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### Measuring core inflation
- Primary measure: the weighted median inflation rate published by the Federal Reserve of Cleveland (median CPI), which strips out the effects of unusually large price changes in certain industries.
- Rationale: median CPI isolates the core component of inflation more effectively than the traditional core measure excluding food and energy prices (XFE), especially during the COVID era when much headline volatility has come from industries other than food and energy.
- September 2022 snapshot: weighted median inflation accounted for 7.0 percentage points of the 8.2 headline inflation rate.
- Operational definition: “headline-inflation shocks” = deviations of headline from median; median (core) plus deviations (headline shocks) sum to headline inflation.
- Alternatives discussed (Appendix): trimmed mean PCE deflator (Federal Reserve Bank of Dallas) and weighted median PCE deflator (Federal Reserve Bank of Cleveland).

### Empirical framework and key variables
- Decomposition: headline inflation = core inflation + headline shocks.
- Core (median) interpreted as a slow-moving component determined by:
  - Expected inflation (ten-year SPF forecasts).
  - Aggregate labor-market slack measured by V/U (ratio of job vacancies to unemployment), averaged over current and previous eleven months (or current and previous three quarters in some exercises).
  - Past headline shocks measured by the average deviation of headline from median over the same averaging window.
- Dependent variable: core inflation gap = core inflation − expected inflation.
- Nonlinearities: cubic functions of V/U and headline shocks H are included to capture nonlinearity and asymmetry in responses.

### Main empirical findings and magnitudes
- Pre-pandemic core inflation equation fits pandemic-period core inflation well (out-of-sample fit using 1985–2019 estimates).
- Drivers of the 2021–2022 increase in core inflation:
  - Rise in V/U to unprecedented levels.
  - Pass-through from adverse headline shocks, with V/U’s role growing over the last year.
- Non-linearity and asymmetry:
  - V/U has larger marginal effects on core inflation when V/U is unusually high or low.
  - Pass-through is strongly asymmetric: negligible for negative shocks (headline below median) and strong for positive shocks.
- Wage channel:
  - Both V/U and past headline shocks strongly affect nominal wage growth (ECI), supporting wage → price transmission.
- Phillips-curve estimates (Table 1, preserving reported coefficients from Monthly 1985–2022, Column (4)):
  - V/U: 9.140*** (2.234)
  - V/U‐squared: ‐10.328*** (2.545)
  - V/U‐cubed: 4.241*** (0.863)
  - H: 0.058 (0.075)
  - H‐squared: 0.089*** (0.019)
  - H‐cubed: 0.031** (0.012)
  - Constant: ‐2.654*** (0.586)
  - Observations: 453; R‐squared: 0.575; Rbar‐squared: 0.569

### Pandemic-era headline shocks: drivers and magnitudes
- Three most important contributors to headline-minus-core deviations during the pandemic:
  - Energy-price shocks.
  - IHS Markit backlogs of work (supply-chain measure).
  - Auto-related price changes (new and used cars, rentals, insurance).
- Headline-minus-core shocks magnitude: 12-month average rose as high as 3.7 percentage points (March 2022) and was 1.4 percentage points in September 2022.
- Simple regressions (Table 2 bivariate Rbar-squared, selected):
  - Energy price inflation: Rbar‐squared = 0.646 (coefficient 0.064*** (0.013), Observations 33)
  - Backlogs of work: Rbar‐squared = 0.429 (coefficient 0.585*** (0.114), Observations 33)
  - Weighted average of car inflation rates: Rbar‐squared = 0.191 (coefficient 0.081*** (0.021), Observations 33)
- Multivariate regression (energy, backlogs, auto): adjusted R‐squared = 0.912.

### Decomposition of the rise in inflation (December 2020 → September 2022)
- 12-month headline inflation: from 1.3 percent (December 2020) to 8.2 percent (September 2022) = 6.9 point rise.
- 12-month core (median) inflation: from 2.3 percent to 7.0 percent = 4.6 percentage point rise.
- Using preferred core and headline-shock equations:
  - Direct plus pass-through effects of headline-inflation shocks account for about 4.6 percentage points of the 6.9 point rise.
  - Contributions within that 4.6 percentage points:
    - Energy-price shocks: 2.7 percentage points.
    - Backlogs of work: 1.7 percentage points.
    - Auto-price shocks: net effect includes significant pass-through from past auto shocks; direct current effect turned negative reflecting partial reversals.
  - For energy and backlogs, ~two thirds of each contribution is current headline inflation and ~one third is pass-through into core.
  - Rise in expected inflation: 0.5 percentage point.
  - Rise in V/U: 2.0 percentage points (nearly a third of total inflation increase).
- Change in annualized one-month core inflation (December 2020 → September 2022): rose 6.4 percentage points (from 1.9 percent to 8.3 percent); V/U’s contribution to that change: 5.0 percentage points.

### Role and counterfactual of the American Rescue Plan (ARP)
- ARP context: $1.9 trillion stimulus passed March 2021.
- Using Barnichon, Oliveira, and Shapiro (2021) estimates: ARP increased V/U by ~0.6 at end-2021 and ~0.5 at end-2022; monthly path linearly interpolated.
- Counterfactual (no ARP) September 2022 results:
  - Monthly core (median CPI) inflation: counterfactual 4.1 percent versus actual 8.3 percent → difference 4.2 percentage points.
  - 12-month core inflation: counterfactual 5.1 percent versus actual 7.0 percent → difference 1.9 percentage points.
  - The 1.9 percentage point difference ≈ 40 percent of the rise in 12-month core inflation from end-2020 to September 2022 and ≈ one quarter of the rise in 12-month headline inflation.
- Caveat: if ARP also worsened supply-chain problems (raising H), these estimates are lower bounds.

### Labor-market tightness, Beveridge curve, and natural-rate implications
- V/U measure: ratio of job vacancies (JOLTS, Barnichon extension) to unemployment; theory: V/U captures bargaining pressures for wages.
- First half of 2022 averages:
  - Unemployment rate = 3.7 percent (slightly above January 2020 level of 3.5).
  - Average V/U = 1.88 (highest since 1951 in Barnichon series).
- Beveridge-curve estimates (log-linear v = a u^b):
  - Pre-pandemic (July 2009 – March 2020): a = 13.9, b = –0.85.
  - Pandemic (April 2020 – August 2022): a = 15.5, b = –0.60.
- Implication: outward shift raises the natural rate of unemployment if persistent.
- Estimated natural rates (assuming long-term expected core inflation is 0.20 percentage points higher than expected headline inflation):
  - Pre-pandemic Beveridge curve: 4.8 percent.
  - Pandemic-era Beveridge curve: 6.5 percent.
  - Estimated pandemic increase in the natural rate: 1.7 points.
- Sensitivity examples:
  - If difference between expected median and expected headline inflation = 0 → natural rates 5.5 percent and 7.8 percent.
  - If difference = 0.4 percentage points → natural rates 4.0 percent and 5.3 percent.

### Inflation expectations: measurement and behavior
- Measure: median ten-year-ahead CPI inflation forecast from SPF.
- Historical (2000–2019): expected inflation averaged 2.36 percent, never deviating by more than 0.3 percentage points from this level.
- Pandemic-era movement: expected inflation rose from 2.2 percent in 2019Q4 to 2.8 percent in 2022Q3.
- Simple expectations model: π^e_t = γ π^e_{t-1} + (1–γ) π_t; estimated γ by era (quarterly):
  - 1985Q1-1998Q1: γ = 0.945
  - 1998Q2-2008Q2: γ = 1.003
  - 2008Q3-2019Q4: γ = 0.991
  - 2020Q1-2022Q3: γ = 0.980
- Interpretation: γ closer to 1 implies tighter anchoring; pandemic-era γ = 0.980 indicates modest weakening of anchoring.

### Scenario simulations and key outcomes (methodology preserved)
- Simulation setup:
  - Start from actual data through September 2022; set monthly headline shocks to zero from October 2022 onward.
  - Monthly steps derive V/U from assumed U and Beveridge curve, compute 12-month averages, compute core inflation gap via monthly Phillips curve (Table 1, column (4)), and update expected inflation via the expectations equation (or a faster reversion in the most optimistic scenario).
  - With zero future headline shocks, monthly headline = core and 12-month headline converges to core by September 2023.
- Unemployment paths considered:
  - FOMC SEP September 2022: unemployment peaks at 4.4 percent in late 2023.
  - IMF October 2022 WEO: unemployment peaks at 5.6 percent in second half of 2024.
  - Summers (2022b) scenario: unemployment rises to 7.5 percent for two years.
- Beveridge-curve scenarios:
  - Pessimistic: pandemic-era Beveridge curve persists.
  - Optimistic: Beveridge curve shifts back toward pre-pandemic position at one quarter of the gap per month starting July (near-complete reversal in ~6 months).
- Expectations scenarios (monthly; starting SPF Sept 2022 = 2.8 percent):
  - Most optimistic: expected inflation reverts toward 2.2 percent one quarter of the way each month.
  - Medium: expected inflation follows γ = 0.980 (quarterly → cube root for monthly).
  - Pessimistic: γ = 0.944 (quarterly).
- Representative outcomes under SEP unemployment path for December 2024:
  - Most optimistic (optimistic Beveridge curve & expectations): core inflation falls to 2.5 percent (peak at 7.5 percent in Dec 2022–Jul 2023 range).
  - Most pessimistic: core inflation = 6.3 percent (peak 8.6 percent).
  - If Beveridge curve reverts to pre-pandemic, December 2024 inflation ranges from 2.5 to 3.9 percent depending on expectations.
  - If pandemic-era Beveridge curve persists, December 2024 inflation always stays above 4.0 percent.
- Under IMF unemployment path (peak 5.6 percent), December 2024 inflation ranges from 2.3 to 4.8 percent.
- Under Summers 7.5 percent unemployment for two years, December 2024 inflation rates cluster around 2.6 percent.
- Fed-equivalent target comparison:
  - Fed’s 2 percent PCE target ≈ 2.6 percent for median CPI per Atlanta Fed dashboard.
  - Most optimistic forecast: median CPI = 2.53 percent in Dec 2024 (slightly below 2.6).
  - In most other scenarios, inflation stays above the Fed-equivalent 2.6 percent.
- Sacrifice ratio: defined as additional unemployment required to reduce inflation by one extra percentage point by Dec 2024; sacrifice ratio increases with larger inflation reductions and is scenario-dependent.

### Why high inflation was such a surprise (diagnosis)
- Primary forecasting errors:
  - Unexpected, persistent headline inflation shocks (supply-chain disruptions; energy-price rise linked to the war in Ukraine).
  - Pre-pandemic modeling flaws: underweighting V/U relative to unemployment, assuming linear/small unemployment effects, neglecting pass-through H.
- Specification experiments show incremental improvements when:
  - Replacing unemployment with V/U.
  - Allowing cubic (nonlinear) functions of V/U.
  - Adding pass-through variable H (also cubic).
- Preferred specification (nonlinear V/U + pass-through H) predicts most of the pandemic-era rise in core inflation when estimated on pre-pandemic data.

### Suggested directions for future research
- Refine measurement of:
  - Core inflation.
  - Labor-market tightness (V/U dynamics and measurement).
  - Inflation expectations.
- Better understand:
  - Non-linear effects of tightness and past headline shocks on core inflation.
  - Causes of Beveridge-curve shifts and mechanisms for anchoring/de-anchoring inflation expectations.

*Source: Section 1–5 and Appendix of IMF Working Paper No. WP/2022/208 (wpiea2022208-print-pdf).*

### Section 1 of this paper describes how we measure core inflation. Our primary measure is the

### wpiea2022208-print-pdf - Section 1 of this paper describes how we measure core inflation. Our primary measure is the

### Measuring core inflation
- Primary measure: the weighted median inflation rate published by the Federal Reserve of Cleveland, which strips out the effects of unusually large price changes in certain industries.
- Rationale: This variable isolates the core component of inflation more effectively than the traditional core measure excluding food and energy prices (XFE), especially during the COVID era when much headline volatility has come from industries other than food and energy.
- September 2022 snapshot: weighted median inflation accounted for 7.0 percentage points of the 8.2 headline inflation rate.
- Operational definition: “headline-inflation shocks” are defined as deviations of headline from median; by construction, median (core) plus deviations (headline shocks) sum to headline inflation.
- Alternative measures discussed (in Appendix): trimmed mean PCE deflator (Federal Reserve Bank of Dallas) and weighted median PCE deflator (Federal Reserve Bank of Cleveland).

### Behavior of core inflation and key empirical framework
- Core inflation (median) interpreted as a relatively slow-moving component determined by inflation expectations and aggregate labor-market slack (textbook Phillips curve).
- Headline shocks: high-frequency movements from large price changes in particular sectors (e.g., energy, travel-related industries, used cars).
- Basic decomposition used: headline inflation = core inflation + headline shocks.
- Primary empirical specification explains core inflation with:
  - Expected inflation (ten-year forecasts from the Survey of Professional Forecasters — SPF).
  - Labor-market tightness measured by the ratio of job vacancies to unemployment (V/U), averaged over current and previous eleven months.
  - Past headline shocks measured by the average deviation of headline from median over current and previous eleven months.
- Dependent variable in equations: the “core inflation gap” = core inflation − expected inflation.

### Main empirical findings
- Pre-pandemic core inflation equation fits pandemic-period core inflation well.
- Drivers of the 2021–2022 increase in core inflation:
  - Rise in the V/U ratio to unprecedented levels.
  - Pass-through from adverse headline shocks, with V/U’s role growing over the last year.
- Non-linearity and asymmetry:
  - Some evidence of non-linearity: large positive marginal effect of V/U on core inflation when V/U is above or below its usual range.
  - Strong evidence of asymmetry in pass-through: negligible for shocks that reduce headline inflation; strong for shocks that increase headline inflation.
- Contribution estimates for policy stimulus:
  - Using Barnichon, Oliveira, and Shapiro (2021) estimates of ARP effects on V/U, the American Recovery Act (the “Biden stimulus”) effect for September 2022 is:
    - 4.2 percentage points on annualized monthly inflation.
    - 1.9 percentage points on 12-month inflation (and rising).
- Wage channel:
  - Both V/U and past headline shocks have strong effects on nominal wage growth, supporting transmission of labor-market tightness and headline shocks into core inflation via wages.

### Pandemic-era headline shocks (Section 3 findings)
- Three most important factors explaining headline minus core deviations during the pandemic:
  - Changes in energy prices.
  - A measure of backlogs of goods and services orders from IHS Markit Economics (captures supply-chain problems).
  - Changes in prices in auto-related industries.
- Decomposition of the 6.9 percentage point rise in headline inflation between end-2020 and September 2022 (from 1.3 percent to 8.2 percent):
  - Combination of direct and pass-through effects from headline-inflation shocks: about 4.6 percentage points of the rise in 12-month inflation.
  - Rise in expected inflation: 0.5 percentage point.
  - Rise in labor-market tightness (measured by V/U): 2.0 percentage points.

### Headline and median CPI dynamics since 2020 (selected statistics)
- Monthly headline CPI: highly volatile — near -10 percent in April 2020; soared to 17.1 percent in June 2022; fell to -0.2 percent in July 2022; was 4.7 percent in September 2022.
- 12-month headline CPI: peaked at 9.1 percent in June 2022; was 8.2 percent in September 2022.
- Median CPI stability: monthly median series never changed by more than 3 percentage points from one month to the next. By September 2021 the 12-month median was below its January 2020 level; by September 2022 the 12-month median reached 7.0 percent.

### Role of expected inflation (Section 4 context)
- Measure used: median ten-year-ahead CPI inflation forecast from the SPF.
- Historical behavior (2000–2019): expected inflation averaged 2.36 percent, never deviating by more than 0.3 percentage points from this level.
- Pandemic-era movement: expected inflation rose from 2.2 percent in 2019Q4 to 2.8 percent in 2022Q3.
- Modeling assumption: core inflation responds one-for-one to movements in long-run expected inflation.

### Labor-market tightness: V/U versus unemployment
- Measure adopted: ratio of job vacancies (V) to unemployment (U) using JOLTS and Barnichon extension.
- The V/U ratio has theoretical appeal as a determinant of wage pressures (search model bargaining threat points).
- Pandemic divergence:
  - First half of 2022 averages: unemployment rate = 3.7 percent (slightly above January 2020 level of 3.5); average V/U = 1.88, the highest since 1951 in the Barnichon series.
  - Implication: unemployment alone understated labor-market tightness; V/U signaled an unusually tight labor market.
- Empirical approach: compare inflation gap (median − expected inflation) to averages of V/U over current and previous three quarters or current and previous 11 months to capture lagged effects.

### Scenarios and costs of disinflation (Sections 4–5)
- Key determinants of future disinflation costs:
  - The Beveridge curve relationship between unemployment and vacancies: an unfavorable (shifted) Beveridge curve implies higher unemployment costs to reduce inflation; normalization toward pre-pandemic Beveridge curve would lower costs.
  - Evolution of long-term inflation expectations: currently well-anchored through most of the pandemic but showing hints of increase during 2022; a substantial upward trend would raise the costs of containing inflation.
- Simulation highlights:
  - Consideration of an unemployment path equal to Fed SEP September 2022 projections, which peak at 4.4 percent in 2023 and 2024.
  - Under optimistic assumptions about both the Beveridge curve and inflation expectations, inflation falls to a level near the Fed’s target by the end of 2024.
  - Under a range of other plausible assumptions, inflation remains well above target.
  - Conclusion: policymakers will likely need to push unemployment higher than the SEP projections if they are determined to meet the inflation goal.

*Source: Section 1–5 of the provided IMF working paper content (wpiea2022208-print-pdf).*

### Appendix considers the relation between the inflation gap and the current level of V/U alone).

### wpiea2022208-print-pdf - Appendix considers the relation between the inflation gap and the current level of V/U alone)

### Relation between the inflation gap and V/U (scatterplots and fitted curves)
- Sample partition and markers used in figures:
  - 1968-1972
  - 1973-1984 (period of high inflation and then disinflation ushered in by the first oil shock)
  - 1985-2019 (includes the Great Moderation of 1985-2007 and the Great Recession and recovery)
  - COVID era 2020-2022 (through 2022 Q3 or September)
- Key observations:
  - The 1973-1984 period shows unusually high inflation gaps and a steep relation between the gap and V/U; attributed to pre-Volcker monetary regime, accommodative policy, unanchored expectations, and CPI housing treatment before 1982.
  - Data from 1985 onward appear consistent with an upward-sloping relation between the inflation gap and V/U that is fairly stable.
  - Late 2021 and 2022 observations are in the upper right of the graphs: significantly higher gaps and a tighter labor market than any time since 1985; recent inflation gaps are roughly consistent with unusually high V/U and the pre-pandemic relation.
  - Recent observations are also fairly consistent with late 1960s overheating (last time V/U levels comparable to 2021-2022), but econometrics in the paper use data starting in 1985.
- Fitted curves and nonlinearity:
  - Figure 4 fits a cubic function of V/U and a Lowess estimator with a bandwidth of 0.8; both produce similar results.
  - The relation is fairly flat for mid-range V/U and steeper when V/U is unusually high or unusually low—V/U has larger marginal effect on core inflation at the tails.
  - Levels of the inflation gap are somewhat above fitted curves in late 2021, but most recent observations are close to the curves.
- Data note:
  - The observation for V in September 2022 is not available as written; it is estimated by assuming V in September equals V in August.

### Importance of core inflation measurement
- Core inflation measured by the weighted median inflation rate is central to the paper's results.
- When core is measured with inflation excluding food and energy (XFE inflation):
  - The relation between the inflation gap and V/U becomes noisier before the pandemic.
  - During the pandemic XFE inflation fluctuates erratically with no clear relation to V/U.
  - Noise arises from large price changes in industries other than food and energy.
- Appendix results: regressions using XFE core show almost no evidence of pass-through from past headline shocks to core, and an equation estimated through 2019 fails to predict the pandemic-era rise in inflation when core is XFE.

### Pass-through from headline-inflation shocks
- Conceptual channels:
  - Wage adjustment: headline inflation increases cost of living and influences wage demands, contributing to core inflation (Blanchard 2022).
  - Input-cost transmission: price changes in sectors contributing to headline shocks increase costs for other goods (ECB 2014).
- Measurement:
  - Headline shocks are measured as the average of headline minus core inflation over the same four-quarter or 12-month period used for V/U.
  - The Appendix finds headline shocks averaged over shorter periods do not explain core inflation as well.
- Magnitude in the pandemic:
  - The 12-month average of headline-minus-core shocks rose as high as 3.7 percentage points (in March 2022) and fell to 1.4 percentage points as of September 2022.

### An equation for core inflation (Phillips-curve specification and estimates)
- Dependent variable: core inflation gap = median minus expected inflation.
- Regressors: four-quarter or 12-month averages of V/U and headline shocks H; allow nonlinearities via cubic functions of V/U and H.
- Estimation details:
  - Data start in 1985.
  - Results reported for 1985-2019 (pre-pandemic) and 1985-2022 (extended to present, 2022 Q3/September).
  - Both quarterly and monthly results are similar.
  - Squared and cubic terms for both V/U and H are statistically significant in most specifications:
    - Joint significance of (V/U)^2 and (V/U)^3: p = 0.053 for monthly 1985-2019 (borderline), p = 0.012 for quarterly 1985-2019, and p < 0.01 for both quarterly and monthly through 2022.
- Shapes and asymmetries (Figure 6):
  - Panel A: fitted inflation-gap as a function of V/U (with H = 0) resembles the bivariate relation in Figure 4—flat in mid-range, steeper at tails.
  - Panel B: effect of H at given V/U is strikingly asymmetric: negative H (headline below median) has negligible effect on core inflation, positive H raises core inflation.
- Explaining pandemic core inflation:
  - Fitted vs actual monthly core inflation gap from 2020 onward (Figure 7):
    - Using the full sample (1985-present) and pre-pandemic (1985-2019) both show fitted and actual values close.
    - Panel B (pre-pandemic estimation) is an out-of-sample forecast that fits well, implying the pandemic experience can be explained by paths of V/U and H and their pre-pandemic estimated effects.
  - Decomposing contributions:
    - Through most of 2021: little contribution from labor-market tightness; large pass-through from headline shocks drove inflation up.
    - In 2022: pass-through diminished and labor-market tightness became the main cause of high core inflation.

### Role of the American Rescue Plan (ARP)
- Context:
  - The ARP passed in March 2021 is the $1.9 trillion stimulus plan with enhanced unemployment benefits and stimulus checks.
  - Hypothesis: ARP stimulated demand, reduced unemployment, increased vacancies, raised V/U, and thus raised inflation via the V/U → inflation relation.
- Procedure:
  - The paper does not estimate ARP effects on the labor market directly; it uses Barnichon, Oliveira, and Shapiro (2021) estimates of ARP effects on V/U.
  - Barnichon et al. use the Ramey and Zubairy (2018) methodology; caveat that pandemic-era lockdowns could reduce consumption response to government spending.
  - Barnichon et al. conclude ARP increased V/U by approximately 0.6 at end-2021 and 0.5 at end-2022; the monthly path is linearly interpolated between these values.
- Counterfactual inflation without ARP:
  - The counterfactual subtracts the inflation effect of the ARP-induced V/U difference using the relation in Figure 6 and assumes expected inflation is unaffected.
  - Results in September 2022:
    - Monthly core (median CPI) inflation is 4.1 percent in the counterfactual versus 8.3 percent actual — a difference of 4.2 percentage points.
    - 12-month core inflation is 5.1 percent in the counterfactual versus 7.0 percent actual — a difference of 1.9 percentage points.
    - The 1.9 percentage point difference equals about 40 percent of the rise in 12-month core inflation from end-2020 to September 2022 and about one quarter of the rise in 12-month headline inflation.
  - Caveat: assuming labor-market tightness is the only ARP channel; if ARP also worsened supply-chain problems (raising H), these estimates are a lower bound.

### Wage inflation and its relation to V/U and H
- Data and measure:
  - Wage inflation measured by growth rate of the employment cost index (ECI), a quarterly series.
  - Wage-inflation gap = wage inflation minus expected inflation.
  - Trend productivity: output per hour in non-farm business sector smoothed with the Hodrick-Prescott filter with λ = 16,000.
- Findings:
  - Scatter plot of the wage-inflation gap against four-quarter average V/U (1968-2022Q2) shows an upward-sloping relationship, somewhat noisy; 1970s not unusual in this relation.
  - One outlier omitted: 1972Q1 (annualized wage increase of 13.2 percent).
  - Estimated wage Phillips curves (cubic in V/U and H, plus trend productivity) produce effects of V/U and H broadly similar to their effects on price inflation, consistent with a wage → price transmission channel.
  - Difference vs price inflation:
    - Estimated effect of V/U on wage inflation is approximately linear, unlike the nonlinear price response; unclear if this is substantive or due to noisy wage data.

### Explaining headline inflation (drivers of headline-minus-core shocks)
- Approach:
  - Monthly deviations of headline from core inflation arise from large sectoral shocks (supply or demand).
  - The paper regresses headline-inflation shocks on many candidate variables (Table 2), and reports simple regressions and multiple regressions.
- Simple regression explanatory power (adjusted R-squared in parentheses), in order of importance:
  - Energy-price shocks (energy price inflation minus median inflation): 0.646
  - IHS Markit index of firms’ backlogs of goods and services orders (measure of supply chain disruptions): 0.429
  - Share of goods in aggregate consumption (captures shift away from services during lockdowns): 0.253
  - Auto-price shocks (weighted average of new and used cars, car rentals, and car insurance inflation minus median inflation): 0.191
- Multiple regression:
  - A regression of headline shocks on energy-price shocks, backlogs of work, and auto-price shocks yields adjusted R-squared = 0.912.
  - When these three variables are included, the goods share is not significant.
- Interpretation and timing (Figure 11):
  - The three variables explain the downward spike in headline inflation at the start of the pandemic and different parts of the subsequent high-inflation experience.
  - Example: auto-related prices important for the summer 2021 run-up (chip shortage); energy prices and backlogs help explain the ten-percentage-point headline shock in March (text truncated).

*Source: wpiea2022208-print-pdf - Appendix considers the relation between the inflation gap and the current level of V/U alone).*

### 2022. Energy prices explain the positive headline shock in June 2022 and the negative shocks from

### wpiea2022208-print-pdf - 2022. Energy prices explain the positive headline shock in June 2022 and the negative shocks from July to September.

### Accounting for the rise in inflation
- 12-month headline inflation in September 2022: 8.2 percent.
- 12-month headline inflation in December 2020: 1.3 percent.
- Difference between these two rates: 6.9 point rise in 12-month inflation.
- 12-month core (median) inflation increased 4.6 percentage points (from 2.3 percent to 7.0 percent) over the same period.
- Using the preferred core inflation equation (Table 1, column (4)) and the preferred headline-shock equation (Table 2B, column (3)), the decomposition yields:
  - Direct plus pass-through effects of headline-inflation shocks account for about 4.6 percentage points of the 6.9 point rise in 12-month inflation.
  - Total contributions to the 4.6 percentage points from headline shocks:
    - Energy-price shocks: 2.7 percentage points.
    - Backlogs of work: 1.7 percentage points.
    - Auto-price shocks: net effect includes a significant pass-through from past auto-price shocks, but the direct effect on current headline inflation has turned negative (reflecting partial reversals of Summer 2021 auto price increases).
  - For energy-price shocks and backlogs, roughly two thirds of each contribution is the effect on current headline inflation and one third is pass-through into core.
  - Rise in expected inflation accounts for 0.5 percentage point of the rise.
  - Contribution of V/U (vacancies-to-unemployed ratio) to the rise in 12-month inflation: 2.0 percentage points (nearly a third of the total inflation increase).
- Decomposing the change in annualized one-month core inflation from December 2020 to September 2022:
  - One-month core inflation rose 6.4 percentage points (from 1.9 percent to 8.3 percent).
  - Contribution of V/U to that change: 5.0 percentage points.
- Calculation steps used in the decomposition:
  - Decompose core inflation into expected inflation, labor market tightness (V/U), pass-through from past headline shocks, and a residual (based on Table 1 column 4).
  - Decompose the pass-through effect into energy shocks, backlogs, auto-price shocks, and a residual using coefficients from the headline-shock equation (Table 2B, Column 3) and 12-month averages of the three variables.
  - Divide current headline-inflation shock into components due to the three variables and a residual using the headline-shock equation.
  - Subtract the average of each component over January–December 2020 from the average over October 2021–September 2022 to derive the reported decomposition; report a single residual combining residuals from steps.

### Why high inflation was such a surprise
- Primary causes of forecasting errors:
  - Unexpected, adverse headline inflation shocks (unusual and persistent supply-chain disruptions; rise in energy prices associated with the war in Ukraine).
  - Flaws in pre-pandemic understanding of inflation dynamics.
- Three intertwined problems with conventional thinking identified:
  - Measurement of labor-market tightness:
    - Economists typically used the deviation of unemployment from its natural rate (as estimated by the Congressional Budget Office) and therefore neglected tightening captured by the job vacancies-to-unemployed ratio V/U.
    - Example: In March 2021, the V/U ratio was 0.9, below its pre-pandemic (January 2020) level of 1.2, with little indication that it would rise to above 2.0 by March 2022.
  - Assumed linear and small effect of unemployment on inflation:
    - Many economists assumed a linear, modest unemployment-inflation effect based on pre-pandemic stable-inflation estimates; extreme tightening was thus expected to have modest inflationary effects (e.g., Ball and others (2021a) predicted that if unemployment fell to 1.5 percent, core inflation would rise only to 2.9 percent).
  - Neglect of pass-through from headline deviations into core:
    - Economists typically assumed deviations of headline inflation from core would not feed into core; ignoring pass-through led to underestimation of risks to core inflation in mid-2021.
- Specification experiments comparing alternative core-inflation equations:
  - Traditional specification: linear in the 12-month deviation of unemployment from CBO natural rate and excludes pass-through variable H — performs poorly during the pandemic (predicts a decrease in inflation in 2020 and almost no increase since then).
  - Stepwise modifications that improve fit:
    - Replace unemployment measure with V/U while maintaining linear relation.
    - Use a cubic rather than linear function of V/U (non-linearity).
    - Add pass-through variable H (first linearly, then as a cubic).
  - The preferred specification (non-linear V/U and pass-through H as cubic) performs well in predicting core inflation during the pandemic; each modification (measure of slack, non-linearity, pass-through) materially improves fit.
  - Appendix results: similar findings when core inflation is measured by median PCE inflation; preferred specification predicts most of the observed rise (with some under-prediction since May 2022).
- Ex-post perspective:
  - Before 2020, inflation equations fit data better with non-linear V/U and with pass-through, but evidence was not striking enough to influence most economists’ models.
  - Movements in V/U were strongly correlated with unemployment historically, and extreme labor-market tightness had not been observed pre-2020.
  - Headline inflation shocks were smaller and less persistent before 2020, making pass-through effects easy to miss.

### Two big questions about the near-term outlook
- Key uncertainties that will determine the cost of containing inflation:
  - Behavior of the Beveridge curve (relation between unemployment rate and vacancy rate).
  - Behavior of inflation expectations.
- Beveridge curve discussion:
  - The Beveridge curve is downward-sloping: a tightening labor market increases vacancies and reduces unemployment.
  - The curve determines the relation between unemployment and V/U and therefore affects the level of unemployment needed to reduce inflation.
  - Figure 14 (described): unemployment and vacancy rates from 2001 through August 2022 show periods of a stable Beveridge curve with discrete shifts over time.
    - Stable from 2001 to 2009, then shifted outward and stable until March 2020.
    - Pandemic shutdown of April 2020 produced an abrupt outward shift in the curve; initially the shift was a jump in the unemployment rate.

*Source: wpiea2022208-print-pdf - 2022. Energy prices explain the positive headline shock in June 2022 and the negative shocks from July to September.*

### 14.7 percent with little change in the vacancy rate; since then, the tightening of the labor market

### wpiea2022208-print-pdf - 14.7 percent with little change in the vacancy rate; since then, the tightening of the labor market

### Beveridge curve and labor-market matching
- The Beveridge curve has shifted outward during the pandemic, indicating the labor market has become less efficient at matching unemployed workers with vacant jobs.
- Possible factors cited:
  - Increased reallocation of workers across firms, as captured by the gross level of hiring (Blanchard, Domash, and Summers).
  - Decreased search intensity of unemployed workers, indicated by a decline in the fraction who actively submit job applications (Briggs, 2022).
- Recent data (August 2022) showed V decreased noticeably with little change in U, but it is too soon to conclude a durable shift back.
- Estimated log-linear Beveridge-curve parameters:
  - Pre-pandemic (July 2009 – March 2020): a = 13.9 and b = –0.85.
  - Pandemic (April 2020 – August 2022): a = 15.5 and b = –0.60.
- The outward shift raises the natural rate of unemployment if persistent; the natural rate will fall if the curve shifts back.

### Relation between unemployment and core inflation (Phillips-curve implications)
- Beveridge curve specified as v = a u^b, a>0, b<0, implies V/U = au^(b-1).
- Substituting V/U into the Phillips curve yields a non-linear unemployment–inflation tradeoff with two shock sources: (i) Beveridge-curve shocks and (ii) traditional cost-push shocks.
- Key non-linear slope examples (pre-pandemic curve):
  - Slope at 8 percent unemployment: -0.28
  - Slope at 6 percent unemployment: -0.31
  - Slope at 4 percent unemployment: -0.67
  - Slope at 3.5 percent unemployment: -2.8
- Pandemic-era curve makes inflation higher for any given unemployment rate, especially when unemployment is low:
  - At 4 percent unemployment, core inflation gap = 0.5 percentage point with the old Beveridge curve, and 3.7 percentage points with the pandemic Beveridge curve.

### Natural rate of unemployment
- Definition: unemployment rate at which actual inflation equals expected inflation; adjust for median vs headline inflation by assuming long-term expected core inflation is 0.20 percentage points higher than expected headline inflation.
- Estimated natural rates:
  - Pre-pandemic Beveridge curve: 4.8 percent.
  - Pandemic-era Beveridge curve: 6.5 percent.
- Estimated pandemic increase in the natural rate: 1.7 points.
- Sensitivity: if the difference between expected median and expected headline inflation is assumed to be zero, natural rates would be 5.5 percent and 7.8 percent; if assumed 0.4 percentage points, natural rates would be 4.0 percent and 5.3 percent.
- Estimates are imprecise and sensitive to calibration, but the outward Beveridge shift increasing the natural rate is a robust result.

### Anchoring of inflation expectations
- Long-term inflation expectations were well-anchored pre-pandemic at the Fed’s target; anchoring can weaken if a large and persistent rise in inflation causes people to revise expectations upward.
- Evidence of modest de-anchoring:
  - Ten-year expected inflation (SPF) rose from 2.2 percent in 2019Q4 to 2.8 percent in 2022Q3.
  - Five-year expectations in the Michigan survey rose from 2.2 percent in December 2019 to 2.9 percent in September 2022.
- Simple expectations model: π^e_t = γ π^e_{t-1} + (1–γ) π_t, with γ capturing anchoring (γ=1 implies fully anchored).
- Estimated γ by era (quarterly frequency):
  - 1985Q1-1998Q1: γ = 0.945 (pre-anchoring).
  - 1998Q2-2008Q2: γ = 1.003 (anchoring regime).
  - 2008Q3-2019Q4: γ = 0.991 (some de-anchoring).
  - 2020Q1-2022Q3: γ = 0.980 (weaker anchoring than pre-pandemic).
- Interpretative point: weaker γ implies expectations respond more to actual inflation; historical regimes guide scenario choices.

### Scenarios for future inflation — unemployment paths and key assumptions
- Unemployment paths considered:
  - FOMC SEP (September 2022): unemployment rate peaks at 4.4 percent at the end of 2023.
  - IMF October 2022 WEO projection: unemployment peaks at 5.6 percent.
  - Summers (2022b) scenario: unemployment rises to 7.5 percent for two years.
- Beveridge-curve scenarios:
  - Pessimistic: Beveridge curve remains at its pandemic-era position (light curve).
  - Optimistic: Beveridge curve shifts back quickly to its pre-pandemic position; operationalized as shifting one quarter of the way toward pre-pandemic each month starting from the pandemic-era curve in July, implying near-complete reversal after about six months.
- Expectations scenarios (monthly frequency; start at SPF September 2022 = 2.8 percent):
  - Most optimistic: expected inflation reverts to 2.2 percent, moving one quarter of the way each month.
  - Medium: expected inflation follows pandemic-era process with γ = 0.980 (quarterly), using cube root of 0.980 for monthly simulations.
  - Pessimistic: γ = 0.944 (quarterly), matching 1985-1998 anchoring parameter (loss of post-1998 anchoring).

### Simulation methodology for inflation paths
- Start from actual data through September 2022; set headline-inflation shocks to zero from October 2022 onward.
- Monthly simulation steps:
  - Use the assumed Beveridge curve and given U to derive V/U and compute the 12-month average of V/U.
  - Compute the 12-month headline shock H given zero monthly shocks starting in October 2022 and actual shocks before that (12-month average declines to zero in September 2023).
  - Given 12-month V/U and H, compute the core inflation gap from the monthly Phillips curve (column (4) of Table 1).
  - Given the core inflation gap and last month’s expected inflation, derive current core inflation and expected inflation via the expectations equation (except in the most optimistic expectations scenario, where expected inflation moves one quarter of the way toward 2.2).
- Aggregation:
  - The monthly series yields a 12-month path of core inflation starting in October 2022.
  - With future headline shocks set to zero, the monthly headline path equals core; the 12-month headline path converges to core in September 2023.

*Source: wpiea2022208-print-pdf*

### 2022. In these forecasts, the unemployment rate rises only modestly over time and peaks in late

### wpiea2022208-print-pdf - 2022. In these forecasts, the unemployment rate rises only modestly over time and peaks in late

### Forecast assumptions and unemployment path construction
- The SEP forecasts the unemployment rate in the fourth quarters of 2022, 2023, and 2024; each fourth-quarter forecast is assigned to November and interpolated monthly, starting from the actual unemployment rate of 3.5 percent in September 2022.
- Under the SEP unemployment path, the unemployment rate peaks in late 2023 at 4.4 percent.
- Alternative unemployment paths considered:
  - IMF World Economic Outlook (October 2022) forecasts: unemployment rises to 5.6 percent in the second half of 2024. Quarterly forecasts are assigned to the middle month of each quarter and interpolated monthly.
  - Summers (2022b) scenario: unemployment rises linearly from its September 2022 level to 7.5 percent in January 2023 and then stays at 7.5 percent through December 2024.

### Simulated core inflation paths (median CPI) and common features
- All simulated core inflation paths:
  - Rise from the current level of 7.0 percent and peak between December 2022 and July 2023.
  - Eventually decline as V/U continues to fall and pass-through effects of past headline shocks dissipate.
- Key scenario outcomes for December 2024 under the SEP unemployment path:
  - Most optimistic assumptions (optimistic Beveridge curve and expectations): core inflation peaks at 7.5 percent and falls to 2.5 percent in December 2024.
  - Most pessimistic assumptions: core inflation peaks at 8.6 percent and is 6.3 percent in December 2024 (0.7 percentage point below the current level).
- Sensitivity to Beveridge curve and expectations:
  - If the Beveridge curve shifts back to its pre-pandemic position, December 2024 inflation ranges from 2.5 to 3.9 percent depending on expectations.
  - If the Beveridge curve does not shift back (pandemic-era Beveridge curve), the December 2024 inflation rate always stays above 4.0 percent.
  - With the pandemic-era Beveridge curve, a peak unemployment rate of 4.4 percent is not sufficient to reduce V/U to a non-inflationary level.

### Comparison to the Fed’s target and measurement nuance
- Forecasts use weighted median CPI for core inflation; the Fed targets 2 percent in PCE.
- According to the Federal Reserve Bank of Atlanta Underlying Inflation Dashboard, the Fed’s target is equivalent to a 2.6 percent target for median CPI inflation given historical differences between median CPI and headline PCE.
- Most optimistic forecast for December 2024: core inflation of 2.53 percent, slightly below the Fed-equivalent target of 2.6 percent.
- In all other scenarios reported, inflation stays above the Fed-equivalent target.

### Higher-unemployment scenarios and their implications
- IMF unemployment path (peaking at 5.6 percent in second half of 2024):
  - December 2024 inflation ranges from 2.3 to 4.8 percent (compared to 2.5 to 6.3 percent under the SEP path).
  - Even with higher unemployment, inflation often remains above the Fed’s target; behavior of inflation expectations is critical.
  - With the more pessimistic Beveridge curve, median CPI inflation falls to 2.9 percent (slightly above the implicit 2.6 percent target) if expected inflation reverts to its pre-pandemic level.
- Two years of 7.5 percent unemployment scenario:
  - Differences across Beveridge-curve and expectations assumptions are relatively small.
  - December 2024 inflation rates are clustered around 2.6 percent, each less than one percentage point away from that level.
  - This unemployment path robustly brings inflation close to the Fed’s goal but entails a painful and prolonged increase in unemployment.
- Sacrifice ratio:
  - Defined here as the additional unemployment required to reduce inflation by an extra percentage point by December 2024.
  - The sacrifice ratio is always larger for a greater reduction in inflation from the level today across the scenarios.

### Quantitative decomposition of past inflation rise and main findings
- Decomposition of the 12-month inflation increase between end-2020 and September 2022:
  - Direct and pass-through effects from headline-inflation shocks account for about 4.6 percentage points of the 6.9 percentage point rise in 12-month inflation.
  - A rise in expected inflation accounts for 0.5 percentage point.
  - A rise in labor market tightness (measured by the ratio of vacancies to unemployment) accounts for 2.0 percentage points.
  - The role of labor market tightness is rising over time.
- Broad forward-looking finding:
  - Fed policymakers’ forecasts that inflation will return to target while unemployment rises only to 4.4 percent are reasonable only under quite optimistic assumptions about both the Beveridge curve and expectations.
  - If either the Beveridge curve or inflation expectations behaves less benignly, reducing inflation is likely to require higher unemployment than the Fed anticipates.

### Suggested directions for future research (as stated)
- Refine measurement of core inflation, labor-market tightness, and inflation expectations.
- Better understand non-linear effects of tightness and past headline shocks on core inflation.
- Investigate why the Beveridge curve shifts and why inflation expectations become anchored or de-anchored.

*Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022208-print-pdf.pdf*

### References

### References

### Major cited works on inflation dynamics, labor markets, and methodology
- Extensive citations on Phillips curve, inflation measurement, and labor-market tightness, including works by:
  - Abraham, Katharine; Haltiwanger, John; Rendell, Lea (2020).
  - Ball, Laurence (multiple papers, 1994–2021).
  - Barnichon, Regis (2010; 2021; 2022).
  - Bernanke, Ben S. (2004).
  - Blanchard, Olivier (2021; 2022).
  - Crump, Richard K.; Eusepi, Stefano; Giannoni, Marc; Şahin, Ayşegül (2022).
  - Domash, Alex; Summers, Lawrence H. (2022).
  - Many others focused on core inflation measures, volatility, pass-through, and supply-chain effects.

### Data sources and measurement notes (as used throughout figures/tables)
- Core inflation is median CPI inflation from the Federal Reserve Bank of Cleveland.
- Long-term expected inflation is the ten-year-ahead CPI inflation forecast from the Survey of Professional Forecasters (SPF).
- “V/U” denotes ratio of vacancies to unemployed (4-quarter or 12-month average).
- Headline-inflation shocks denote the difference between headline and median CPI inflation (monthly annualized).
- Backlogs of work variable is taken from IHS Markit Economics.
- Energy, food, and auto-related price inflation variables are created by subtracting median inflation from sectoral inflation rates; these are in monthly annualized terms.

### Key empirical results and model estimates (preserving reported values)
- Table 1. Phillips Curve Estimates: Median CPI Inflation
  - Model columns: (1) Quarterly 1985‐2019, (2) Quarterly 1985‐2022, (3) Monthly 1985‐2019, (4) Monthly 1985‐2022.
  - V/U coefficients:
    - Column (1): V/U 11.039*** (3.645)
    - Column (2): V/U 9.024*** (2.120)
    - Column (3): V/U 9.553** (4.297)
    - Column (4): V/U 9.140*** (2.234)
  - V/U‐squared coefficients:
    - Column (1): ‐13.261** (5.485)
    - Column (2): ‐10.083*** (2.383)
    - Column (3): ‐10.879* (6.435)
    - Column (4): ‐10.328*** (2.545)
  - V/U‐cubed coefficients:
    - Column (1): 5.541** (2.530)
    - Column (2): 4.032*** (0.789)
    - Column (3): 4.439 (2.958)
    - Column (4): 4.241*** (0.863)
  - Headline-inflation shock (H) terms:
    - H:
      - Column (1): 0.021 (0.068)
      - Column (2): 0.031 (0.074)
      - Column (3): 0.010 (0.073)
      - Column (4): 0.058 (0.075)
    - H‐squared:
      - Column (1): 0.155*** (0.041)
      - Column (2): 0.081*** (0.016)
      - Column (3): 0.128*** (0.035)
      - Column (4): 0.089*** (0.019)
    - H‐cubed:
      - Column (1): 0.054*** (0.019)
      - Column (2): 0.026** (0.010)
      - Column (3): 0.053*** (0.017)
      - Column (4): 0.031** (0.012)
  - Constant terms:
    - Column (1): ‐3.026*** (0.747)
    - Column (2): ‐2.616*** (0.557)
    - Column (3): ‐2.759*** (0.879)
    - Column (4): ‐2.654*** (0.586)
  - Observations and fit:
    - Observations: 140, 151, 420, 453 (columns 1–4 respectively)
    - R‐squared: 0.512, 0.761, 0.284, 0.575 (columns 1–4 respectively)
    - Rbar‐squared: 0.490, 0.751, 0.274, 0.569 (columns 1–4 respectively)
  - Note: Newey-West standard errors with 4 lags (quarterly) and 12 lags (monthly) are reported in parentheses; ***, **, * denote significance at 1, 5, and 10 percent.

- Table 2. Explaining Headline Inflation Shocks, 2020-2022
  - Bivariate regressions (columns and reported coefficient, standard error):
    - Energy price inflation: Coefficient 0.064*** (0.013); Observations 33; R‐squared 0.657; Rbar‐squared 0.646
    - Food price inflation: Coefficient ‐0.233 (0.274); Observations 33; R‐squared 0.041; Rbar‐squared 0.010
    - Harper Charter Rate: Coefficient 0.000 (0.001); Observations 33; R‐squared 0.026; Rbar‐squared ‐0.0056
    - Baltic Dry: Coefficient 0.002*** (0.001); Observations 33; R‐squared 0.216; Rbar‐squared 0.191
    - Supplier delivery times: Coefficient ‐0.172*** (0.056); Observations 33; R‐squared 0.138; Rbar‐squared 0.110
    - FRBNY Supply Chain Index: Coefficient 0.887* (0.459); Observations 33; R‐squared 0.043; Rbar‐squared 0.0125
    - Backlogs of work: Coefficient 0.585*** (0.114); Observations 33; R‐squared 0.447; Rbar‐squared 0.429
    - Goods share of real consumption: Coefficient 2.115*** (0.640); Observations 33; R‐squared 0.277; Rbar‐squared 0.253
    - Weighted average of car inflation rates: Coefficient 0.081*** (0.021); Observations 33; R‐squared 0.216; Rbar‐squared 0.191
    - COVID-19 intervention stringency: Coefficient 0.033 (0.042); Observations 33; R‐squared 0.019; Rbar‐squared ‐0.0127
  - Selected multivariate regressions (columns and reported coefficient, standard error):
    - Energy price inflation:
      - Coefficient 0.051*** (0.012); Observations 33; R‐squared 0.785; Rbar‐squared 0.771
      - Coefficient 0.047*** (0.010); Observations 33; R‐squared 0.827; Rbar‐squared 0.809
      - Coefficient 0.056*** (0.008); Observations 33; R‐squared 0.920; Rbar‐squared 0.912
      - Coefficient 0.055*** (0.008); Observations 33; R‐squared 0.922; Rbar‐squared 0.911
    - Backlogs of work:
      - Coefficient 0.346*** (0.082); Observations 33; R‐squared 0.785; Rbar‐squared 0.771
      - Coefficient 0.291*** (0.063); Observations 33; R‐squared 0.827; Rbar‐squared 0.809
      - Coefficient 0.208*** (0.062); Observations 33; R‐squared 0.920; Rbar‐squared 0.912
      - Coefficient 0.203*** (0.061); Observations 33; R‐squared 0.922; Rbar‐squared 0.911
    - Durable goods share of real consumption:
      - Coefficient 0.896** (0.408) in one specification; 0.215 (0.277) in another.
    - Weighted average of car inflation rates:
      - Coefficient 0.068*** (0.007); 0.064*** (0.008) in selected specifications.
    - Constant terms in multivariate specs: ‐17.802*** (4.123); ‐50.165*** (15.901); ‐11.563*** (3.252); ‐19.682* (10.375).
  - Note: Huber-White standard errors in parentheses; ***, **, * denote significance at 1, 5, and 10 percent. Newey-West standard errors are not reported for short sample.

### Quantitative decompositions and counterfactuals (preserving reported magnitudes)
- Figure 12. Accounting for the Rise in Headline Inflation (December 2020 to September 2022)
  - Total rise in 12-month headline inflation: 6.94 percentage points (from 1.28 percent to 8.22 percent).
  - Total rise in 12-month core (median) CPI inflation: 4.63 percentage points (from 2.34 percent to 6.98 percent).
  - Decomposition contributions (percentage points) shown in figure (bar positions preserved visually):
    - Expected inflation: 0.53
    - V/U: 2.02
    - Energy prices: 0.87
    - Backlogs of work: 1.83
    - Auto prices: 0.56
    - Residual: 1.14
    - Headline inflation shock pasthrough: 0.53
    - Additional small negative adjustments: ‐0.32, ‐0.21 (as displayed)
  - Based on estimates in Table 1 (column 4) and Table 2B (column 3).

- Figure 8. Counterfactual Scenario Without American Rescue Plan
  - Impact on V/U comes from Barnichon, Oliveira, and Shapiro (2021).
  - Core inflation (median CPI) monthly annualized and 12-month averages are shown comparing Actual vs Counterfactual for 2021–2023; graphical results indicate higher V/U and higher core inflation in actual relative to counterfactual (values shown visually).

### Model fit, dynamics, and scenario inputs (figures and captions)
- Figures 1–6 document:
  - The divergence between headline and core (median) CPI inflation during 2020–2022, and the role of headline-inflation shocks.
  - Scatterplots and fitted relationships of the inflation gap (median minus long-term SPF expectations) against V/U for 1968–2022, with separate panels for quarterly and monthly data and smoothed fits (Lowess, cubic).
  - Figure 6 presents fitted inflation-gap responses to V/U and to headline-inflation shocks with 95 percent confidence bands (monthly data).
- Figure 7 compares actual median inflation gap 2020–2022 to fitted values from Phillips Curve models estimated on full sample (1985–2022) and pre-pandemic sample (1985–2019), and shows fitted series assuming no headline inflation shocks.
- Figures 9–11:
  - Show wage inflation gaps vs V/U and joint estimated responses of wage and price inflation gaps to slack (V/U) and headline shocks (1985–2022).
  - Figure 11 shows actual vs fitted headline-inflation shocks (regression in Column 3, Table 2) and time-series relationships of headline shocks with energy-price shocks, backlogs of work, and auto-related price shocks.
- Figures 13–16:
  - Compare model variants (linear vs nonlinear slack measures, inclusion of pass-through H) and illustrate behavior of the Beveridge Curve and inflation-expectations anchoring (parameter γ indicates degree of anchoring).
- Figures 17–19 present scenario simulations for core (median) CPI inflation:
  - Conditional on September 2022 FOMC unemployment forecasts (Figure 17).
  - A pessimistic Beveridge Curve with drifting expectations (γ = 0.98) scenario (Figure 18).
  - Conditional projections under alternative unemployment paths, including IMF staff unemployment path and a “Higher unemployment” path assuming 7.5 percent unemployment during 2023 and 2024 as suggested by Summers (2022b) (Figure 19).
  - Horizontal dashes in these figures mark a 2.6 percent target for median CPI based on 2 percent PCE target as reported on Federal Reserve Bank of Atlanta Underlying Inflation Dashboard.
  - Specific scenario point labels shown on figures include: 6.34, 2.53, 4.79, 2.34, 3.26, 1.62 (as plotted values corresponding to scenario outcomes).

### Notes on interpretation embedded in captions
- Inflation gap is consistently defined as the difference between median (or wage) inflation and long-term SPF inflation expectations.
- Monthly inflation measures in figures are often annualized.
- Several figures separate sample periods (e.g., 1968-1972; 1973-1984; 1985-2019; 2020-2022) when plotting historical relationships.
- Confidence intervals (95 percent) are reported in fitted-panels where indicated.

*Understanding U.S. Inflation During the COVID Era — Working Paper No. WP/2022/208*

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_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022208-print-pdf.pdf_