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}
}

A Granular Investigation on the Stability of Money Demand

Mon, 30 Sep 2024 00:00:00 +0000

The Instability of U.S. Money Demand After 1980 Is a Measurement Artifact

TL;DR: The post-1980 breakdown of U.S. money demand functions is not evidence that households changed their preferences for monetary assets — it is evidence that simple-sum aggregation stopped tracking monetary services once interest-bearing deposits mattered. Chen and Valcarcel (2024, Macroeconomic Dynamics) show that Divisia monetary aggregates paired with their own user costs deliver a stable cointegrating money demand function across both the 1980 DIDMCA deregulation break and the post-2008 zero-lower-bound period. The T-bill yield, by contrast, loses all information content after 2008. At the asset level, 29 of 40 granular tests show correct-sign cointegration between individual monetary components and their own user costs.

Key Concepts

Measurement-not-preference verdict: The paper’s bottom-line conclusion: post-1980 money demand instability comes from how money is measured, not from households’ changing preferences over monetary assets. Chen and Valcarcel (2024) .
User-cost sufficiency for money demand: The finding that Divisia real user costs, but not the T-bill yield, maintain cointegration with monetary aggregates through the 1980 deregulation and post-GFC zero-lower-bound periods. Chen and Valcarcel (2024) .
Granular money-demand cointegration: Bilateral cointegration between each disaggregated monetary asset (currency, demand deposits, savings, repos, CP, etc.) and its own CFS user cost. The paper is the first to run this exercise historically. Chen and Valcarcel (2024) .

Q1. Why is the U.S. money demand function unstable after 1980?

The instability is a measurement artifact of simple-sum aggregation, not a change in households’ preferences for monetary assets. Simple-sum M2 and M3 treat interest-bearing deposits as perfect substitutes for non-interest-bearing currency, which breaks down after the 1980 Depository Institutions Deregulation and Monetary Control Act legalized interest on checkable accounts.

The instability itself is well-documented. Friedman and Kuttner (1992) show that postwar time-series relationships between money and nominal income weaken sharply when the sample extends into the 1980s , and Ball (2001) rejects a stable long-run M1 demand once the sample extends to 1996 . Choi and Jung (2009) locate two structural breaks in 1959-2000 simple-sum data . The standard explanation has been financial innovation inducing preference change.

Chen and Valcarcel (2024) show the instability is instead about measurement . Using CFS Divisia M2 and M3 with Barnett (1980) aggregation — which weights monetary assets by their expenditure shares via user costs — the cointegrating relationship between money and output survives straddling 1980. Andrews-Ploberger and Bai-Perron structural break tests locate the break in Divisia balances around 1980:Q2, consistent with DIDMCA’s institutional timing, but the relationship itself reconstitutes in the post-1980 subsample when user costs are used as the opportunity cost.

This is the measurement-not-preference verdict: the 1980 break shows up because simple-sum aggregation stops tracking monetary services once interest-bearing deposits matter; it does not show up in properly aggregated money.

Four Measurement Combinations for U.S. Money Demand

Dimension	Simple-sum + T-bill	Simple-sum + user cost	Divisia + T-bill	Divisia + user cost
Theoretical coherence	Weak. Equal weights on heterogeneous assets; T-bill is the price of a substitute, not of money.	Weak on quantities; coherent on price.	Coherent on quantities; weak on price.	Fully coherent. Barnett (1980) aggregation paired with Barnett (1978) user cost.
Full-sample cointegration (M2)	Fails in both functional forms.	Intermittent — cointegrates under some Johansen specs, not others.	Robust. Cointegrates under all four Johansen (1995) specifications.	Robust. Cointegrates under all four specifications, correct sign, both semi-log and double-log.
Post-1980 subsample (M2)	Fails in semi-log form. Wrong sign in some trend specs.	Fails in 3 of 4 Johansen specifications.	Cointegrates under constant specs only; wrong sign under trend specs.	Robust across all specs, correct sign.
Post-GFC subsample (M3, M4)	Not applicable — simple-sum abandoned for this era.	Not applicable.	Fails under all specs (T-bill stuck near zero).	Robust across all specs, with higher elasticity estimates than pre-GFC.
Asset-level (granular) cointegration	Most components fail or show wrong sign with T-bill.	Not the paper’s focus.	Most components fail or show wrong sign.	29 of 40 specifications show correct sign (Chen & Valcarcel 2024 ).
What it takes as the break event	Money demand itself breaks in 1980.	Break arises from quantity side.	Break arises from price side (T-bill loses information post-1980 and post-2008).	No break — measurement-not-preference verdict. Apparent instability is an aggregation/measurement artifact.
Named concept	—	—	—	User-cost sufficiency for money demand · Granular money-demand cointegration (Chen & Valcarcel 2024 )

Q2. Does Divisia money demand remain stable across the 1980 DIDMCA break?

Yes — the cointegration between Divisia M2 (or M3) and its own user cost holds in both the pre-1980:Q2 and post-1980:Q2 subsamples, across all four Johansen (1995) deterministic-trend specifications. Simple-sum aggregates do not pass this subsample test.

The pre-1980 result is not itself surprising. Belongia (1996) established that replacing simple-sum with Divisia indexes reverses the qualitative conclusions of several influential money studies , and Serletis and Gogas (2014) found cointegration between Divisia aggregates and the T-bill yield in a Johansen (1991) framework . Belongia and Ireland (2019) estimate a stable Divisia M2 and MZM demand over 1967-2019 .

Chen and Valcarcel (2024) extend this by explicitly straddling the 1980:Q2 DIDMCA break and testing across all four Johansen (1995) deterministic-trend specifications — restricted constant, unrestricted constant, restricted trend, unrestricted trend. Key results:

Divisia M2 with user cost of M2: significant cointegration, correct-sign coefficient, all four specifications, both subsamples.
Divisia M3 with user cost of M3: significant cointegration under three of four specifications post-1980.
Simple-sum M2 with user cost of M2: loses cointegration in three of four specifications post-1980.
Simple-sum M3 with user cost of M3: never cointegrates post-1980.

The sharper-than-usual contrast with simple-sum comes from testing multiple Johansen specifications rather than picking one. This is the user-cost sufficiency for money demand result, part one.

Q3. Does the T-bill yield cointegrate with monetary aggregates after the Great Financial Crisis?

No — the three-month T-bill yield loses cointegration with Divisia M3 and Divisia M4 in the post-2008:Q3 subsample, because the yield was pinned near zero for roughly seven years. Divisia user costs do not suffer this information loss because user costs, while compressed, remained well above zero throughout.

Anderson, Bordo, and Duca (2017) document the Great Recession as a major stress test for M2 velocity models , and Lucas and Nicolini (2015) argue that adding money-market deposit accounts to M1 restores stability of the money-interest-rate relationship through the zero-lower-bound period . Mattson and Valcarcel (2016) show Divisia M4 user costs compressed but stayed positive after 2008, while the federal funds rate collapsed .

Chen and Valcarcel (2024) split the sample at 2008:Q3 and test cointegration for Divisia M3 and Divisia M4 . Results:

Pre-GFC sample (1967:Q1-2008:Q3): Divisia M3 and Divisia M4 cointegrate with the T-bill yield under all Johansen specifications, with correct sign.
Post-GFC sample (2008:Q4-2020:Q1): neither Divisia M3 nor Divisia M4 cointegrates with the T-bill yield under any specification.
Post-GFC sample, using the user cost of Divisia M3/M4 instead: cointegration holds under all specifications, with correct sign, and the magnitude of the elasticity is higher than pre-GFC.

The T-bill breakdown is not about the monetary aggregates — it is about the interest rate losing signal when pinned at the effective lower bound. This is the user-cost sufficiency for money demand result, part two.

Q4. Are Divisia user costs better than the T-bill yield as the opportunity cost of holding money?

Yes — on both theoretical and statistical grounds. The user cost is the spread between a benchmark asset’s yield and the asset’s own interest return, which is the textbook opportunity cost of holding a monetary asset. The T-bill yield is the price of a monetary substitute, not of money itself. Statistically, Divisia user costs maintain cointegration through the 1980 and 2008 breaks; the T-bill yield does not.

The theoretical case traces to Barnett (1978), who derived the user cost for each monetary asset under aggregation theory , and Barnett (1980) formalized Divisia monetary aggregation . The statistical case builds on Belongia and Ireland (2014), who argue the Barnett critique — that inconsistent aggregation distorts inference — remains as relevant as when first articulated , and on Belongia and Ireland (2019), who develop a money-in-the-utility model with interest-bearing deposits that predicts a stable Divisia demand function .

Chen and Valcarcel (2024) make the direct statistical comparison . Divisia M2 and Divisia M3 cointegrate with their own user costs under all Johansen (1995) specifications in the full sample and across subsamples straddling 1980 and 2008. The same aggregates cointegrate less reliably with the T-bill yield, and not at all in the post-2008 subsample. Simple-sum M2 and M3 fail both tests.

One more practical point: unit-root tests are consistent with Divisia user costs being level-stationary around a deterministic trend, while the T-bill yield is not level-stationary under any of the DF-GLS specifications. This is consistent with Belongia and Ireland’s (2019) observation of low-frequency stochastic trends in user costs that are swamped by transitory volatility in market rates .

Q5. Which individual monetary assets cointegrate with their own user costs?

Currency, demand deposits, savings deposits, small time deposits, large time deposits, overnight and term repos, institutional money market funds, and the aggregate of commercial paper plus T-bill balances all cointegrate with their own CFS user costs in at least two of four Johansen (1995) specifications, with the correct sign. Only the less-established innovations — other checkable deposits and retail money market funds — show weak or no cointegration. This is the granular money-demand cointegration finding.

CFS provides user costs for each monetary asset separately following Barnett, Liu, Mattson, and van den Noort (2013) . This makes it possible, in principle, to run cointegration tests on each (asset quantity, asset user cost) pair — but to the paper’s knowledge, this had not been done historically before Chen and Valcarcel (2024) .

Numbers from the paper (double-log specification, full sample):

Of 40 estimates (10 asset pairs x 4 Johansen specifications), 29 show the expected negative user-cost elasticity of demand with the correct sign.
Nine specifications fail to find cointegration.
Only two show an inverted sign (both trend specifications for small time deposits).

By contrast, when the same asset quantities are paired with the T-bill yield (semi-log specification), most pairs fail to cointegrate, and those that do often show the wrong sign. For example, savings deposits and repos cointegrate with the T-bill yield but with positive coefficients — inconsistent with a money demand interpretation.

The asset-level result buttresses the aggregate finding: information content for money demand runs through the price duals, not through a generic short rate. Newer assets that emerged as a direct consequence of 1980s deregulation (OCDs, retail money-market funds) are the ones whose demand is hardest to pin down historically — consistent with the structural-break timing.

Q6. Should I use semi-log or double-log money demand specification for Divisia aggregates?

Use the semi-log form (interest rate in levels) for the full sample and the pre-GFC sample. Use the double-log form (log interest rate) when the sample includes the post-2008 zero-lower-bound period, because log transformations accommodate the nonlinearity induced by near-zero rates better than semi-log.

The two canonical functional forms are the Cagan (1956) semi-log form and the Meltzer (1963) double-log form . Bae, Kakkar, and Ogaki (2006) argue the double-log form better accommodates the liquidity-trap region , and Hendrickson (2014) re-evaluates money demand with Divisia across both forms .

Chen and Valcarcel (2024) use both forms . In the full sample, the semi-log form delivers strong cointegration between Divisia M2/M3 and their user costs across all Johansen specifications, with the elasticity estimates stable around 6.5-10 (semi-elasticities). The double-log form also works well and tends to be slightly more robust to the choice of Johansen deterministic-trend assumption.

For samples straddling the GFC, the double-log form is the better default. The paper estimates Divisia M3/M4 demand as a function of the log of their user costs from 2008:Q4 to 2020:Q1 and finds significant cointegration with correct sign for all Johansen specifications; the semi-log form with the T-bill yield fails in the same sample.

Q7. Is money demand instability evidence of a structural change in preferences?

No. The evidence is more consistent with the “measurement-not-preference” reading: once the proper aggregation (Divisia) and the proper opportunity cost (asset-specific user cost) are used, the long-run demand for money is stable across the 1980 DIDMCA deregulation and the post-2008 zero-lower-bound period.

The preference-change story dates to Friedman and Kuttner (1992) and Bernanke and Blinder (1992) , whose finding that simple-sum money aggregates lose their link to nominal income after 1980 drove much of macroeconomics toward pure interest-rate frameworks. Many subsequent papers interpreted the post-1980 breakdown as evidence that financial innovation had changed how households allocate monetary balances — an implied preference shift.

The measurement reading has accumulated support. Belongia (1996) reversed several prominent null results by substituting Divisia for simple-sum . Lucas and Nicolini (2015) restored stability by adding MMDAs to M1 , pointing to the 1982 Regulation Q weakening as the source of the apparent break. Barnett, Ghosh, and Adil (2022) find stable demand for broad Divisia money across multiple countries . Jadidzadeh and Serletis (2019) reject simple-sum aggregation assumptions using a disaggregated demand system .

Chen and Valcarcel (2024) make the cleanest version of this case by running the subsample test on both the aggregate index and its components, both before and after 1980, using both the T-bill yield and the Divisia user cost, across all Johansen (1995) specifications. The result: simple-sum breaks, Divisia does not; T-bill breaks after 2008, user costs do not. The authors conclude that “the instability of money demand is a matter of measurement rather than a consequence of a structural change in agents’ preference for monetary assets.” That is the measurement-not-preference verdict.

Q8. How do I run a Johansen cointegration test of Divisia money demand on my own data?

Six steps in any econometrics package (R, Stata, EViews, Python with statsmodels). Chen and Valcarcel (2024) follow the Johansen (1995) framework — the practical recipe:

Pull quarterly (or monthly) data on real money balances, real income, and the relevant opportunity cost. For Divisia, use the matching CFS aggregate and its own real user cost (not the T-bill yield).
Take logs of money and income. Try both the semi-log form (user cost in levels) and the double-log form (log user cost).
Run unit-root tests (ADF, DF-GLS) on each series. Most monetary aggregates and real income are I(1); user costs typically test as level-stationary, while T-bill yields fail unit-root tests post-2008.
Select VAR lag length via AIC/BIC/HQIC on the levels system.
Estimate the Johansen VECM under all four deterministic-trend specifications: restricted constant, unrestricted constant, restricted trend, unrestricted trend. A result holding across all four is robust; a result conditional on one is fragile.
Test cointegration rank with trace and maximum-eigenvalue tests. Confirm the sign on the user-cost coefficient is negative.

Subsample test for structural breaks: Re-run the entire procedure on pre-1980Q2 and post-1980Q2 samples for the DIDMCA break, and pre-2008Q3 vs. post-2008Q3 for the ELB break. Cointegration that survives both subsample splits is what supports the measurement-not-preference verdict .

Related questions: Should I use semi-log or double-log? · Where do I get the matching Divisia user costs?

Q9. Where do I download CFS Divisia aggregates, user costs, and component-level series?

All from the Center for Financial Stability’s AMFM page at centerforfinancialstability.org/amfm_data.php , updated monthly. CFS publishes Divisia M1, M2, M3, M4-, and M4 aggregates, each accompanied by its corresponding real user cost — the opportunity cost variable Chen and Valcarcel (2024) show is the correct partner for cointegration tests.

What to download:

Divisia monetary services indexes (DMSI): monthly levels of DM1, DM2, DM3, DM4-, DM4. Use logs for cointegration work.
Real user costs (DMSI_UC): the matching real user cost for each aggregate. Use levels for semi-log specifications, logs for double-log.
Component-level data: 15 monetary asset series (currency, demand deposits, OCDs, savings, retail and institutional MMFs, small and large time deposits, repos, CP, T-bills) each with its own user cost. These are what the granular money-demand cointegration tests use.

Companion U.S. macro data — real personal income, PCE price index, three-month T-bill yield — are from FRED . CFS Divisia goes back to January 1967, matching the Belongia and Ireland (2019) sample . Barnett, Liu, Mattson, and van den Noort (2013) document the user-cost construction , and Mattson and Valcarcel (2016) show user costs stayed positive through 2008–2015 while the federal funds rate collapsed — exactly the reason user costs work where the T-bill fails.

Related questions: What user cost do I use for Divisia M4? · How are user costs constructed?

Q10. Do the Divisia money demand stability results hold for other countries?

Yes — Divisia demand stability has been documented for the UK, Eurozone, Japan, Canada, and several emerging markets, and the qualitative finding generalizes: simple-sum aggregates break with financial deregulation, Divisia aggregates do not. The portability of this result is strong support for the measurement-not-preference verdict in Chen and Valcarcel (2024) — if the U.S. instability were preference-driven, similar institutional features should not produce the same Divisia-versus-simple-sum gap elsewhere.

Cross-country evidence:

U.K.: Belongia and Ireland’s (2014) New Keynesian formalization uses U.K. data alongside the U.S., and CFS-style Divisia for the U.K. shows stable demand patterns.
Multi-country: Barnett, Ghosh, and Adil (2022) document stable demand for broad Divisia money across multiple countries , reinforcing the pattern.
Mexico: Colunga-Ramos and Valcarcel (2024) construct Mexican Divisia M4 and show monetary identification works ; a follow-on money-demand cointegration paper is the natural extension.

For researchers in countries without an official Divisia series, the Barnett (1980) construction is well-documented. The required inputs — component quantities and a benchmark yield — are typically in national monetary statistics.

Related questions: How is Divisia constructed for countries without an official series? · Does the post-2008 user-cost-sufficiency result hold abroad?

Q11. What does a stable Divisia money demand imply for monetary policy frameworks like NGDP targeting or money-growth rules?

It removes the strongest empirical objection to money-quantity-based policy frameworks. The standard case against rules like Friedman’s k-percent rule or NGDP targeting has been that “money demand is unstable” — making any money-quantity target a moving target. Chen and Valcarcel (2024) show this objection rests on simple-sum aggregation and on using the T-bill yield as the opportunity cost ; with Divisia aggregates and matching user costs, the long-run demand relationship is stable across the 1980 DIDMCA break and the post-2008 ELB.

Implications for policy design:

Money-growth rules become operational again. Belongia and Ireland’s (2022) theoretical case for a money-growth rule responding gradually to inflation and output requires a stable demand function as a precondition; the stability is now empirically supported.
NGDP targeting becomes more credible. If real money demand is stable, then nominal NGDP can be controlled via a Divisia M4 instrument with predictable elasticity, even at the ELB.
Operational policy monitoring. For central banks not formally adopting a money-quantity rule, Divisia M4 growth alongside the policy rate provides a robust real-time measure of monetary stance, particularly through ELB periods where the rate alone loses content.

The point is not that money-growth rules are necessarily optimal — that depends on the loss function, transmission lags, and exogenous shocks — but that the empirical precondition for considering them is now met.

Related questions: What does a money-growth policy rule look like operationally? · How does Divisia M4 perform through the ELB?

Data and Code

The CFS Divisia monetary aggregates and their real user costs used in Chen and Valcarcel (2024) are from the Center for Financial Stability’s AMFM program . Other series — PCE price index, real personal income, three-month Treasury yield — are from FRED . Sample period: January 1967 - March 2020, monthly.

Replication files are available on request. Contact: zhengyang.chen@uni.edu .

Citation

Chen, Zhengyang, and Victor J. Valcarcel. 2024. “A Granular Investigation on the Stability of Money Demand.” Macroeconomic Dynamics. https://doi.org/10.1017/S1365100524000427

@article{chenvalcarcel2024granular,
 title={A Granular Investigation on the Stability of Money Demand},
 author={Chen, Zhengyang and Valcarcel, Victor J.},
 journal={Macroeconomic Dynamics},
 year={2024},
 publisher={Cambridge University Press},
 doi={10.1017/S1365100524000427}
}

Divisia Monetary Aggregates | Robin Chen