodity prices, futures data, and similar variables have no accepted structural equation in the consensus New Keynesian model the authors want to preserve.

Identification does not rely on a Cholesky ordering or on any delayed-reaction exclusion restriction on the policy indicator — even though the indicator is still ordered first in the system. Instead, the authors use a rational-expectations forecast-revision relationship, E_t v_{t+j} − E_{t-1} v_{t+j} = S_v Ψ^j D e_t, to express the monetary policy shock as a specific linear combination of the reduced-form VAR’s residuals (e.g., ε_t^MP = e_t^i − (φ_π S_π Ψ^{h_π} D e_t)′ − (φ_y S_y Ψ^{h_y} D e_t)′ for the interest-rate rule, and an analogous expression for the money-growth rule). This recovers the structural shock without ever estimating the structural parameters themselves, and the approach is explicitly frequentist rather than Bayesian.

The model is a three-equation “consensus” New Keynesian system, estimated simultaneously: a forward-looking monetary policy rule (written either as a Taylor-type interest-rate rule or as a money-growth rule), a forward-looking IS equation, and a Phillips-curve AS equation. The policy rule responds to expected future inflation and output (E_t π_{t+h_π}, E_t y_{t+h_y}) rather than to current or lagged values, which is what lets the RE-SVAR be “forward-looking” without adding extra observable variables to the system. The paper treats this three-equation structure as a deliberate simplification: it is not modular in the way a standard reduced-form VAR is, since each additional variable would need its own fully specified structural equation.

Rather than estimating the policy rule’s response coefficients and expectation horizons, the paper conducts a “pseudo-calibration” grid search across 241,865 combinations of them — 61 values each of φ_π and φ_y (0 to 4, in 1/15 increments), 13 inflation-expectation horizons (h_π = 0,…,12 months), and 5 output-expectation horizons (h_y = 0,…,4 months) — producing a “cloud” of structural impulse responses rather than a single point estimate. A response is counted as a “puzzle” if, following an expansionary shock to the policy indicator, the output or inflation response goes negative at any point within the first year afterward; a “joint” puzzle is recorded if either variable (or both) misbehaves this way. The lag length is fixed at 8 (AIC-selected for the shadow-rate specification) across all specifications.

In the main October 1988-February 2020 sample, the shadow federal funds rate produces output puzzles in 98.68% of the 241,865 specifications and inflation puzzles in 99.13%, so only 0.87% (2,109 specifications) survive a no-joint-puzzle criterion; switching the policy indicator to Divisia M4 growth cuts output puzzles to 4.02% and inflation puzzles to 4.13%, so 95.85% (231,825 specifications) survive. The gap between the two indicators is thus very large across virtually the entire grid of plausible forward-looking policy-rule parameterizations, not just at a single calibrated point.

The finding holds up across a coarser 25,137-specification grid, a post-Global-Financial-Crisis effective-lower-bound sample (December 2008-February 2020), a long historical sample back to January 1967, a narrower Divisia M2 aggregate, and a PCE-based inflation measure — with one partial exception. Even in the shadow rate’s best-performing subsample (the post-GFC ELB period), its output and inflation puzzle rates remain high, at 72% and 93% respectively, versus low single digits for the Divisia aggregates in the corresponding comparisons. The exception is the long historical sample (1967-2020) under PCE inflation, where the output-puzzle advantage of Divisia over the shadow rate narrows considerably: 53.3% for the shadow rate versus 56.0% for DM4 and 47.9% for DM2. The authors also report that extending the sample through April 2022 yields qualitatively similar conclusions, but flag that COVID-period estimates may be severely distorted and so exclude that period from the main analysis.

Yes — longer inflation-expectation horizons substantially help Divisia M4 but barely move the shadow rate’s puzzle incidence. At a 6-month inflation-expectation horizon, 17,973 of 241,865 DM4 specifications are non-puzzling compared with only 195 for the shadow rate; at a 12-month horizon the gap widens further, to 18,430 non-puzzling DM4 specifications versus just 5 for the shadow rate. This pattern is consistent with the paper’s broader claim that Divisia money growth is better able to embed forward-looking policy behavior than the federal funds rate.

Adding the Gilchrist-Zakrajsek (2012) excess bond premium (EBP) as a fourth variable, using Divisia M4 as the policy indicator over a July 1979-February 2020 sample (dictated by EBP data availability), still yields a high no-joint-puzzle survival rate of 81.45% (196,988 of 241,865 specifications). In this extended system, median responses to IS shocks show industrial production and CPI both rising, while median responses to AS shocks show industrial production falling and CPI rising — patterns the authors describe as broadly consistent with textbook New Keynesian predictions. The EBP’s own structural equation, however, is specified in a deliberately unrestricted form, since the authors note there is “a lack of theoretical foundation for a law of motion” for the excess bond premium.

The authors flag four main limitations: the RE-SVAR is non-modular, its validity is conditional on the underlying three-equation theory being sensible, its frequentist grid-search approach does not report cloud-median responses (following Inoue and Kilian’s 2022 critique of doing so from a Bayesian perspective), and the excess bond premium’s structural equation in the four-variable extension is underspecified for lack of theory. On non-modularity, they write that adding variables to the RE-SVAR requires a fully specified structural equation for each one, unlike a standard recursive VAR where variables can simply be appended — an explicit disadvantage relative to modular n-variable recursive SVARs. On the theoretical dependency, they state plainly that “if the theoretical model we construct our RE-SVAR from is not sensible, it renders the whole enterprise a non-starter.” The sample also excludes the COVID period, on the grounds that pandemic-era estimates may be severely distorted even though results extending through April 2022 are reported as qualitatively similar.