## wpiea2025094-print-pdf — Section 2–6 summary

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---

### Decomposition approach and scope
- Headline CPI inflation decomposed into:
  - Core inflation: weighted median inflation rate from the Federal Reserve Bank of Cleveland.
  - Headline shocks: headline minus core inflation, reflecting large relative-price changes in particular industries.
- Sample and series:
  - Main series analyzed from 1985 through March 2025, with focused analysis since 2020.
  - PCE deflator considered in the Appendix with broadly similar results.

### Stylized facts since 2020
- Rise and retreat:
  - The inflation surge over 2021–2022 reflected both rising core inflation and positive headline shocks; subsequent disinflation reflected declining core inflation and negative headline shocks.
- Key numeric milestones:
  - Twelve-month core inflation in March 2025: 3.5.
  - Fitted twelve-month core inflation in March 2025:
    - 3.1 (estimates based on sample ending September 2022 — out-of-sample conditional forecast after that date).
    - 3.2 (estimates based on sample extended through March 2025).
  - Expected inflation (median ten-year-ahead SPF):
    - 2.2 percent in 2019:Q4.
    - Peaked at 2.95 percent in 2022:Q4.
    - 2.3 percent in 2025:Q1.
  - Vacancy-to-unemployment ratio (V/U):
    - Peaked at 2.0 in March 2022.
    - Historical average 0.6 over 1985–2019.
    - 1.1 in February 2025 (and 1.1 in December 2019).
    - Unemployment rate never rose above 4.2 percent during the disinflation period.
  - Headline shocks (twelve-month average):
    - Positive from March 2021 to December 2022, peaking at 3.7 percentage points in March 2022.
    - Trough of -3.2 in June 2022.
    - -1.1 percentage points in March 2025.
  - Twelve-month core inflation peak: 7.0 percent in February 2023.

### Three drivers analyzed informally (Section 3)
- Inflation expectations:
  - Rose modestly in 2021–2022 relative to actual inflation; rise peaked at 2.95 percent in 2022:Q4 and reversed from 2023:Q1 back toward pre-pandemic levels by 2025:Q1.
- Labor market tightness (V/U):
  - V/U rose rapidly in 2021, peaked in March 2022 at 2.0 (an unusually high level relative to the 1985–2019 average of 0.6), then fell steadily through mid-2024 to 1.1 in February 2025.
  - Beveridge curve: stable 2009–March 2020; dramatic outward shift at pandemic start; movement back toward pre-pandemic curve by end-2024.
  - The rise and fall in V/U is identified as the most important factor behind core inflation behavior in subsequent econometric work.
- Headline shocks:
  - Three explanatory variables for headline shocks: relative energy inflation (sector minus median), relative auto-related inflation (sector minus median), and firms’ backlogs of work (IHS Markit survey; neutral level = 50).
  - All three moved to inflationary levels late 2020–2022 and then moved disinflationary as relative energy and auto prices reversed and supply chains normalized.
  - Empirically, these variables explain the timing and sign of headline shocks from 2020–2025.

### Core inflation equation (Section 4)
- Specification:
  - π = πe + f(V/U) + g(H) + ε
    - π = annualized monthly core inflation.
    - πe = expected inflation (SPF median, ten-year-ahead).
    - V/U = average of V/U over the twelve months t-11 to t.
    - H = average of headline shock (headline minus median) over t-11 to t.
  - f(•) and g(•) are specified as cubic functions to allow for possible non-linearities and asymmetries.
- Estimated relations and interpretation:
  - f(V/U) is steepest at unusually high or low V/U and flatter in between; steep slope for V/U above 1.5 reflects high core inflation when V/U was in that range in 2022–2023.
  - g(H) is sharply asymmetric: positive headline shocks pass through into higher core inflation, but negative shocks do not have a significant effect.
  - Extending the sample through March 2025 modestly reduces the estimated steepness at high V/U but does not materially change the asymmetry in pass-through from H.
- Fit and forecasting:
  - Fitted values from the equation (both full sample and sample ending September 2022) follow actual core inflation closely over 2020–2025.
  - Fitted twelve-month core inflation in March 2025: 3.1 (shorter sample), 3.2 (longer sample); actual 3.5 percent.
  - Data revisions (seasonal adjustments) led to some differences relative to earlier published fitted values.
- Decomposition of fitted core inflation (full sample 1985–2025):
  - The rise in twelve-month core inflation (peak 7.0 percent in February 2023) reflected:
    - Pass-through from headline shocks (important early in the rise).
    - Rising V/U (increasingly important as the rise continued).
    - Rising expected inflation contributed modestly.
  - The pass-through effect faded in early 2023; core inflation remained high at that point because V/U was at its peak.
  - Subsequent decline in core inflation was driven primarily by falling V/U and modestly by falling expected inflation.
  - Negative headline shocks in 2023–2025 did not materially lower core inflation because the estimated pass-through is asymmetric (negative H has little effect on core).

### Headline shocks equation (Section 5)
- Regression results:
  - Headline shocks regressed on the three factors (energy relative inflation, auto-related relative inflation, backlogs of work).
  - Coefficient estimates similar for January 2020–September 2022 and for the extended sample; R-squared = 0.95 for both samples.
  - Fitted values from either sample closely track actual headline shocks through March 2025.
- Factor contributions (decomposition based on full-sample fitted values):
  - Energy prices: the most important factor—positive contributions June 2020–June 2022, especially spikes after Russia’s invasion of Ukraine in February 2022; then mostly negative contributions as energy prices subsided.
  - Auto prices: material contributor during the rising-inflation period, especially April–June 2021 (chip shortage).
  - Backlogs of work: modest contribution after controlling for auto and energy prices; rose above neutral (50) then fell below it.
  - Residuals (idiosyncratic industry shocks): mostly negative during the rising-inflation period and included large negative residuals from falls in airline fares and apparel prices (February and September 2021); during falling inflation, largest negative residual in November 2023 reflected a fall in the relative price of food at home.

### Overarching interpretation and conclusions (Section 6)
- Central interpretation:
  - The rise in twelve-month inflation from January 2020 to the peak in June 2022 is explained primarily by positive headline shocks, which raised inflation directly and by passing through into core inflation.
  - By early 2023, the pass-through effects of those shocks had faded and inflation remained high because V/U remained extremely high.
  - The decline in inflation since early 2023 reflects falling V/U and negative headline shocks.
  - Movements in energy and auto-related prices have been the most important sources of headline shocks.
- Empirical robustness:
  - The framework and equations developed in the 2022 paper continue to explain inflation dynamics through March 2025, with high explanatory power for headline shocks (R-squared = 0.95) and a close fit for core inflation (fitted values closely tracking actual series).

### Interpretation of post‑COVID US inflation (additional discussion)
- Role of V/U:
  - The framework attributes a central role to the vacancy-to-unemployment ratio (V/U) in explaining inflation dynamics since the COVID-19 crisis.
- Comparison with other studies:
  - Alignment with Benigno and Eggertsson (2023) in emphasizing V/U; contrast with Bernanke and Blanchard (2024), who attribute more of the runup to supply shortages and sectoral price increases.
  - Dao and others (2024) highlight that this framework:
    - Allows V/U to have a nonlinear relationship with inflation.
    - Permits V/U, as a broad measure of macroeconomic conditions, to affect prices directly, not only via wage inflation.

### Recent data and labor-market implications (as of March 2025)
- Key values:
  - Twelve-month core (median) inflation stood at 3.5 percent.
  - Vacancy-to-unemployment ratio (V/U) stood at 1.1.
  - Historical average of V/U over 1985-2019 is 0.6.
- Policy implication:
  - Reducing inflation to the Federal Reserve’s target would—other things equal—require further cooling of the labor market, pushing V/U closer to 0.6.
- Beveridge curve dynamics:
  - Since 2023, an inward shift of the Beveridge curve allowed vacancies to fall, reducing V/U and inflation, without a substantial rise in unemployment.
  - Continued benign disinflation without much unemployment increase would require the Beveridge curve to shift inward relative to its pre-pandemic position; reversing the outward shift of 2020-2022 is not sufficient.
  - If the Beveridge curve stabilizes at its current position or shifts outward again, normalization of V/U will require movement along the curve, implying a significant rise in unemployment and a fall in vacancies.

### Headline shocks and asymmetry
- Future shocks could be disinflationary (e.g., decline in international energy prices) or inflationary (e.g., tariffs and supply‑chain effects).
- Key empirical finding on asymmetry: inflationary shocks pass through into core inflation while disinflationary shocks do not.
- Timing of return to target depends on:
  - Which shocks occur,
  - The evolution of V/U, and
  - Whether long-term inflation expectations remain at target-consistent levels.

### Empirical estimates — Median CPI Inflation (Table 1)
- Sample periods: January 1985-September 2022 (Column 1); January 1985-March 2025 (Column 2).
- V/U (coefficient):
  - Column (1): 8.393***
  - Column (2): 6.564***
- V/U-squared:
  - Column (1): -8.957***
  - Column (2): -6.271***
- V/U-cubed:
  - Column (1): 3.513***
  - Column (2): 2.410***
- H (headline-inflation shock):
  - Column (1): 0.071
  - Column (2): 0.048
- H-squared:
  - Column (1): 0.090***
  - Column (2): 0.085***
- H-cubed:
  - Column (1): 0.028**
  - Column (2): 0.033***
- Constant:
  - Column (1): -2.542***
  - Column (2): -2.190***
- Observations:
  - Column (1): 453
  - Column (2): 483
- R-squared:
  - Column (1): 0.553
  - Column (2): 0.622
- Adjusted R-squared:
  - Column (1): 0.547
  - Column (2): 0.617
- Notes: V/U denotes ratio of vacancies to unemployed (12-month average). H denotes headline-inflation shock (12-month average). Newey-West standard errors with 12 lags reported in the paper.

### Empirical drivers of headline inflation shocks (Table 2)
- Sample periods: January 2020-September 2022 (Column 1); January 2020-March 2025 (Column 2).
- Energy price inflation:
  - Column (1): 0.081***
  - Column (2): 0.079***
- Backlogs of work:
  - Column (1): 0.005
  - Column (2): 0.008**
- Auto-related inflation:
  - Column (1): 0.087***
  - Column (2): 0.087***
- Constant:
  - Column (1): -0.034*
  - Column (2): -0.050***
- Observations:
  - Column (1): 33
  - Column (2): 63
- R-squared:
  - Column (1): 0.952
  - Column (2): 0.950
- Adjusted R-squared:
  - Column (1): 0.947
  - Column (2): 0.947
- Notes: Relative energy and auto-related inflation variables are created by subtracting median inflation from energy and auto-related inflation respectively. Backlogs of work variable is taken from IHS Markit Economics, with the neutral value of 50 subtracted.

### Data revisions and vintage effects (Appendix A)
- Seasonal adjustment revisions in 2023 materially changed seasonally adjusted CPI inflation for 2022.
- Revisions reduced average core (weighted median) inflation over June-September 2022 from 8.3 to 7.6.
- Vintage effects on estimated relationships:
  - Relationship between core inflation and V/U is somewhat steeper at high V/U when estimated with current-vintage data through September 2022 than with data through March 2025.
  - Relationship is steepest when using the older data from the earlier paper; updating the data through September 2022 made a larger difference than extending the sample through March 2025.
  - Older data imply higher fitted values for core inflation in late 2022 and 2023 because they yield a steeper relationship at high V/U.

### PCE deflator results (Appendix B)
- Main exercises repeated using the PCE deflator (core measured by the weighted median PCE inflation rate from the Federal Reserve Bank of Cleveland).
- General conclusion: Results for PCE deflator inflation are similar to those for CPI inflation.
- Table B1 — Phillips Curve Estimates: Median PCE Inflation (coefficients and fit)
  - Column (1) — 1985-2022:
    - V/U: 6.470*** (1.777)
    - V/U-squared: -7.360*** (2.124)
    - V/U-cubed: 3.083*** (0.734)
    - H: 0.153 (0.113)
    - H-squared: 0.151*** (0.021)
    - H-cubed: 0.039* (0.024)
    - Constant: -1.757*** (0.417)
    - Observations: 453
    - R2: 0.460
    - Adjusted R2: 0.452
  - Column (2) — 1985-2025:
    - V/U: 5.450*** (1.591)
    - V/U-squared: -5.413*** (1.789)
    - V/U-cubed: 2.061*** (0.599)
    - H: 0.219* (0.125)
    - H-squared: 0.159*** (0.023)
    - H-cubed: 0.037 (0.024)
    - Constant: -1.600*** (0.400)
    - Observations: 482
    - R2: 0.476
    - Adjusted R2: 0.469
- Key PCE findings:
  - Slack (V/U) has a strongly nonlinear relationship with the PCE inflation gap; all V/U polynomial terms statistically significant at the 1 percent level in both samples.
  - H-squared is statistically significant at the 1 percent level in both samples; H is significant at the 10 percent level in Column (2) (0.219*); H-cubed is weakly significant in Column (1) (0.039*).
  - Model fit improves modestly when extending the sample to February/March 2025.

*IMF Working Paper — The Rise and Retreat of US Inflation: An Update (content unit: wpiea2025094-print-pdf).*

### Section 2 of this paper reviews the behavior of CPI inflation since 2020. As in our earlier work, we decompose

### wpiea2025094-print-pdf — Section 2–6 summary

### Decomposition approach and scope
- Headline CPI inflation is decomposed into:
  - Core inflation: weighted median inflation rate from the Federal Reserve Bank of Cleveland.
  - Headline shocks: headline minus core inflation, reflecting large relative-price changes in particular industries.
- Sample and series:
  - Main series analyzed from 1985 through March 2025, with focused analysis since 2020.
  - PCE deflator considered in the Appendix with broadly similar results.

### Stylized facts since 2020
- Rise and retreat:
  - The inflation surge over 2021–2022 reflected both rising core inflation and positive headline shocks; subsequent disinflation reflected declining core inflation and negative headline shocks.
- Key numeric milestones:
  - Twelve-month core inflation in March 2025: 3.5.
  - Fitted twelve-month core inflation in March 2025:
    - 3.1 (estimates based on sample ending September 2022 — out-of-sample conditional forecast after that date).
    - 3.2 (estimates based on sample extended through March 2025).
  - Expected inflation (median ten-year-ahead SPF):
    - 2.2 percent in 2019:Q4.
    - Peaked at 2.95 percent in 2022:Q4.
    - 2.3 percent in 2025:Q1.
  - Vacancy-to-unemployment ratio (V/U):
    - Peaked at 2.0 in March 2022.
    - Historical average 0.6 over 1985–2019.
    - 1.1 in February 2025 (and 1.1 in December 2019).
    - Unemployment rate never rose above 4.2 percent during the disinflation period.
  - Headline shocks (twelve-month average):
    - Positive from March 2021 to December 2022, peaking at 3.7 percentage points in March 2022.
    - Trough of -3.2 in June 2022.
    - -1.1 percentage points in March 2025.
  - Twelve-month core inflation peak: 7.0 percent in February 2023.

### Three drivers analyzed informally (Section 3)
- Inflation expectations:
  - Rose modestly in 2021–2022 relative to actual inflation; rise peaked at 2.95 percent in 2022:Q4 and reversed from 2023:Q1 back toward pre-pandemic levels by 2025:Q1.
- Labor market tightness (V/U):
  - V/U rose rapidly in 2021, peaked in March 2022 at 2.0 (an unusually high level relative to the 1985–2019 average of 0.6), then fell steadily through mid-2024 to 1.1 in February 2025.
  - Beveridge curve: stable 2009–March 2020; dramatic outward shift at pandemic start; movement back toward pre-pandemic curve by end-2024.
  - The rise and fall in V/U is identified as the most important factor behind core inflation behavior in subsequent econometric work.
- Headline shocks:
  - Three explanatory variables for headline shocks: relative energy inflation (sector minus median), relative auto-related inflation (sector minus median), and firms’ backlogs of work (IHS Markit survey; neutral level = 50).
  - All three moved to inflationary levels late 2020–2022 and then moved disinflationary as relative energy and auto prices reversed and supply chains normalized.
  - Empirically, these variables explain the timing and sign of headline shocks from 2020–2025.

### Core inflation equation (Section 4)
- Core inflation specification:
  - π = πe + f(V/U) + g(H) + ε
    - π = annualized monthly core inflation.
    - πe = expected inflation (SPF median, ten-year-ahead).
    - V/U = average of V/U over the twelve months t-11 to t.
    - H = average of headline shock (headline minus median) over t-11 to t.
  - f(•) and g(•) are specified as cubic functions to allow for possible non-linearities and asymmetries.
- Estimated relations and interpretation:
  - The estimated f(V/U) is steepest at unusually high or low V/U and flatter in between; steep slope for V/U above 1.5 reflects high core inflation when V/U was in that range in 2022–2023.
  - g(H) is sharply asymmetric: positive headline shocks pass through into higher core inflation, but negative shocks do not have a significant effect.
  - Extending the sample through March 2025 modestly reduces the estimated steepness at high V/U but does not materially change the asymmetry in pass-through from H.
- Fit and forecasting:
  - Fitted values from the equation (both full sample and sample ending September 2022) follow actual core inflation closely over 2020–2025.
  - Fitted twelve-month core inflation in March 2025: 3.1 (shorter sample), 3.2 (longer sample); actual 3.5 percent.
  - Data revisions (seasonal adjustments) led to some differences relative to earlier published fitted values.
- Decomposition of fitted core inflation (full sample 1985–2025):
  - The rise in twelve-month core inflation (peak 7.0 percent in February 2023) reflected:
    - Pass-through from headline shocks (important early in the rise).
    - Rising V/U (increasingly important as the rise continued).
    - Rising expected inflation contributed modestly.
  - The pass-through effect faded in early 2023; core inflation remained high at that point because V/U was at its peak.
  - Subsequent decline in core inflation was driven primarily by falling V/U and modestly by falling expected inflation.
  - Negative headline shocks in 2023–2025 did not materially lower core inflation because the estimated pass-through is asymmetric (negative H has little effect on core).

### Headline shocks equation (Section 5)
- Regression results:
  - Headline shocks regressed on the three factors (energy relative inflation, auto-related relative inflation, backlogs of work).
  - Coefficient estimates similar for January 2020–September 2022 and for the extended sample; R-squared = 0.95 for both samples.
  - Fitted values from either sample closely track actual headline shocks through March 2025.
- Factor contributions (decomposition based on full-sample fitted values):
  - Energy prices: the most important factor—positive contributions June 2020–June 2022, especially spikes after Russia’s invasion of Ukraine in February 2022; then mostly negative contributions as energy prices subsided.
  - Auto prices: material contributor during the rising-inflation period, especially April–June 2021 (chip shortage).
  - Backlogs of work: modest contribution after controlling for auto and energy prices; rose above neutral (50) then fell below it.
  - Residuals (idiosyncratic industry shocks): mostly negative during the rising-inflation period and included large negative residuals from falls in airline fares and apparel prices (February and September 2021); during falling inflation, largest negative residual in November 2023 reflected a fall in the relative price of food at home.

### Overarching interpretation and conclusions (Section 6)
- Central interpretation:
  - The rise in twelve-month inflation from January 2020 to the peak in June 2022 is explained primarily by positive headline shocks, which raised inflation directly and by passing through into core inflation.
  - By early 2023, the pass-through effects of those shocks had faded and inflation remained high because V/U remained extremely high.
  - The decline in inflation since early 2023 reflects falling V/U and negative headline shocks.
  - Movements in energy and auto-related prices have been the most important sources of headline shocks.
- Empirical robustness:
  - The framework and equations developed in the 2022 paper continue to explain inflation dynamics through March 2025, with high explanatory power for headline shocks (R-squared = 0.95) and a close fit for core inflation (fitted values closely tracking actual series).

*IMF Working Paper — Section summary as provided in the source content.*

### 2022. In our view, the good fit of the framework in explaining the disinflation since 2022 strengthens the

### wpiea2025094-print-pdf - 2022. In our view, the good fit of the framework in explaining the disinflation since 2022 strengthens the

### Interpretation of post‑COVID US inflation
- The framework attributes a central role to the vacancy-to-unemployment ratio (V/U) in explaining inflation dynamics since the COVID-19 crisis.
- The paper’s interpretation aligns with Benigno and Eggertsson (2023) in emphasizing V/U; it differs from Bernanke and Blanchard (2024), who attribute more of the runup to supply shortages and sectoral price increases.
- Dao and others (2024) suggest differences arise because this framework:
  - Allows V/U to have a nonlinear relationship with inflation.
  - Permits V/U, as a broad measure of macroeconomic conditions, to affect prices directly, not only via wage inflation.

### Recent data and current conditions (as of March 2025)
- Twelve-month core (median) inflation stood at 3.5 percent.
- Vacancy-to-unemployment ratio (V/U) stood at 1.1.
- Historical average of V/U over 1985-2019 is 0.6.
- Implication: Reducing inflation to the Federal Reserve’s target would—other things equal—require further cooling of the labor market, pushing V/U closer to 0.6.

### Beveridge curve dynamics and implications for labor market adjustment
- Since 2023, an inward shift of the Beveridge curve allowed vacancies to fall, reducing V/U and inflation, without a substantial rise in unemployment.
- For continued benign disinflation without much unemployment increase would require the Beveridge curve to shift inward relative to its pre-pandemic position; reversing the outward shift of 2020-2022 is not sufficient.
- If the Beveridge curve stabilizes at its current position or shifts outward again, normalization of V/U will require movement along the curve, implying:
  - A significant rise in unemployment, and
  - A fall in vacancies.

### Headline shocks and asymmetry
- Future headline shocks could be disinflationary (e.g., decline in international energy prices) or inflationary (e.g., tariffs and supply‑chain effects).
- Key finding on asymmetry: inflationary shocks pass through into core inflation while disinflationary shocks do not.
- Therefore, the timing of return to target depends on:
  - Which shocks occur,
  - The evolution of V/U, and
  - Whether long-term inflation expectations remain at target-consistent levels.

### Empirical estimates — Median CPI Inflation (Table 1)
- Sample periods: January 1985-September 2022 (Column 1); January 1985-March 2025 (Column 2).
- V/U (coefficient):
  - Column (1): 8.393***
  - Column (2): 6.564***
- V/U-squared:
  - Column (1): -8.957***
  - Column (2): -6.271***
- V/U-cubed:
  - Column (1): 3.513***
  - Column (2): 2.410***
- H (headline-inflation shock):
  - Column (1): 0.071
  - Column (2): 0.048
- H-squared:
  - Column (1): 0.090***
  - Column (2): 0.085***
- H-cubed:
  - Column (1): 0.028**
  - Column (2): 0.033***
- Constant:
  - Column (1): -2.542***
  - Column (2): -2.190***
- Observations:
  - Column (1): 453
  - Column (2): 483
- R-squared:
  - Column (1): 0.553
  - Column (2): 0.622
- Adjusted R-squared:
  - Column (1): 0.547
  - Column (2): 0.617
- Notes: V/U denotes ratio of vacancies to unemployed (12-month average). H denotes headline-inflation shock (12-month average). Newey-West standard errors with 12 lags reported in the paper.

### Empirical drivers of headline inflation shocks (Table 2)
- Sample periods: January 2020-September 2022 (Column 1); January 2020-March 2025 (Column 2).
- Energy price inflation:
  - Column (1): 0.081***
  - Column (2): 0.079***
- Backlogs of work:
  - Column (1): 0.005
  - Column (2): 0.008**
- Auto-related inflation:
  - Column (1): 0.087***
  - Column (2): 0.087***
- Constant:
  - Column (1): -0.034*
  - Column (2): -0.050***
- Observations:
  - Column (1): 33
  - Column (2): 63
- R-squared:
  - Column (1): 0.952
  - Column (2): 0.950
- Adjusted R-squared:
  - Column (1): 0.947
  - Column (2): 0.947
- Notes: Relative energy and auto-related inflation variables are created by subtracting median inflation from energy and auto-related inflation respectively. Backlogs of work variable is taken from IHS Markit Economics, with the neutral value of 50 subtracted.

### Data revisions and vintage effects (Appendix A)
- Seasonal adjustment revisions in 2023 materially changed seasonally adjusted CPI inflation for 2022.
- Revisions reduced average core (weighted median) inflation over June-September 2022 from 8.3 to 7.6.
- Vintage of CPI data affects the estimated relationship between core inflation and V/U:
  - Relationship is somewhat steeper at high V/U when estimated with current-vintage data through September 2022 than with data through March 2025.
  - Relationship is steepest when using the older data from the earlier paper; updating the data through September 2022 made a larger difference than extending the sample through March 2025.
  - Older data imply higher fitted values for core inflation in late 2022 and 2023 because they yield a steeper relationship at high V/U.

### PCE deflator results (Appendix B)
- The paper repeats the main empirical exercises using the PCE deflator (core measured by the weighted median PCE inflation rate from the Federal Reserve Bank of Cleveland).
- General conclusion: Results for PCE deflator inflation are similar to those for CPI inflation.

*Source: IMF Working Paper — The Rise and Retreat of US Inflation: An Update (content unit: wpiea2025094-print-pdf).*

### Appendix A: Data Vintages

### wpiea2025094-print-pdf - Appendix A: Data Vintages

### Appendix A — Data vintages and model sensitivity
- Figures A1–A3 compare current data and a September 2022 vintage dataset:
  - Figure A1: Median CPI inflation (monthly annualized rate, percent) — visual comparison of current data vs. September 2022 vintage; vertical line indicates September 2022.
  - Figure A2: Estimated functions comparing fitted values across vintages:
    - Panel A: fitted values for constant term and V/U terms based on monthly equation estimated with current data for January 1985 – September 2022 (Table 1, Column 1) and January 1985 – March 2025 (Table 1, Column 2), and on January 1985 – September 2022 using the Ball, Leigh, and Mishra (2022) vintage dataset.
    - Panel B: fitted values for headline inflation shock (H) terms.
    - Dotted lines indicate 95 percent confidence intervals.
    - Inflation gap denotes monthly annualized median CPI inflation minus long-term inflation expectations.
  - Figure A3: Predictions for median CPI inflation across data vintages:
    - Predicted values based on monthly equation estimated with current data for January 1985 – September 2022 (Table 1, Column 1) and for January 1985 – March 2025 (Table 1, Column 2), and on January 1985 – September 2022 using the Ball, Leigh, and Mishra (2022) vintage dataset.
    - Vertical line indicates September 2022.

### Key points from Appendix A visuals
- The appendix emphasizes model sensitivity to data vintage by:
  - Showing divergences between actual series in current data and the September 2022 vintage for median CPI inflation (monthly annualized and twelve-month inflation).
  - Reporting fitted relationships between inflation gap and two drivers: V/U (ratio of vacancies to unemployed, 12-month average) and headline inflation shock H (12-month average), with 95 percent confidence intervals.
  - Presenting model-predicted inflation paths estimated on different sample end dates and vintage datasets to illustrate robustness and vintage-driven differences.

*Source: wpiea2025094-print-pdf — Appendix A: Data Vintages (IMF Working Paper No. WP/2025/094).*

### Appendix B: PCE Inflation

### PCE inflation series and decomposition (Figures B1–B5)
- Figure B1: PCE inflation series, 2020–2025 (percent)
  - Series plotted: Headline, Headline-inflation shocks, Core inflation (weighted median PCE from FRB Cleveland).
  - Vertical line indicates September 2022.
  - Panels show both monthly annualized inflation and twelve-month inflation.
- Figure B2: Estimated PCE inflation gap as function of slack (V/U) and headline inflation shocks (H)
  - Panel A: fitted values for constant term and V/U terms from Table B1 estimates.
  - Panel B: fitted values for H terms.
  - Dotted lines show 95 percent confidence intervals.
  - Inflation gap denotes monthly annualized median PCE inflation minus long-term inflation expectations.
- Figure B3: Predictions for median PCE inflation since the pandemic — comparison across sample periods
  - Predicted values based on monthly equation estimated for January 1985 - September 2022 (Table B1, Column 1) and January 1985 – February 2025 (Table B1, Column 2).
  - Panels show monthly annualized inflation and twelve-month inflation.
- Figure B4: Accounting for the rise and retreat of core PCE inflation (percent)
  - Contributions decomposed into: Pass-through (pass-through of headline shocks to core inflation), Expectations (contribution of long-term SPF inflation expectations to core inflation), and V/U (ratio of vacancies to unemployed).
  - Plots display monthly annualized and twelve-month contributions and core inflation series for 2020–2025.
- Figure B5: Accounting for the rise and retreat of headline PCE inflation (percent)
  - Contributions decomposed into: Headline shock, Passthrough, Expectations (contribution of long-term SPF inflation expectations to core (weighted median) inflation), and V/U.
  - Plots show monthly annualized and twelve-month contributions and headline inflation series for 2020–2025.

### Table B1 — Phillips Curve Estimates: Median PCE Inflation
- Model specification notes:
  - V/U denotes ratio of vacancies to unemployed (twelve-month average).
  - H denotes headline-inflation shock (twelve-month average).
  - Columns (1) and (2) are based on samples January 1985–September 2022, and January 1985–February 2025 respectively.
  - Newey-West standard errors with 12 lags in parentheses. ***, **, * denote significance at 1, 5, and 10 percent.
- Coefficient estimates and inference (preserve exact values):
  - Column (1) — 1985-2022:
    - V/U: 6.470*** (1.777)
    - V/U-squared: -7.360*** (2.124)
    - V/U-cubed: 3.083*** (0.734)
    - H: 0.153 (0.113)
    - H-squared: 0.151*** (0.021)
    - H-cubed: 0.039* (0.024)
    - Constant: -1.757*** (0.417)
    - Observations: 453
    - R2: 0.460
    - Adjusted R2: 0.452
  - Column (2) — 1985-2025:
    - V/U: 5.450*** (1.591)
    - V/U-squared: -5.413*** (1.789)
    - V/U-cubed: 2.061*** (0.599)
    - H: 0.219* (0.125)
    - H-squared: 0.159*** (0.023)
    - H-cubed: 0.037 (0.024)
    - Constant: -1.600*** (0.400)
    - Observations: 482
    - R2: 0.476
    - Adjusted R2: 0.469

### Key findings from Appendix B
- Slack (V/U) has a strongly nonlinear relationship with the PCE inflation gap:
  - Positive linear V/U coefficients and negative squared coefficients with positive cubic terms in both samples; all V/U polynomial terms are statistically significant at the 1 percent level in both columns.
- Headline-inflation shock (H) shows:
  - H-squared is statistically significant at the 1 percent level in both samples.
  - H is not significant in Column (1) but is significant at the 10 percent level in Column (2) (0.219*).
  - H-cubed is weakly significant in Column (1) at the 10 percent level (0.039*) and not significant in Column (2).
- Model fit:
  - R2 increases slightly from 0.460 to 0.476 when extending the sample to February 2025.
  - Observations increase from 453 to 482 when extending the sample.

*Source: wpiea2025094-print-pdf — Appendix B: PCE Inflation (IMF Working Paper No. WP/2025/094).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025094-print-pdf.pdf_