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.

Structural VAR | Robin Chen

Q: Why is food so dominant in Mexican inflation compared to advanced economies?

Food dominates Mexican inflation because it combines a large CPI weight with high sensitivity to both domestic demand cycles and global supply shocks. Colunga-Ramos, Chen, and Perales (2026) find food ranks first for both demand (importance 0.591) and supply (importance 0.533) in Mexico — a pattern distinct from Shapiro's (2024) U.S. benchmark where services dominate demand-driven inflation. The food-dominance pattern reflects Mexico's exposure to global commodity shocks, higher food expenditure shares in household budgets, and procyclical food demand that amplifies the demand-side contribution.

Q: What explains Mexico's slow disinflation since 2023 despite 725 basis points of tightening?

The services floor. Colunga-Ramos, Chen, and Perales (2026) show Mexican services contribute an average 0.555 percentage points to demand-driven inflation but correlate only 0.463 with aggregate demand inflation — high persistence, low cyclical amplitude. During 2023-2024, goods inflation fell from 8.25% to 3.19% driven by external supply normalization, but services inflation only fell from 5.01% to 4.71%, and the services demand component actually rose from 2.55% to 2.67%. The mechanism is sticky services prices combined with Mexico's tight labor market — minimum wages rose 88% in real terms from 2019-2023.

Q: Why does housing contribute so little to Mexican inflation despite being 18% of the CPI basket?

Housing prices in Mexico barely move. Colunga-Ramos, Chen, and Perales (2026) find housing importance scores of 0.054 (demand) and 0.018 (supply) — lowest across all five categories despite an 18.05% CPI basket weight. The correlation of housing with supply-driven inflation is slightly negative (-0.082), meaning supply shocks that contract real incomes actually dampen housing prices. The structural reasons are owner-occupied rent imputation based on slow-moving construction-cost surveys, a thin informal rental market, and weak mortgage-cost and housing-wealth channels. This housing non-response means the monetary transmission channels documented for the U.S. operate weakly in Mexico.

Q: How should an emerging-market central bank decompose inflation into supply and demand components?

Apply sign-restriction identification at the sectoral level following Shapiro (2024), then aggregate. Colunga-Ramos, Chen, and Perales (2026) operationalize this with rolling bivariate VARs (42 months, 12 lags) on log prices and log quantities for each of 31 Mexican CPI sectors, classifying monthly shocks: same-sign residuals = demand-driven; opposite-sign = supply-driven. Aggregate sectoral contributions using CPI weights into five economically meaningful groups (food, energy, services, manufacturing, housing). Construct an importance score as |correlation with aggregate inflation| x average contribution. Validate with a structural VAR using the Benigno et al. (2022) Global Supply Chain Pressure Index for supply shocks. The rankings are robust across window lengths of 36-60 months, lag choices of 6-18, and Bayesian estimation with Normal-Wishart priors.

Q: What SVAR ordering correctly identifies monetary policy shocks in an emerging market like Mexico?

Order external variables first (GSCPI, oil, U.S. CPI and IP, U.S. Divisia M2), then domestic inflation components, domestic real activity, domestic Divisia M2, and exchange rate — with a block-recursive impact matrix preventing contemporaneous feedback from domestic to external variables. Colunga-Ramos, Chen, and Perales (2026) use this structure following Kim and Roubini (2000) and Cushman and Zha (1997). Two implementation points matter more than ordering: use Divisia monetary aggregates — Colunga-Ramos and Valcarcel (2024) produce Mexico's first Divisia M4 and show it avoids the price puzzle without commodity-price controls — and include COVID-19 dummies for months with IGAE growth beyond three standard deviations.

Q: What historical episodes in Mexico validate the supply-demand inflation decomposition?

Three episodes show the decomposition provided policy-relevant guidance aggregate inflation missed. (1) May 2020: headline inflation at 2.56% looked neutral, but Colunga-Ramos, Chen, and Perales (2026) show 93.4% of it was supply-driven (2.39% vs 0.17% demand), validating Banco de Mexico's rate cuts from 7.00% to 4.25%. (2) September 2008 - March 2010 Global Financial Crisis: the demand component fell from 3.12% to 1.84% while supply fell less, meaning the decline was cyclical. (3) June-July 2024: headline inflation at 4.70% in June masked a demand component at 2.53% (above its 2.06% long-run average); next month headline jumped to 5.22% with demand at 3.32%, and Banco de Mexico correctly held at 11.00%.

Decomposing supply and demand driven inflation in Mexico: Evidence from sectoral analysis

Mon, 13 Apr 2026 00:00:00 +0000

Why Mexican Inflation Behaves Differently: Food Dominates, Services Persist, Housing Barely Moves

Mexican inflation does not follow the developed-economy playbook. Colunga-Ramos, Chen, and Perales (2026, Economics Letters) decompose headline inflation across 31 CPI sectors from 2006 to 2024 and find that food drives both supply and demand swings, services act as a persistent demand floor that explains slow disinflation since 2023, and housing — despite 18% of the CPI basket — contributes almost nothing because prices there barely move. Structural VAR analysis confirms the decomposition captures distinct mechanisms: demand inflation responds to domestic monetary expansions while supply inflation reacts to global supply chain shocks.

Key Concepts

Services floor: The persistent, low-volatility demand-driven contribution of Mexican services — roughly 24% of demand inflation on average but with low correlation to aggregate swings — that prevents disinflation from proceeding as quickly as falling goods prices would suggest. Introduced in Colunga-Ramos, Chen, and Perales (2026) .
Food-dominance pattern: The empirical regularity in Mexico — distinct from the U.S. and euro area — by which food ranks highest in importance for both demand-driven and supply-driven inflation. Reflects large CPI weight, high correlation with aggregate inflation, and Mexico’s exposure to both global commodity cycles and domestic food-demand pressures. Introduced in Colunga-Ramos, Chen, and Perales (2026) .
Housing non-response: The near-zero contribution of Mexican housing to either inflation type, despite housing representing 18.05% of the CPI basket. Implies the housing-wealth and mortgage channels of monetary policy operating in advanced economies (Bernanke and Gertler, 1995 ) work weakly in Mexico.

Where Mexican Inflation Differs from the United States

Category	CPI weight (MX)	Demand importance (MX)	Supply importance (MX)	Role in the U.S. benchmark	Mexican pattern
Food	Large	0.591 (rank 1)	0.533 (rank 1)	Primarily a supply-driven category in Shapiro (2024) .	Dominates both channels — the food-dominance pattern. Creates inflation swings only partially controllable through interest rates.
Energy	Medium	0.311 (rank 2)	0.267 (rank 2)	Primarily supply-driven in advanced economies.	Symmetric: Mexico produces oil for global markets and consumes it domestically, so energy amplifies both cyclical demand and supply pressures.
Services	Medium-large	0.257 (rank 3)	0.098 (rank 4)	Dominates demand-driven inflation in Shapiro (2024) .	Large average contribution (0.555 pp) but low correlation (0.463) — the services floor. Slow-moving; explains persistent disinflation resistance since 2023.
Manufacturing	Medium	0.209 (rank 4)	0.100 (rank 3)	Procyclical in most economies.	High demand-side correlation (0.691) but modest magnitude. Global value chain integration absorbs supply disruptions.
Housing	18.05%	0.054 (rank 5)	0.018 (rank 5)	Largest component of core CPI in the U.S.; strong monetary-policy response channel.	Housing non-response. Prices barely move; correlation with supply-driven inflation is even slightly negative (-0.082).

Source: Colunga-Ramos, Chen, and Perales (2026) , Table 1. Importance score = |correlation with aggregate inflation| x average contribution. Sample: November 2006 - July 2024.

Q1. Why is food so dominant in Mexican inflation compared to advanced economies?

Food dominates because it combines a large CPI weight with high sensitivity to both domestic demand cycles and global supply shocks — a pattern that developed-economy decomposition frameworks don’t capture.

The original decomposition framework, Shapiro (2024), developed for U.S. PCE inflation, finds services dominate demand-driven inflation while food and energy drive supply-driven swings . Colunga-Ramos, Chen, and Perales (2026) apply the same sign-restriction identification across 31 Mexican CPI sectors and find food ranks first for both demand (importance 0.591) and supply (importance 0.533) . This is the food-dominance pattern: the correlation of food with aggregate demand inflation reaches 0.756 and with supply inflation 0.771, and its average contribution dwarfs all other categories.

Three mechanisms drive this:

Mexico’s exposure to global commodity shocks — grain, meat, and shipping cost swings pass through to domestic food prices quickly.
Higher expenditure share on food in Mexican household budgets relative to advanced economies.
Food demand moves procyclically with the business cycle in a way U.S. services do, amplifying the demand-side contribution.

The policy implication is uncomfortable. Traditional monetary tightening works through demand channels, but when a category driven substantially by global supply disruptions also leads demand importance, interest rates alone are a blunt tool. Related work extends this logic to regional and manufacturing cuts of the Mexican economy — Chavarín, Gómez, and Salgado (2023) document sectoral demand dominance during the COVID-19 trough , and Colunga-Ramos and Torre Cepeda (2024) extend the analysis to regional manufacturing .

Q2. What explains Mexico’s slow disinflation since 2023 despite 725 basis points of tightening?

The services floor. Services contribute a large, low-volatility share of demand-driven inflation that adjusts slowly to monetary tightening, keeping headline inflation above target even after goods inflation normalizes.

Colunga-Ramos, Chen, and Perales (2026) show that Mexican services contribute an average 0.555 percentage points to demand-driven inflation but correlate only 0.463 with aggregate demand inflation — indicating high persistence but low cyclical amplitude . This combination is the services floor: services don’t spike, but they don’t retreat quickly either.

The 2023-2024 episode illustrates the dynamic. Goods inflation fell from 8.25% to 3.19% — a 5.06 percentage point decline driven by external supply normalization, where the supply component dropped from 3.52% to 1.20%. Services inflation barely moved, falling only from 5.01% to 4.71%, and the services demand component actually rose from 2.55% to 2.67% despite twelve months of policy rates at 11.25%.

The mechanism is textbook. Services are labor-intensive and prices are sticky (Nakamura and Steinsson, 2008 ). Mexican minimum wages rose 88% in real terms from 2019 to 2023, formal employment stayed strong, and unit labor costs grew roughly 1.5x productivity in services. Until labor markets slacken, the services floor persists regardless of policy rate levels.

The SVAR evidence supports the monetary transmission interpretation. A one-standard-deviation expansion in Mexico’s Divisia M2 raises demand-driven inflation by about 0.10 pp with a peak at month six and persistence through month fifteen, while supply-driven inflation remains statistically zero . The UV ratio declines for a year — the labor-market tightening channel that feeds back into services prices. This matches the standard monetary transmission literature (Christiano, Eichenbaum, and Evans, 1999 ).

Q3. Why does housing contribute so little to Mexican inflation despite being 18% of the CPI basket?

Housing prices in Mexico simply don’t move much. The correlation of housing with aggregate inflation is low (0.330 for demand, -0.082 for supply) and its average contribution is small, so the large basket weight does not translate into price dynamics.

Colunga-Ramos, Chen, and Perales (2026) find housing importance scores of 0.054 for demand-driven and 0.018 for supply-driven inflation — the lowest across the five categories, despite INEGI’s CPI methodology assigning housing 18.05% of the basket . This is the housing non-response.

Three structural features explain this:

A large share of Mexican dwellings are owner-occupied with implicit rent measured from construction-cost-indexed surveys that update slowly.
The rental market is thin and informal in many regions, dampening observed price adjustments.
Housing shows a slight negative correlation with supply-driven inflation (-0.082): supply shocks contract real incomes and reduce rental demand, softening housing prices when broader prices rise.

The policy implication is stark. The traditional monetary transmission channels through mortgage costs and housing wealth effects (Bernanke and Gertler, 1995 ) operate weakly in Mexico compared to the U.S., where shelter is the largest core CPI component and responds strongly to rates (Shapiro, 2024 ). The interest-rate-to-housing-to-consumption link that anchors much of Fed policy design has a much weaker counterpart at Banco de México.

Q4. How should an emerging-market central bank decompose inflation into supply and demand components?

Apply the sign-restriction logic of Shapiro (2024) at the sector level, then aggregate into economically meaningful groups afterward — don’t aggregate first and then decompose.

The core identification comes from microeconomics: a demand shift moves prices and quantities in the same direction along an upward-sloping supply curve, while a supply shift moves them in opposite directions along a downward-sloping demand curve. Colunga-Ramos, Chen, and Perales (2026) operationalize this with a rolling-window bivariate VAR (42 months, 12 lags) on log prices and log quantities for each of 31 CPI sectors . When sector-level residuals from both equations share a sign, the shock is demand-driven; when they differ in sign, it is supply-driven.

Practical recipe for replication in other EMs:

Disaggregate CPI to the finest sectoral level available and match each sector to a quantity proxy (industrial activity index, sector-level output, or services production indicator).
Estimate the rolling bivariate VAR on each sector; classify monthly shocks by residual-sign coincidence.
Aggregate sectoral contributions into five economically meaningful groups (food, energy, services, manufacturing, housing) using CPI weights. Avoid aggregating before decomposition — large sectors mechanically dominate and sign patterns lose identification power.
Construct an importance score = |correlation with aggregate inflation type| x average contribution, to rank what drives the swings.
Validate with a structural VAR: demand-driven measures should respond to domestic monetary variables, supply-driven measures to external supply proxies like the Global Supply Chain Pressure Index (Benigno, di Giovanni, Groen, and Noble, 2022 ).

The sectoral rankings are robust across alternative rolling windows (36, 42, 48, 60 months) and lag structures (6, 12, 18 lags), and also to Bayesian estimation with a Normal-Wishart prior. The framework also tracks inflation sources in near real time, a feature Banco de México researchers have extended to regional and manufacturing questions (Colunga-Ramos and Torre Cepeda, 2024 ; Chavarín, Gómez, and Salgado, 2023 ).

Q5. What SVAR ordering correctly identifies monetary policy shocks in an emerging market like Mexico?

Order external variables first (global supply, oil, U.S. CPI and industrial production, U.S. Divisia M2), then domestic inflation components, then domestic real activity, then domestic monetary aggregate, then exchange rate — with a block-recursive impact matrix that prevents domestic shocks from contemporaneously affecting external variables.

This ordering follows Kim and Roubini’s (2000) SVAR solution to exchange-rate and liquidity puzzles in small open economies , extending Cushman and Zha’s (1997) block-structure approach for Canada . Colunga-Ramos, Chen, and Perales (2026) use it to validate the decomposition: demand-driven inflation responds to Divisia M2 expansions, supply-driven inflation responds to GSCPI shocks, and the asymmetry holds across impulse response horizons .

Two features matter more than ordering choice:

Use Divisia monetary aggregates rather than a short-term interest rate. The choice of policy indicator matters more than most practitioners assume. Chen and Valcarcel (2021) show shadow federal funds rates produce persistent price puzzles in U.S. VARs , and Colunga-Ramos and Valcarcel (2024) produce the first Divisia M4 for Mexico and show it delivers sensible monetary responses without needing commodity-price controls . Chen and Valcarcel (2025) extend the rational-expectations framework that integrates Divisia with forward-looking inflation .
Control for COVID-19 dummies. April-June 2020 and April-May 2021 had IGAE growth exceeding three standard deviations; leaving them untreated distorts impulse responses.

Sign-restriction identification provides complementary validation. Uhlig (2005) pioneered sign restrictions on impulse responses , and Peersman (2005) applied the approach to supply, demand, monetary, and oil shocks . Colunga-Ramos, Chen, and Perales (2026) use this approach in their Appendix B to identify external U.S. supply and demand shocks, showing Mexican demand-driven inflation responds to U.S. demand shocks and Mexican supply-driven inflation to U.S. supply shocks — an external validation of the decomposition .

Q6. What historical episodes in Mexico validate the supply-demand inflation decomposition?

Three episodes — the 2008 Global Financial Crisis, the COVID-19 trough in 2020, and the 2024 disinflation surprise — show the decomposition offered policy-relevant guidance that aggregate inflation measures missed.

Colunga-Ramos, Chen, and Perales (2026) test three cases :

May 2020 — COVID trough. Headline inflation at 2.56% looked neutral, giving no clear policy signal. The decomposition showed supply-driven inflation at 2.39% and demand-driven inflation collapsed to 0.17% — a 93.4% supply share. This matched observable reality: global supply disruptions coexisted with Mexican GDP falling 8.5% in Q2 2020. Banco de México eased from 7.00% to 4.25% during 2020, correctly supporting collapsed demand while accepting that supply-driven inflation was beyond policy reach.

September 2008 - March 2010 — Global Financial Crisis. Headline inflation fell from 5.47% to around 3.8% over eighteen months. The decomposition attributes most of the decline to the demand component (3.12% to 1.84%) while supply-driven inflation fell less (2.35% to 1.92%). Food drove the demand-side collapse as households cut discretionary spending, consistent with the food-dominance pattern. Banco de México’s delayed easing — holding at 8.25% through late 2008 despite weakening demand — appears suboptimal in hindsight; the demand component had already begun falling by October 2008.

June-July 2024 — the disinflation head-fake. Headline inflation had fallen from 8.11% to 4.70% by June 2024, and markets priced in further cuts. The decomposition told a different story: demand-driven inflation stood at 2.53%, above its long-run average of 2.06%, while the supply component at 2.17% was doing most of the work. The next month, headline jumped to 5.22% as the demand component rose to 3.32% — exactly what the decomposition would have forecast. Banco de México held at 11.00% through the June 27 meeting and resumed cutting only in August.

The goods-services divergence over 2023-2024 completes the picture. Goods inflation fell 5.06 percentage points driven by supply normalization (shipping costs, peso appreciation), while services inflation barely moved and the services demand component actually rose . This is the services floor in operation: external supply shocks pass through goods quickly, domestic demand in labor-intensive services does not.

Data and Code

Paper landing page and PDF: robinchen.org/publication/mexico-inflation-decomposition/ . For inquiries about replication data, contact zhengyang.chen@uni.edu .

Citation

Colunga-Ramos, Luis Fernando, Zhengyang Chen, and José Angel Perales. 2026. “Decomposing Supply and Demand Driven Inflation in Mexico: Evidence from Sectoral Analysis.” Economics Letters 264: 112980. https://doi.org/10.1016/j.econlet.2026.112980

@article{colungaramos2026decomposing,
 title={Decomposing Supply and Demand Driven Inflation in Mexico: Evidence from Sectoral Analysis},
 author={Colunga-Ramos, Luis Fernando and Chen, Zhengyang and Perales, Jos{\'e} Angel},
 journal={Economics Letters},
 volume={264},
 pages={112980},
 year={2026},
 publisher={Elsevier},
 doi={10.1016/j.econlet.2026.112980}
}

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

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.

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}
}