Does the conclusion that Divisia M4 outperforms the federal funds rate depend on sample, price index, or aggregate choice?

No. Chen and Valcarcel (2025) verify the result across three samples (1967–2020, 1988–2020, 2008–2020), two price indexes (CPI and PCE), and two Divisia aggregates (M2 and M4). The Wu-Xia shadow rate produces 72–99% output puzzles and 93–99% price puzzles across all 12 combinations; Divisia M4 produces 2–24% output puzzles and 2–7% price puzzles (with one ambiguous cell in the historical PCE sample where both indicators struggle). The pattern is consistent with Keating et al. (2019) on pre/post-GFC stability and with Chen and Valcarcel (2024) on the stability of Divisia money demand.

Modeling Inflation Expectations in Forward-Looking Interest Rate and Money Growth Rules

Q: Why does the federal funds rate fail as a monetary policy indicator in low-dimensional SVARs?

It generates output and price puzzles across virtually the entire parameter space once forward-looking rational expectations are enforced. Chen and Valcarcel (2025) find 99.13% price puzzles and 98.68% output puzzles across 241,865 parameter combinations in the 1988–2020 sample using the Wu-Xia shadow federal funds rate , with only 2,109 combinations producing non-puzzling responses. The pattern is robust across three samples, both CPI and PCE, and aligns with prior methodology-independent findings in Chen and Valcarcel (2021) using a TVP-FAVAR.

Q: Why does a forward-looking money growth rule with Divisia M4 produce sensible responses?

Because broad Divisia aggregates internalize substitution effects across monetary assets that simple-sum measures and short-rate indicators discard, and the growth rate of Divisia M4 carries information through the effective lower bound. Chen and Valcarcel (2025) find 95.85% no-joint-puzzle responses with Divisia M4 in the 1988–2020 sample — 231,825 surviving IRFs out of 241,865. This extends the evidence from Keating et al. (2019) and Belongia and Ireland (2014) into a fully rational-expectations framework, with the underlying stability of Divisia money demand separately established in Chen and Valcarcel (2024) .

Q: How should researchers handle forward-looking horizons in the policy reaction function?

Iterate over them rather than estimate them, and report response clouds rather than single median IRFs. Chen and Valcarcel (2025) use a grid of h_pi in 0–12 months and h_y in 0–5 months combined with phi_pi and phi_y each in increments of 1/15, generating 241,865 distinct SVAR specifications. The motivation traces to Batini and Haldane (1999) on the flexibility of forecast-targeting rules, and the reporting practice to Inoue and Kilian (2022) on the limits of median response summaries.

Q: What is the non-modularity of the RE-SVAR approach?

Non-modularity means every added variable requires its own fully specified structural equation — you cannot append commodity prices or factors to improve fit. Chen and Valcarcel (2025) argue this is a feature: identification validity rests on the theoretical construct itself, not on the restriction scheme. Section 7 of the paper demonstrates extension to a four-variable system with the Gilchrist-Zakrajšek (2012) excess bond premium, which requires a sequential IV procedure and two additional restrictions for global identification per Rubio-Ramírez, Waggoner and Zha (2010) .

Q: How should one interpret response clouds from 241,865 SVARs rather than a single impulse response function?

As a joint distribution over structural IRFs, with the no-joint-puzzle share as the primary summary statistic. Inoue and Kilian (2022) argue that median Bayesian IRFs can mislead when the joint distribution contains sign reversals. Chen and Valcarcel (2025) report the survival share directly (95.85% for Divisia M4 vs. 0.87% for the shadow federal funds rate in the modern sample), slice the cloud by horizon or by policy coefficient, and avoid median responses of the full cloud. The framework connects naturally to set-identification in Rubio-Ramírez, Waggoner and Zha (2010) .

Wed, 15 Jan 2025 00:00:00 +0000

A low-dimensional SVAR can directly embed rational expectations — and once it does, a forward-looking money growth rule with Divisia M4 delivers puzzle-free monetary transmission where the federal funds rate fails across 99% of specifications

Chen and Valcarcel (2025) propose the RE-SVAR: an internal instrumental-variable procedure that directly embeds forward-looking rational expectations into a three-variable consensus AS–IS–MP system. Searching over 241,865 forward-horizon and policy-coefficient combinations, the Wu-Xia shadow federal funds rate generates price puzzles in 99.13% of specifications; Divisia M4 as the policy indicator delivers puzzle-free responses in 95.85%.

Five named concepts anchored in this paper

RE-SVAR: Rational expectations-augmented structural vector autoregression. A low-dimensional SVAR that directly embeds forward-looking rational expectations via an internal instrumental-variable procedure, without mapping from a DSGE.
Response clouds (cloud of structural IRFs): The set of 241,865 impulse responses generated by grid-searching forward-looking horizons and policy-rule coefficients, with each combination producing a separate realization of the SVAR.
No-joint-puzzle response: The survival criterion: an IRF that avoids both the output puzzle and the price puzzle within the first year post-shock.
Low-dimensional forward-lookingness: The paper's methodological claim: forward-looking behavior can be modeled inside a three-variable AS–IS–MP consensus system without appending factors or unobservables.
Non-modularity of RE-SVAR: The property that each added variable requires a fully specified structural equation; you cannot simply append commodity prices, Greenbook forecasts, or factors without a theoretical construct.

How can rational expectations be embedded directly into a low-dimensional SVAR without mapping from a DSGE?

Through an instrumental-variable procedure internal to the SVAR that exploits the forecast-revision identity implied by rational expectations, applied to a fully specified consensus AS–IS–MP system.

The standard options have been unsatisfactory. Backward-looking recursive SVARs, in the tradition of Christiano, Eichenbaum and Evans's Handbook of Macroeconomics chapter, impose a delayed-reaction assumption through Cholesky ordering but struggle to accommodate forward-lookingness. The mapping approach — finding conditions under which a DSGE can be represented as a VAR or VARMA — requires lag truncation or dimension reduction that defeats the point. DSGEs themselves are RE-consistent but come with laws of motion for unobservables that constrain the parameter space in ways the textbook consensus model does not require.

Chen and Valcarcel (2025) propose a third path — the RE-SVAR — that stays within a three-variable consensus model and derives the structural monetary policy shock as a linear combination of reduced-form residuals using the forecast-revision identity. Taking a stand on the policy-rule coefficients and horizons (rather than estimating them) produces a unique structural shock for each parameter combination — a pseudo-calibration that yields response clouds rather than a single IRF.

Why this matters operationally:

No Cholesky ordering and no delayed-reaction assumption.
No unobserved state variables or moving-average components.
The three-variable system remains directly comparable to the textbook AS–IS–MP model, with each equation having a structural interpretation.
Forward-looking horizons (h_π, h_y) are parameters you iterate over, not constants you estimate.

The trade-off: the method is not modular. Adding a variable requires a fully specified structural equation for it — which the paper demonstrates for the Gilchrist-Zakrajšek excess bond premium in Section 7 but which rules out ad hoc inclusion of commodity prices or Greenbook forecasts.

RE-SVAR vs. Standard SVAR Approaches to Monetary Policy Identification
Dimension	Recursive SVAR (delayed reaction)	FAVAR / Factor-augmented	Proxy SVAR (external instruments)	RE-SVAR (Chen & Valcarcel 2025)
Core identification	Cholesky ordering with policy indicator ordered after economic activity; imposes delayed reaction.	Large information set spanned by principal-component factors; recursive identification within the factor VAR.	High-frequency monetary surprises used as external instruments for structural policy shock.	Forecast-revision identity applied to a fully specified AS–IS–MP system; shock is a linear combination of reduced-form residuals.
Key references	Christiano, Eichenbaum & Evans (1999), Hanson (2004)	Bernanke, Boivin & Eliasz (2005), Boivin, Kiley & Mishkin (2010)	Gertler & Karadi (2015), Kuttner (2001)	Chen & Valcarcel (2025); foundations in Blanchard & Perotti (2002)
Handles forward-looking expectations	No — inherently backward-looking; requires appending forward-looking variables.	Partially — factors can proxy for forward-looking information but lack structural interpretation.	Implicitly — high-frequency surprises embed forward-looking market expectations.	Yes — forward horizons h_π, h_y are parameters of the policy rule; RE restriction is internal.
Dimensionality	Small-to-medium (typically 6–8 variables); grows with information-set fixes.	High (100+ variables summarized by 3–5 factors).	Small-to-medium, augmented by external instrument.	Low (3–4 variables); strictly bounded by the number of structural equations available.
Modularity	High — append variables as needed.	High — scale factors up or down.	Medium — add instruments; adding endogenous variables remains standard.	None — each added variable requires its own structural equation.
Identification validity rests on	Restriction scheme (Cholesky ordering).	Approximating the true information set with a factor structure.	Validity and relevance of the external instrument.	Theoretical credibility of the consensus AS–IS–MP model itself.
Price puzzle incidence in low-dimensional form	Pervasive without commodity-price augmentation; still present even with it in many samples.	Generally resolved, but Boivin, Kiley & Mishkin (2010) show sensitivity to specification.	Generally resolved at short horizons; longer-horizon responses vary.	Resolved with Divisia M4 (<4%); unresolved with Wu-Xia shadow rate (>98%).
Works through the effective lower bound	Only with shadow-rate construction (e.g., Wu & Xia 2016).	Yes, via shadow rate or factors.	Yes, via high-frequency surprises.	Yes — Divisia growth rate is unbounded; Keating et al. (2019) document pre/post-GFC stability.
Named concept	Block-recursive identification	Information-sufficient factor identification	High-frequency external-instrument identification	RE-SVAR · Response clouds · Non-modularity (Chen & Valcarcel 2025)

Why does the federal funds rate fail as a monetary policy indicator in low-dimensional SVARs?

It generates the price puzzle and the output puzzle across virtually the entire parameter space once forward-looking rational expectations are enforced. In Chen and Valcarcel's modern sample, 99.13% of 241,865 parameter combinations produce at least one puzzling response within the first year after a federal funds rate shock.

The price puzzle — first documented by Eichenbaum (1992), who noted that the price level rises rather than falls after a contractionary interest rate shock — has been treated for three decades as a problem of information insufficiency. The standard fix, from Christiano, Eichenbaum and Evans (1999), is to augment the VAR with commodity prices. Hanson (2004) showed this fix is unreliable: many alternative indicators with strong inflation-forecasting power fail to resolve the puzzle, and the puzzle is particularly resistant in pre-1979 samples.

Chen and Valcarcel (2025) reveal that once rational expectations are embedded directly and the researcher searches over the full space of forward-looking policy-rule parameters, the price puzzle is not an incidental feature of particular specifications — it is the dominant outcome. Using the Wu and Xia (2016) shadow federal funds rate to span the effective lower bound period, the paper finds 98.68% output puzzles and 99.13% price puzzles across 241,865 realizations in the 1988–2020 sample. Only 2,109 combinations — less than 1% — produce non-puzzling responses in both industrial production and inflation. Chen and Valcarcel (2021) reached a similar conclusion with an entirely different methodology (TVP-FAVAR), suggesting the federal funds rate's weakness as a low-dimensional policy indicator is methodology-independent.

The interpretation: absent an augmented information set — factors à la Bernanke, Boivin and Eliasz's FAVAR, futures data, or Greenbook forecasts — the federal funds rate cannot carry the forward-looking information content required to identify monetary policy shocks in a consensus three-variable system.

Why does a forward-looking money growth rule with Divisia M4 produce sensible responses where the federal funds rate fails?

Because broad Divisia monetary aggregates internalize substitution effects across monetary assets that simple-sum measures and short-rate indicators discard — and because the growth rate of Divisia M4 is not bound to zero, it carries information through the effective lower bound period that the federal funds rate cannot.

The theoretical case for Divisia over simple-sum M2, established by Barnett (1980) with the derivation of the monetary services index from Diewert's index theory and reinforced by Belongia and Ireland (2014) in their New Keynesian formalization of the Barnett critique, is that a CES aggregate of interest-bearing and non-interest-bearing assets tracks the true monetary aggregate almost perfectly to second order. Keating, Kelly, Smith and Valcarcel (2019) show in a block-recursive SVAR that Divisia M4 resolves the price puzzle for both pre- and post-GFC samples, while Belongia and Ireland (2022) argue theoretically that a money growth rule responding to inflation and output gradually delivers stabilization comparable to an estimated Taylor rule.

Chen and Valcarcel (2025) extend this evidence into a fully forward-looking rational-expectations framework. In the same 1988–2020 sample where the shadow federal funds rate generates 99% puzzles, Divisia M4 as the policy indicator produces 95.85% no-joint-puzzle responses — 231,825 surviving IRFs out of 241,865. The output-puzzle rate drops to 4.02% and the price-puzzle rate to 4.13%. The pattern holds across CPI and PCE price indexes and across historical (1967–2020), modern (1988–2020), and post-ELB (2008–2020) samples, with narrower Divisia M2 performing comparably to the broader Divisia M4. Notably, at the longest expectation horizon considered (h_π = 12 months), fewer than 1% of Divisia specifications exhibit puzzles while 99.9% of shadow-rate specifications do.

Why the asymmetry is structural and not merely empirical:

Divisia M4 reflects substitution across a broader set of monetary assets than the segmented federal funds market, giving it richer information content per unit of variation.
The money growth rule remains operational through the ELB period — where even the Wu-Xia shadow rate is a constructed object — which matters for samples that straddle 2008–2015.
The long-run relationship between Divisia aggregates and economic activity is stable (Chen and Valcarcel 2024), consistent with its role as a forward-looking policy indicator.

How should researchers handle forward-looking horizons in the policy reaction function?

Iterate over them rather than estimate them — and report response clouds for different horizon choices rather than a single median IRF. Chen and Valcarcel's grid of h_π ∈ {0, 1, …, 12} months for inflation and h_y ∈ {0, 1, …, 5} months for output, combined with φ_π, φ_y ∈ [0, 4] in increments of 1/15, generates 241,865 distinct SVAR specifications from a single underlying model.

The theoretical motivation comes from Batini and Haldane (1999), who argued that forward-looking rules with flexibility over both the forecast horizon and the feedback parameter are the right analog to Svensson's flexible inflation-forecast-targeting rule. Estimating h_π and h_y requires either Fed-internal data (Greenbook forecasts, as in Orphanides (2001) on real-time monetary policy rules) or heavy structural assumptions.

Chen and Valcarcel (2025) exploit this flexibility to show that the qualitative conclusion — Divisia dominates the shadow federal funds rate in producing sensible responses — is invariant to which horizon assumption you make. More specifically, for the money growth specification the number of no-joint-puzzle responses increases with the horizon (from 88.4% at h_π = 1 to 99.1% at h_π = 12), while for the federal funds rate specification it decreases (from 2.1% at h_π = 1 to 0.03% at h_π = 12). The two indicators thus differ not only in level but in how they behave as forward-lookingness intensifies.

Practical implication: any paper reporting a single IRF from a forward-looking policy rule is reporting one realization from a response cloud. The distributional features matter because Inoue and Kilian (2022) argue against reporting median responses when the joint distribution of IRFs contains the policy-relevant information.

What is the non-modularity of the RE-SVAR approach, and why does it matter for applied work?

Non-modularity means that every variable added to the system requires its own fully specified structural equation — you cannot simply append variables to improve fit, as is routine in standard empirical VARs. This is the principal cost of the RE-SVAR framework, and the main reason it constrains itself to low-dimensional consensus models.

The contrast with standard practice is sharp. Standard VAR specifications treat the information set as expandable: Christiano, Eichenbaum and Evans (1999) add commodity prices, Bernanke, Boivin and Eliasz (2005) add 120+ factors in their FAVAR, Hanson (2004) surveys numerous alternative predictors, and Gertler and Karadi (2015) augment with high-frequency monetary surprises as external instruments. Each addition is defensible statistically — more information should improve identification — but often lacks a theoretical construct within the consensus macroeconomic model.

Chen and Valcarcel (2025) argue the non-modularity is a feature, not a bug: the identification validity depends on the suitability of the underlying theoretical structure, not on the restriction scheme. Section 7 of the paper demonstrates how to add the Gilchrist-Zakrajšek (2012) excess bond premium as a fourth variable — but this requires writing out a fourth structural equation, establishing a sequential IV procedure for each additional parameter, and verifying that the Rubio-Ramírez, Waggoner and Zha (2010) rank condition for global identification is satisfied.

Implication for applied researchers:

If your question requires adding commodity prices, Greenbook forecasts, or a factor for forward-looking expectations, the RE-SVAR is not the tool; a standard VAR with external instruments or a FAVAR is.
If your question is about whether the consensus AS–IS–MP model can carry forward-looking dynamics on its own, the RE-SVAR is specifically designed for that test, and the non-modularity guarantees you cannot cheat by adding variables with no structural role.

How should one interpret response clouds from 241,865 SVARs rather than a single impulse response function?

As a joint distribution over structural IRFs, where each point in the parameter grid is a distinct identification of the same underlying model. The cloud is the object of inference; any single IRF is a point in it.

The approach parallels the Bayesian posterior-over-impulse-responses literature but uses a frequentist grid rather than posterior draws. Inoue and Kilian (2022) argue that summarizing Bayesian VAR inference with median responses is misleading when the joint distribution contains features — such as multi-modality or sign reversals across plausible parameter regions — that a median collapses.

Chen and Valcarcel (2025) handle this in three ways:

Report the no-joint-puzzle share directly. The survival rate — 95.85% for Divisia M4, 0.87% for the shadow federal funds rate in the modern sample — is itself a summary statistic that preserves the joint distribution's information without collapsing to a point estimate.
Slice the cloud by horizon. Fixing h_π at different values (1, 3, 6, 12 months) and reporting median responses within each slice reveals how forward-lookingness interacts with indicator choice.
Slice by policy coefficient. Fixing φ_π = 1.5 (the Taylor (1993) classic value) and reporting median responses reveals which subsets of the cloud correspond to empirically relevant parameter choices.

This treatment provides a natural connection to set-identified SVAR literature (Rubio-Ramírez, Waggoner and Zha 2010) and to sign-restriction approaches such as Uhlig (2005): the response cloud is the identified set under the rational-expectations restriction combined with the parameter grid, and the no-joint-puzzle responses are the subset satisfying textbook sign restrictions as well.

Does the conclusion that Divisia M4 outperforms the federal funds rate depend on the specific sample, price index, or Divisia aggregate?

No — the dominance of Divisia money over the shadow federal funds rate is robust across three samples (1967–2020, 1988–2020, 2008–2020), two price indexes (CPI and PCE), and two Divisia aggregates (M2 and M4).

Chen and Valcarcel (2025) report Table 1 across all 12 combinations. A condensed summary:

Sample	Price	Wu-Xia FFR output puzzle	Wu-Xia FFR price puzzle	DM4 output puzzle	DM4 price puzzle
1988–2020	CPI	99.5%	99.4%	3.7%	3.8%
1988–2020	PCE	99.6%	99.4%	23.7%	4.2%
2008–2020	CPI	72.0%	93.0%	2.4%	1.6%
2008–2020	PCE	90.8%	96.1%	9.1%	5.1%
1967–2020	CPI	98.9%	98.8%	3.9%	4.1%
1967–2020	PCE	53.3%	94.7%	56.0%	7.4%

The single ambiguous cell is the 1967–2020 sample with PCE inflation, where both indicators show elevated output-puzzle rates — but even there, Divisia's price-puzzle rate (7.4%) is an order of magnitude below the shadow rate's (94.7%). The robustness is consistent with Keating et al. (2019), who find similar pre/post-GFC stability of money growth rules in a block-recursive setting. The narrower Divisia M2 performs comparably to Divisia M4 across all cells, consistent with Kelly, Barnett and Keating (2011) on the liquidity effects of broader Divisia aggregates. Chen and Valcarcel (2024) separately establish that the underlying money-demand relationships for Divisia aggregates are cointegrated and stable in modern samples, reinforcing that the SVAR results are not driven by spurious regression dynamics.

How do I implement the RE-SVAR procedure on my own data?

The implementation has five steps once you have a balanced panel of inflation, output, and a policy indicator: write down the AS–IS–MP consensus model with the forward-looking horizons you want to test, derive the forecast-revision identity for each equation, set up the IV procedure that yields the structural policy shock as a linear combination of reduced-form residuals, grid-search over the policy-rule parameters (φ_π, φ_y) and horizons (h_π, h_y), and compute impulse responses for each grid point. Chen and Valcarcel (2025) provide the full derivation in Sections 3–4.

The non-trivial step is the IV procedure itself. The forward-looking AS–IS–MP system implies a contemporaneous restriction between the structural policy shock and the reduced-form residuals through the rational-expectations forecast-revision identity. The structural shock for each grid point is a known linear combination of residuals — no estimation needed for the contemporaneous identification; only the lag dynamics need a reduced-form VAR.

Compute budget: With (h_π ∈ {0…12}) × (h_y ∈ {0…5}) × (φ_π ∈ [0,4] at 1/15) × (φ_y ∈ [0,4] at 1/15) = 241,865 specifications. Each grid point requires only matrix algebra applied to one reduced-form VAR — total runtime is minutes on a laptop. Adding a fourth variable multiplies cost: each new variable requires its own structural equation, its own IV step, and verification that the Rubio-Ramírez, Waggoner and Zha (2010) rank condition for global identification holds. The paper demonstrates the four-variable extension for the Gilchrist-Zakrajšek excess bond premium in Section 7.

Related questions: What is non-modularity? · How should horizons be handled?

What minimum data set is required to estimate an RE-SVAR with a forward-looking policy rule?

Three variables: a price index, a real activity measure, and a policy indicator — all monthly, ideally over a sample of at least 20 years. The RE-SVAR is deliberately low-dimensional and does not require commodity prices, factors, Greenbook forecasts, or futures data — the non-modularity property means each additional variable must come with a structural equation, so the minimum data set is the minimum model.

Recommended series for U.S. work, matching Chen and Valcarcel (2025):

Price: CPI or PCE deflator (the paper uses both and shows results are robust).
Activity: Industrial production index (monthly availability is the binding constraint).
Policy indicator (rate specification): Wu and Xia (2016) shadow federal funds rate.
Policy indicator (money specification): Divisia M4 (or M2) from CFS AMFM, in growth rates.
Sample length: The paper estimates over 1967–2020, 1988–2020, and 2008–2020 — the three-sample comparison gives the cleanest robustness test across structural breaks.

For non-U.S. work, the procedure does not require Greenbook-style internal forecasts, which sidesteps the Orphanides (2001) real-time-data problem — the rational-expectations restriction is inside the model, not imposed via external forecasts.

Can the RE-SVAR framework be extended to open-economy or international policy rules?

Yes, with two caveats: each open-economy variable (real exchange rate, foreign output, foreign rate) needs its own structural equation, and the rank condition for global identification must be re-verified for the larger system. This is the same non-modularity constraint that limits the framework's flexibility — but it is precisely what makes the open-economy extension principled rather than ad hoc.

The standard open-economy SVAR template comes from Cushman and Zha (1997) for Canada and Kim and Roubini (2000) for the G7, both using block-recursive identification with external variables ordered first. The RE-SVAR analog would write a forward-looking IS equation augmented by a real-exchange-rate term, derive the forecast-revision identity for each equation, and add a monetary block for the foreign central bank.

Practical entry points for researchers wanting to attempt this: for Eurozone monetary policy identification, Belongia and Ireland's (2022) money-growth-rule framework provides the theoretical anchor; for Mexico, Colunga-Ramos and Valcarcel (2024) construct a Mexican Divisia M4 that could serve as the policy indicator in an RE-SVAR adapted for a small open economy. The framework is, in principle, portable to these settings, though each extension requires verifying the identification conditions for the expanded system.

Related questions: What is non-modularity? · How is the RE-SVAR implemented?

What does the RE-SVAR evidence imply for central banks considering money-growth rules?

It implies that money-growth rules are more robust to forward-looking dynamics than interest-rate rules in low-dimensional consensus models — the opposite of the standard view that interest-rate rules are modern best practice and money-growth rules are historical curiosities. Chen and Valcarcel (2025) document that as the policy-rule's forward-looking horizon h_π increases from 1 to 12 months, the no-joint-puzzle share for Divisia M4 rises from 88.4% to 99.1%, while for the Wu-Xia shadow rate it falls from 2.1% to 0.03%. The asymmetry is structural and survives across price indices, sample periods, and aggregation tiers.

For applied central-bank work, three concrete implications:

Operational monitoring should include Divisia M4 growth alongside the policy rate, since the rate loses identifying content as the policy regime becomes more forward-looking.
Communication strategy: forward guidance and transparency are part of the reason the short-rate indicator fails — they are facts about the modern monetary regime that the monetary aggregate accommodates, not problems to walk back.
Post-QE normalization: Divisia M4's sensitivity to Treasury and repo holdings makes it a better real-time indicator of policy stance than the policy rate alone as central banks unwind balance sheets.

This complements Belongia and Ireland's (2022) theoretical case for money-growth rules, who argue that a rule responding gradually to inflation and output can deliver stabilization comparable to an estimated Taylor rule.

Data and reproducibility

Monetary policy indicator (shadow rate): Wu and Xia (2016) shadow federal funds rate, monthly.
Divisia monetary aggregates: Center for Financial Stability — AMFM dataset, Divisia M2 and M4.
Macroeconomic data: FRED (CPI, PCE, industrial production, unemployment).
Sample: Three samples — 1967–2020, 1988–2020, 2008–2020, monthly frequency.
Software: Custom RE-SVAR procedure; grid of 241,865 specifications from h_π ∈ {0,…,12}, h_y ∈ {0,…,5}, φ_π, φ_y ∈ [0,4] at increments of 1/15.
Open access: UNI ScholarWorks · SSRN preprint · Journal of Economic Dynamics and Control

Related publications

Chen and Valcarcel (2021), JEDC — methodology-independent evidence that the federal funds rate fails in low-dimensional settings (TVP-FAVAR approach).
Chen and Valcarcel (2024), Macroeconomic Dynamics — cointegration and stability of Divisia money demand; establishes the long-run foundation for the policy indicator results here.

Cite as: Chen, Z., & Valcarcel, V. J. (2025). Modeling inflation expectations in forward-looking interest rate and money growth rules. Journal of Economic Dynamics and Control, 170, 104999. https://doi.org/10.1016/j.jedc.2024.104999

@article{chenvalcarcel2025resvar,
 author = {Chen, Zhengyang and Valcarcel, Victor J.},
 title = {Modeling inflation expectations in forward-looking
 interest rate and money growth rules},
 journal = {Journal of Economic Dynamics and Control},
 volume = {170},
 pages = {104999},
 year = {2025},
 doi = {10.1016/j.jedc.2024.104999},
 url = {https://doi.org/10.1016/j.jedc.2024.104999}
}

Monetary Transmission in Money Markets: The Not-So-Elusive Missing Piece of the Puzzle

Wed, 11 Aug 2021 00:00:00 +0000

In a Modern U.S. Sample, the Federal Funds Rate Is No Longer a Reliable Monetary Policy Indicator — but a Broad Divisia Monetary Aggregate Is

TL;DR: The price puzzle — contractionary monetary policy raising prices in VAR models — has resisted every standard fix in post-1988 U.S. data. Chen and Valcarcel (2021, Journal of Economic Dynamics and Control) show that swapping the Wu-Xia shadow rate for Divisia M4 resolves the puzzle without any ad hoc fixes, and reveals a post-2008 flight-to-safety pattern in which less-liquid money markets respond more strongly than currency and demand deposits. The problem was never the omitted information — it was the indicator itself.

Key Concepts

Modern-sample price puzzle: The post-1988 incarnation of the price puzzle that, unlike the historical version, is not resolved by the Christiano-Eichenbaum-Evans remedies (commodity prices, fed funds futures, forward rates). Coined by Chen and Valcarcel (2021) .
Divisia-sufficiency: The result that, in a modern-sample VAR, replacing the short-term rate with a Divisia monetary aggregate is by itself sufficient to restore theory-consistent responses of prices and output, even without commodity prices or futures data. Chen and Valcarcel (2021) .
Post-crisis flight-to-safety transmission: The finding that post-2008, less-liquid assets (IMMFs, large time deposits, repos, commercial paper, T-bills) respond with larger magnitudes than currency and demand deposits to an expansionary Divisia M4 shock — the opposite of the contractionary, liquidity-preserving pattern produced by shadow-rate shocks. Chen and Valcarcel (2021) .

Q1. Why does the U.S. price puzzle persist in modern-sample VARs even with commodity prices and futures data?

The price puzzle persists in post-1988 U.S. data because the federal funds rate — conventionally augmented with commodity prices, fed funds futures, or forward rates — has lost much of its identifying power for monetary policy shocks in an environment of heightened Fed transparency, forward guidance, and a near-zero neutral rate. The problem is not the omitted information; it is the indicator itself.

Christiano, Eichenbaum and Evans established that including commodity prices in a recursive VAR eliminates the price puzzle in a sample spanning 1965-1995 , and Kuttner introduced the use of fed funds futures data to separate anticipated from unanticipated target changes . Brissimis and Magginas argued that augmenting VARs with forward-looking variables such as futures and forward rates resolves the puzzle . Bernanke, Boivin and Eliasz proposed factor-augmented VARs as a more comprehensive information-set fix .

Chen and Valcarcel (2021) show that every one of these fixes fails in a 1988-2020 sample . Across 23 iterations of the federal funds rate specification — combining real output measures (IP, CFNAI, monthly RGDP), price levels (PCE, CPI, core variants), commodity prices (CRB, IMF), and federal funds futures or forward rates — price puzzles remain pervasive, both in time-varying-parameter VARs and in constant-parameter counterparts. This is the modern-sample price puzzle.

Consistent with this, Barakchian and Crowe find that monetary policy post-1988 became more forward-looking, invalidating the identifying assumptions in conventional methods , and Ramey’s Handbook synthesis confirms the preponderance of puzzles across post-1983 identification schemes .

Why the standard fixes fail: A neutral federal funds rate with enough room for material movement is a prerequisite for the short-rate indicator to work. The post-2008 effective-lower-bound period, combined with decades of increasingly transparent Fed communication and forward guidance, has squeezed the unanticipated component of federal funds rate movements toward zero — the thing SVARs need to identify a shock.

Three Approaches to Monetary Policy Indicator in a Modern U.S. Sample (1988-2020)

Dimension	Short Rate + Commodity Prices (CEE 1999)	Short Rate + Futures/Forward Rates (Brissimis-Magginas 2006)	Divisia M4 (Chen-Valcarcel 2021)
Core claim	Commodity prices proxy the Fed’s forward-looking information set and resolve the price puzzle.	Forward-looking variables (fed funds futures, forward rates) reflect market expectations of policy and resolve the price puzzle.	The short rate has lost identifying power in the modern sample; a Divisia monetary aggregate is the correct policy indicator.
Key references	Christiano, Eichenbaum & Evans (1999) , Bernanke, Boivin & Eliasz (2005)	Kuttner (2001) , Cochrane & Piazzesi (2002) , Brissimis & Magginas (2006) , Gertler & Karadi (2015)	Belongia (1996) , Belongia & Ireland (2014) , Keating, Kelly, Smith & Valcarcel (2019) , Chen & Valcarcel (2021)
Testable prediction	Including commodity prices eliminates the price puzzle across samples.	Including futures or forward rates eliminates the price puzzle.	Divisia M4 as the indicator eliminates the price puzzle without commodity prices or futures.
Empirical verdict in modern sample (1988-2020)	Fails. Price puzzle persists across 23 iterations of the federal funds rate specification with commodity prices .	Fails. Price puzzle remains even with 30-day fed funds futures, CRB or IMF commodity indices, or forward rates constructed from overnight repo spreads .	Succeeds. Divisia M4 resolves the puzzle across 23 specifications, including three-variable VARs with no commodity prices and no futures .
Policy transmission to money markets	Puzzlingly contractionary responses for currency, deposits, repos, CP, T-bills post-2008.	Same contractionary puzzles as commodity-prices specification; futures/forward rates do not rescue transmission.	Sensible expansionary responses; less-liquid assets respond more strongly than currency/DDs post-2008 (flight-to-safety).
Sample-period applicability	Works for historical samples (1960s-1990s); breaks down after 1988.	Works to varying degrees in historical samples; breaks down after 1988.	Designed for the modern sample; also works historically (Keating, Kelly, Smith & Valcarcel 2019 ).
Named concept	CEE identification / commodity-prices fix	Forward-looking-variables identification	Divisia-sufficiency · Modern-sample price puzzle · Post-crisis flight-to-safety transmission (Chen & Valcarcel 2021 )

Q2. Does replacing the federal funds rate with a Divisia monetary aggregate resolve the price puzzle in a modern sample?

Yes. Replacing the Wu-Xia shadow federal funds rate with Divisia M4 (or the narrower Divisia M2) produces sensible, theory-consistent price responses in every specification Chen and Valcarcel examine — including three-variable VARs that contain no commodity prices and no futures data. This is Divisia-sufficiency: the Divisia aggregate does the heavy lifting by itself.

The foundation for this result rests on the Barnett critique. Belongia demonstrated that replacing simple-sum aggregates with Divisia indexes reverses the qualitative inference of four out of five influential studies on the effects of money , and Belongia and Ireland formalized within a New Keynesian model that “measurement matters” — a Divisia quantity tracks the true monetary aggregate almost perfectly while simple-sum does not . Keating, Kelly, Smith and Valcarcel extended this to a VAR framework, showing Divisia M4 identification delivers plausible responses free of price, output, and liquidity puzzles in a historical sample .

Chen and Valcarcel (2021) extend the Divisia result to the post-1988 modern sample . Across three-variable TVP-VARs and larger TVP-FAVARs, specifications with DM4 or DM2 as the indicator yield:

A gradual (and correctly-signed) price level response consistent with New Keynesian sticky-price predictions.
Theory-consistent real output responses across PCE, CPI, core price measures, and three alternative output indicators.
Resolution that holds even when commodity prices and federal funds futures are excluded from the VAR — unlike the Christiano-Eichenbaum-Evans recipe, Divisia does not require these crutches.
Quantitatively larger post-2008 price responses for DM4 than for DM2, consistent with DM4 capturing a wider array of the monetary shocks that eventually pass through to prices.

This aligns with Belongia and Ireland’s finding of a stable Divisia money demand relationship in the modern sample , which is the microfounded underpinning for why a Divisia aggregate can serve as a policy indicator.

Q3. How does the transmission of monetary policy to money markets differ between the federal funds rate and Divisia M4 after 2008?

After 2008, expansionary federal funds rate shocks generate puzzlingly contractionary money-market responses — balances in currency, demand deposits, savings, repos, commercial paper, and T-bills all fall. Expansionary Divisia M4 shocks, by contrast, produce sensible expansionary responses, and the less-liquid assets (IMMFs, large time deposits, repos, CP, T-bills) respond with larger magnitudes than the highly liquid ones. Chen and Valcarcel call this post-crisis flight-to-safety transmission.

The standard VAR approach places money below interest rates and output. Bernanke, Boivin and Eliasz’s FAVAR treatment orders the rate indicator last and restricts monetary assets not to respond within the period , while Keating, Kelly, Smith and Valcarcel instead order the indicator before the monetary block, allowing money markets to respond freely to policy . Chen and Valcarcel adopt the latter block-recursive approach, letting 14 different deposits and money-market instruments respond unrestricted.

The results are stark . Under the Wu-Xia shadow federal funds rate:

Currency, demand deposits, and OCDs respond negatively to an expansionary shock, particularly after 2008.
Savings at banks and thrifts — counterintuitively — also contract.
IMMFs, repos, and T-bills show large negative responses post-crisis, which is the opposite sign from theory.

Under Divisia M4, the same specifications yield:

Sensible positive responses for currency and demand deposits.
Larger positive responses for savings at banks and thrifts (consistent with higher household personal saving after 2008).
Even larger positive responses for less-liquid assets — IMMFs, LTDs, repos, CP, T-bills — commensurate with savings rather than with currency.

The post-2008 magnitude pattern across asset classes is consistent with a flight-to-safety channel: households moved into savings, firms moved into less-liquid but safer instruments (time deposits, repos against Treasury collateral), and the Fed’s large-scale asset purchases mechanically expanded Treasury holdings in the monetary aggregate.

Q4. Can commodity prices or federal funds futures rescue the short-rate specification in a modern sample?

No. Commodity prices (both CRB and IMF indices), the 30-day federal funds futures rate, and the Brissimis-Magginas overnight-repo-spread forward rate all fail to resolve the modern-sample price puzzle when the Wu-Xia shadow federal funds rate is the indicator. The puzzle-fix-fails-in-modern-data pattern holds across 23 specifications.

Christiano, Eichenbaum and Evans concluded that including commodity prices was needed to resolve the puzzle in a 1965-1995 sample , and Cochrane and Piazzesi argued that high-frequency identification from daily target-change surprises avoids the omitted-variable problem of monthly VARs . Brissimis and Magginas advocated specifically for federal funds futures or forward rates in a recursive VAR , while Gertler and Karadi popularized the use of high-frequency surprises as external instruments in proxy SVARs .

Chen and Valcarcel test all of these within a common TVP-FAVAR framework and find the price puzzle remains . The envelope of impulse responses across 23 different federal funds rate specifications — crossing three output measures, four price indices, two commodity indices, and futures/forward rate variants — shows a generally pervasive price puzzle throughout the 1988-2020 sample, with no specification consistently escaping it. This matches the Barakchian-Crowe finding that a forward-looking Fed invalidates post-1988 identifying assumptions and Ramey’s broader synthesis .

The takeaway for practitioners: If your sample begins in the late 1980s or later and you must use a short-term rate, expect puzzles. If you use Divisia M4 instead, the puzzles disappear even without commodity prices or futures.

Q5. Should I use the Wu-Xia shadow federal funds rate to identify monetary policy shocks in a post-2008 sample?

Use it with caution. The Wu-Xia shadow rate extends the federal funds series through the effective-lower-bound period, but it generates persistent price puzzles in modern-sample VARs and the resulting shocks transmit implausibly through money markets. Its sensitivity to minor modeling choices adds further reason for caution.

Wu and Xia proposed the shadow rate to summarize the macroeconomic stance of policy during the effective-lower-bound period , and it has been widely adopted. Krippner, however, demonstrates that shadow short-rate estimates are sensitive to minor estimation choices, and those sensitivities propagate into wide variations in inferred UMP effects . Keating, Kelly, Smith and Valcarcel earlier showed that incidences of the price puzzle are exacerbated in SVARs that include various shadow interest rates for a modern sample .

Chen and Valcarcel (2021) find the shadow rate produces puzzling price responses across 23 specifications spanning 1988-2020, with the puzzle emerging as early as three months post-shock and persisting at 60-month horizons . The responses for slices at December 2008, November 2010, and September 2012 — the starts of QE1, QE2, and QE3 — all show price puzzles for the Wu-Xia specification while the DM4 and DM2 specifications at the same dates show theory-consistent, quantitatively large price responses.

Practical guidance for a modern-sample VAR:

If you need a rate indicator, document the puzzle and treat the effective lower bound period as a structural break rather than a continuous series.
Consider Divisia M4 as the policy indicator. The “post-1984” Great Moderation break in macro dynamics and the Monetary Control Act of 1980 are good reasons to begin samples in the late 1980s, where Divisia performs well.
If you need an external instrument, Arias, Caldara and Rubio-Ramírez’s agnostic sign-restriction identification of the systematic component offers an alternative to high-frequency surprise methods.
For event studies around quantitative tightening or balance-sheet normalization, Smith and Valcarcel demonstrate that short-rate indicators miss first-order financial-market effects that become visible through careful daily-frequency analysis .

Q6. What is the Divisia monetary aggregate and why does it matter for monetary policy identification?

Divisia monetary aggregates, developed by William Barnett, weight each component of the money stock by its user cost — recognizing that currency, demand deposits, savings, money-market funds, and T-bills provide different flows of liquidity services and have different opportunity costs. Simple-sum aggregates (M1, M2) treat all components as perfect substitutes, which is both theoretically wrong and empirically disabling.

The theoretical case is the Barnett critique: simple-sum aggregates violate aggregation theory by adding assets that are not perfect substitutes. Belongia showed empirically that replacing simple-sum with Divisia reverses the qualitative inference of four of five influential monetary studies . Belongia and Ireland formalized the Barnett critique inside a New Keynesian model, demonstrating that a Divisia quantity tracks the theoretically correct monetary services aggregate almost perfectly while simple-sum does not . They later showed that interest rates and Divisia money jointly provide the best measurement of monetary policy stance .

Belongia and Ireland also document a stable cointegrating money demand function for Divisia M2 and MZM over 1967-2019 — including the financial innovations of the 1980s and the post-2008 period — which undermines the long-standing claim that money demand is inherently unstable .

Chen and Valcarcel (2021) operationalize these insights for modern-sample monetary policy identification. They use the Center for Financial Stability’s Divisia series at three levels of aggregation : Divisia M1 (currency, demand deposits, OCDs at banks and thrifts); Divisia M2 (DM1 + savings deposits, retail money-market funds, small time deposits); and Divisia M4 (DM2 + institutional money-market funds, large time deposits, repurchase agreements, commercial paper, and 3-month T-bills — 15 components total, the broadest U.S. monetary aggregate currently available).

Why Divisia M4 is the right choice for modern-sample VARs:

Its 15-component breadth captures the post-1980 financial ecosystem — repos, institutional money funds, commercial paper — that narrow aggregates miss.
It properly weights each component by user cost, respecting the Barnett critique.
In Chen-Valcarcel’s block-recursive identification, it generates theory-consistent responses without commodity prices or futures data.
It exhibits a stable cointegrating money demand relationship over the full modern period.

Q7. How do I estimate a TVP-FAVAR with Divisia M4 as the policy indicator?

The workflow has four moving parts: a block-recursive ordering with Divisia M4 before the monetary block, a stochastic-volatility TVP state space estimated via Primiceri-style MCMC, factors extracted from a panel of monthly macro indicators, and a clean sample-break treatment for 2008.

Chen and Valcarcel (2021) walk through the exact specification , and the practical pipeline distills to:

Construct a balanced monthly panel (1988m1–2020m12) of macro indicators (industrial production, employment, prices, financial conditions) and standardize each series.
Extract 3–5 principal-component factors from the panel and place them as the slow-moving block.
Order Divisia M4 before the money-market block (currency, demand deposits, OCDs, savings, IMMFs, large time deposits, repos, CP, T-bills) — the block-recursive logic from Keating, Kelly, Smith and Valcarcel (2019) .
Estimate the TVP coefficients with Primiceri’s stochastic-volatility MCMC sampler , using Del Negro and Primiceri’s corrigendum to the ordering of steps .
Report impulse-response slices at specific calendar dates (the paper uses December 2008, November 2010, September 2012) rather than averaging over the sample.

Two practical warnings: the sampler is sensitive to the prior on the variance of the time-varying coefficients (Primiceri’s defaults are a reasonable baseline), and TVP-VARs with stochastic volatility require a large number of post-burn-in draws to stabilize the IRF distributions.

Related questions: Should I use DM4 or DM2 in a modern-sample VAR? · How do I extract principal-component factors for a TVP-FAVAR?

Q8. Where do I download Divisia monetary aggregate data and which vintage should I use?

The Center for Financial Stability’s Advances in Monetary and Financial Measurement program at centerforfinancialstability.org/amfm_data.php is the authoritative source for U.S. Divisia monetary aggregates and their user costs, updated monthly in three aggregation tiers (DM1, DM2, DM4) alongside component-level quantities and matching user costs.

What to pull, by research question:

Macro VARs with a broad monetary indicator → Divisia M4 growth rate, monthly, log-differenced; for replication of Chen and Valcarcel (2021) and Chen and Valcarcel (2024) .
Money demand cointegration → Divisia M2 or M3 level, monthly or quarterly, paired with the matching real user cost.
Asset-level liquidity questions → the 15 component series and their individual user costs, following Barnett, Liu, Mattson and van den Noort (2013) .
Through-the-ELB samples → Divisia growth, not the Wu-Xia shadow rate, because the user-cost dual remains positive through the ELB while the federal funds rate is pinned to zero .

Vintage note: CFS revises the historical series when component definitions change. For published-paper replication, freeze a vintage and document the download date; for new research, use the latest vintage.

Related questions: How do I construct a Divisia index from scratch if my country isn’t covered? · Should I use DM4 or DM2 for my VAR?

Q9. Does the Divisia approach to monetary policy identification apply to other countries?

Yes — Divisia monetary aggregates have been constructed for the U.K., Eurozone, Mexico, India, China, and several emerging markets, and the pattern of Divisia outperforming short-rate indicators recurs. The portability of the result is itself evidence that the failure of short-rate identification is a general property of late-cycle, transparent, ELB-touching monetary regimes.

For Mexico, Colunga-Ramos and Valcarcel (2024) construct the first Divisia M4 for the Mexican economy and show it delivers sensible monetary responses without commodity-price augmentation , reproducing the Chen-Valcarcel (2021) finding outside the U.S. Colunga-Ramos, Chen, and Perales (2026) use Mexican Divisia M2 in a sectoral inflation decomposition that validates monetary-versus-supply identification at the sector level . For broader EM coverage, Barnett, Ghosh, and Adil (2022) document stable broad-Divisia money demand across multiple countries . The U.K. Divisia series supports demand stability and policy identification work parallel to the U.S. evidence in Belongia and Ireland (2019) .

Practical takeaway for non-U.S. work: if your country has an aggregation-theoretic Divisia series, use it as the policy indicator. If not, the Barnett (1980) procedure requires only component-level quantities and a benchmark yield — both of which are typically in central-bank statistics. The framework is, in principle, portable to any setting with this minimum data, though constructing a country-specific Divisia series is itself a publishable contribution.

Related questions: How is Divisia M4 constructed in countries without an official series? · Does the post-crisis flight-to-safety pattern appear in Eurozone money markets?

Q10. What does Chen-Valcarcel (2021) imply for empirical work on QE, QT, or the Wu-Xia shadow rate?

Three concrete implications for any paper currently using the Wu-Xia shadow rate to identify unconventional monetary policy effects.

First, impulse responses estimated off the shadow rate are likely contaminated by the modern-sample price puzzle, regardless of whether commodity prices or futures are included as controls. Second, the contamination is particularly acute for money-market and credit-market outcomes, where short-rate shocks generate implausibly contractionary responses for currency, savings, repos, and T-bill balances post-2008. Third, the cleanest fix is to switch the policy indicator to Divisia M4; the second-cleanest is to combine a daily-frequency event-study approach with Smith and Valcarcel’s (2023) framework for quantitative-tightening event studies , which documents balance-sheet effects invisible to monthly short-rate SVARs.

For QE event studies specifically, Chen and Valcarcel (2021) report time-varying IRFs at the QE1, QE2, and QE3 starting dates and find that Divisia M4 delivers theory-consistent price responses while Wu-Xia delivers price puzzles. For applied work using high-frequency surprises as instruments, Chen (2026) shows that pre-FOMC financial conditions already absorb most of the predictable component ; the cleaner combined approach identifies off Divisia M4 in the structural VAR and uses financial-conditions-purged surprises as a robustness instrument.

Related questions: How do I purge high-frequency surprises for SVAR identification? · What is the right monetary policy indicator for QE event studies?

This paper connects to Chen’s broader research program on monetary policy identification. Chen (2026, Journal of Macroeconomics) extends the identification question to high-frequency monetary policy surprises, showing that the Fed responds primarily to financial conditions while adopting a “wait-and-see” stance on recent economic data. Chen (2025, Journal of Economic Dynamics and Control) examines forward-looking monetary policy rules and their implications for inflation expectations.

Data and Replication

All data and code for Chen and Valcarcel (2021) are available at robinchen.org . The paper uses:

Center for Financial Stability Divisia Monetary Aggregates (monthly, M1/M2/M4)
Wu-Xia shadow federal funds rate
14 money-market component series (currency, demand deposits, OCDs, savings, IMMFs, LTDs, repos, CP, T-bills, and more)
CRB and IMF commodity price indices
30-day federal funds futures rate

Citation

Chen, Zhengyang, and Victor J. Valcarcel. 2021. “Monetary Transmission in Money Markets: The Not-So-Elusive Missing Piece of the Puzzle.” Journal of Economic Dynamics and Control 131: 104214. https://doi.org/10.1016/j.jedc.2021.104214

@article{chenvalcarcel2021,
 title={Monetary Transmission in Money Markets: The Not-So-Elusive Missing Piece of the Puzzle},
 author={Chen, Zhengyang and Valcarcel, Victor J.},
 journal={Journal of Economic Dynamics and Control},
 volume={131},
 pages={104214},
 year={2021},
 publisher={Elsevier},
 doi={10.1016/j.jedc.2021.104214}
}

Price Puzzle | Robin Chen