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---

### Robustness: non-linearity and controls
- Recent theoretical and empirical work emphasizes non-linearity in Phillips curves (Ball et al., 2022; Benigno and Eggertsson, 2023; Dao et al., 2024).
- Labor shortage definition: vacancy-to-unemployment ratio exceeds 1 (Benigno and Eggertsson, 2023).
- Subsample analysis (High V/U vs Low V/U):
  - First stage results are similar across both subsamples.
  - Second stage results are larger in the presence of a labor shortage, particularly over the 6- to 12-month horizon.
- Polynomial terms:
  - Quadratic term of log vacancy-to-unemployment ratio estimated negative.
  - Cubic term estimated positive.
  - Neither quadratic nor cubic terms statistically significant (results not reported here).
- Additional controls:
  - Controlling for household income and housing wealth at the MSA level does not change results; findings are robust to these controls.
  - Controlling for time-varying economic conditions at the regional level using region×time fixed effects also leaves results robust and mitigates spatial correlation and spillovers.

### Comparison with recent literature
- Benigno and Eggertsson (2023):
  - Pass-through from log vacancy-to-unemployment ratio to core inflation over a 1-quarter horizon:
    - 4.7 percentage points with labor shortage (when vacancy-to-unemployment ratio is less than 1).
    - 0.5 percentage points without labor shortage.
  - Current study comparison:
    - Pass-through of 5.2 percentage points with labor shortage.
    - Pass-through of 3.6 percentage points without labor shortage.
- Ball et al. (2022):
  - Their results imply an increase in vacancy-to-unemployment ratio from 0.5 to 1.5 leads to a 3.5 percentage point rise in core inflation.
  - Current study: a similar change implies a 5.0 percentage point increase in services inflation (excluding housing).
- Interpretation:
  - Results are comparable to, although slightly larger than, recent aggregate-data estimates.
  - Possible reasons for differences: variations in data and methods; services are more labor-intensive and less tradable than goods, implying stronger pass-through for services inflation.
  - Hazell et al. (2022) theoretical result: slope of the regional Phillips curve is steeper than the aggregate Phillips curve when labor market conditions are persistent.
- Note on aggregate Phillips curve estimation:
  - In a monetary union regional Phillips curve, long-run inflation expectations are captured by time fixed effects.

### External validity: data and alternative specifications
- Data source advantages:
  - Homebase wage data provide granular information needed for MSA-level Phillips curve estimation and high-frequency capture of post-pandemic wage movements.
  - National Homebase wage trends closely align with CPS and CES (Dvorkin and Isaacson, 2022; Chen and Lee, 2024).
  - At the state level, monthly wage changes from Homebase and CES are highly correlated (Chen and Lee, 2024).
- Re-estimation with official data (QCEW):
  - MSA-quarter level QCEW sample period: Q1 2019 to Q4 2022 (baseline).
  - First stage coefficient ranges from 26.3 to 38.2 over the 1- to 4-quarter horizon.
  - Second stage coefficient ranges from 0.13 to 0.26 over the same horizon.
  - Combined pass-through peaks at 7.8 percentage points at the 3-quarter horizon.
  - Magnitudes similar to baseline but coefficients are less precisely estimated with QCEW.
  - Figure 12 at 1-quarter horizon: model closely predicts actual inflation; local labor market tightness identified as key driver since mid-2021.
- Aggregate Phillips curve exercises (Appendix A.2):
  - Log vacancy-to-unemployment ratio coefficient is significantly positive, implying an upward-sloping Phillips curve.
  - Coefficient time profile:
    - 0.6 at the 3-month horizon.
    - Between 1.5 and 2.0 at the 12-month horizon and beyond.
  - Rolling regressions with a 3-year window show substantial time variation in the coefficient.
  - Fitted values from the rolling window regression closely track actual inflation surge and decline during 2021-2023.
  - A model that imposes V/U = 1 (“Fitted w/o ln(V/U)”) predicts:
    - Peak inflation 1 percentage point lower.
    - Inflation falling below 2 percent by early 2023, contrary to observed data.

### Key empirical findings and implications
- Main mechanism:
  - Service sector wage growth is an important channel for services inflation through local labor market tightness.
- Strength of links:
  - Pass-through from tight labor market to service sector wage growth is strong.
  - Pass-through from service sector wage growth to services inflation is strong.
- Time period of prominence:
  - Local labor market tightness emerged as a key driver of inflation between Q3 2022 and Q1 2023.
- Policy-relevant implication:
  - Effects of overheated labor markets can be persistent when labor market tightness endures.
  - Substantial wage growth may impede efforts to curb inflation.

### Summary statistics (sample characteristics)
- Sample: monthly panel of MSAs over January 2019 to December 2022; observations weighted by average size of the labor force in each MSA at 2019.
- Reported variables (Obs, Mean, Std. Dev., Min, Median, Max):
  - Services inflation (excl. housing): 906; 3.30; 2.41; -3.18; 2.87; 10.03
  - Wage growth: 906; 0.95; 17.22; -66.33; 3.40; 63.33
  - Bartik shock: 906; 0.41; 0.60; -1.27; 0.52; 1.23
  - Vacancy-to-unemployment ratio: 906; 1.07; 0.54; 0.15; 1.04; 3.53
  - Labor productivity: 906; 1.94; 3.10; -4.50; 2.10; 7.70
  - Headline shock: 906; 0.54; 1.03; -1.81; 0.35; 3.89
  - Log household income per capita: 906; 11.15; 0.19; 10.65; 11.13; 11.73
  - Log housing wealth: 906; 4.97; 0.23; 4.40; 4.97; 5.55
- Measurement notes:
  - Inflation measured as year-on-year logarithmic difference ln(prices)i,t − ln(prices)i,t−12 for MSA i and month t.
  - Wage growth denotes year-on-year wage growth in the service sector from Homebase measured by logarithmic difference ln(wage)i,t − ln(wage)i,t−12.
  - Instrument: Bartik shock for the log vacancy-to-unemployment ratio, defined in equation (2).
  - Controls: labor productivity (BLS) proxies local labor demand; headline inflation shocks defined as the difference between year-on-year headline and core excluding food and energy inflation; housing wealth defined as product of Census homeownership rates and the Freddie Mac House Price Index.

### Model specification and estimation details
- LP-IV model estimated at the MSA level. Key equations:
  - y_h_i,t = α_h_2,t + η_h_2,i + β_h_2 b w_i,t + Σ_{k=1}^K γ_h_2,k w_i,t−k + Σ_{k=0}^K δ_h_2,k y_i,t−k + γ_h_2 X_i,s,t + u_h_2,i,t
  - w_i,t = α_h_1,t + η_h_1,i + β_h_1 Shock_i,t + Σ_{k=1}^K γ_h_1,k w_i,t−k + Σ_{k=0}^K δ_h_1,k y_i,t−k + γ_h_1 X_i,s,t + u_h_1,i,t
- Notation and settings:
  - i, s, and t denote MSA, state, and month respectively.
  - h denotes the estimation horizon; specific figures use h = 3 (3-month horizon) and h = 1,...,H for horizon plots.
  - K = 3 (controls include lags up to 3).
  - y_i,t is year-on-year inflation measured as ln(prices)_i,t − ln(prices)_i,t−12 for services excluding housing.
  - w_i,t is year-on-year wage growth in the service sector from Homebase measured as ln(wage)_i,t − ln(wage)_i,t−12.
  - Wage growth is instrumented using the Bartik shock for the log vacancy-to-unemployment ratio (ln(V/U)) defined in equation (2).
  - Controls X_i,s,t include labor productivity (from BLS), headline inflation shocks (difference between year-on-year headline and core excluding food and energy inflation), and other state-year level controls (log household income per capita, log housing wealth defined as product between Census homeownership rates and Freddie Mac House Price Index, and residuals from regressions for previous horizons).
  - Observations are weighted by the average size of the labor force in each MSA at 2019.
  - Standard errors u_h_i,t are clustered at the MSA level.
  - The solid line in figures plots point estimates; dashed lines show the 90 percent confidence interval.
- Sample periods:
  - Main monthly panel: January 2019 to December 2022.
  - Share-of-inflation calculation window: January 2021 to March 2023.
  - Subsample average for share calculation: average fitted value for ln(V/U) over January 2021 to March 2023, as a share of average actual services inflation over same period.

### Decomposition, MSA-level predictions, and heterogeneity
- Fitted values and bar contributions in figures:
  - Solid line: actual inflation.
  - Dashed line: fitted values from the LP-IV model.
  - Bars show contribution of each independent variable (combined effect from first and second stages).
  - “ln(V/U)” plots the fitted value from the Bartik shock Shock_i,t for the log vacancy-to-unemployment ratio.
  - “Lag Wage Growth” uses fitted values from w_i,t−k.
  - “Lag Inflation” uses fitted values from y_i,t−k.
  - “MSA FE” from α_h_1,t and α_h_2,t.
  - “Time FE” from η_h_1,t and η_h_2,t.
  - “Headline Shock” and “Productivity” come from controls in X_i,s,t.
- MSAs with plotted Actual vs Fitted and decompositions include:
  - Boston-Cambridge-Newton, MA-NH; New York-Newark-Jersey City, NY-NJ-PA; Philadelphia-Camden-Wilmington, PA-NJ-DE-MD; Chicago-Naperville-Elgin, IL-IN-WI; Detroit-Warren-Dearborn, MI; Minneapolis-St.Paul-Bloomington, MN-WI; St. Louis, MO-IL; Washington-Arlington-Alexandria, DC-VA-MD-WV; Miami-Fort Lauderdale-West Palm Beach, FL; Atlanta-Sandy Springs-Roswell, GA; Tampa-St. Petersburg-Clearwater, FL; Baltimore-Columbia-Towson, MD; Dallas-Fort Worth-Arlington, TX; Houston-The Woodlands-Sugar Land, TX; Phoenix-Mesa-Scottsdale, AZ; Denver-Aurora-Lakewood, CO; Los Angeles-Long Beach-Anaheim, CA; San Francisco-Oakland-Hayward, CA; Riverside-San Bernardino-Ontario, CA; Seattle-Tacoma-Bellevue, WA; San Diego-Carlsbad, CA.

### Share of services inflation explained by ln(V/U)
- Calculation:
  - Share for each MSA = average fitted value for ln(V/U) over January 2021 to March 2023 divided by average actual services inflation (excluding housing) over same period.
- Figure 7 lists MSAs (ordered visually) with their shares; MSAs named explicitly in the figure notes include:
  - Philadelphia, Detroit, Chicago, Baltimore, Boston, Denver, Phoenix, Washington, St. Louis, New York, San Francisco, Miami, Houston, Minneapolis, Seattle, Tampa, Los Angeles, Dallas, Atlanta, San Diego, Riverside.
- Black dashed line in figure denotes the average across MSAs.

### Subsample and robustness analyses (figures and notes)
- Labor-shortage subsample (Figure 8):
  - Subsamples split by whether MSA faced a labor shortage defined as V_i,t / U_i,t > 1.
  - LP-IV subsample specification and monthly panel: January 2019 to December 2022.
  - Plotted outputs show services inflation response (second stage) and first-stage regression coefficients over horizons 0 to 12 months for Low V/U and High V/U groups.
- MSA-level models with region-time fixed effects and additional controls (Figures 9 and 10):
  - Specification adds α_h_r,t region-time fixed effects and includes additional controls at state-year level (log household income per capita, log housing wealth).
  - MSAs mapped to states based on principal state; states categorized into four Census Bureau designated regions.
  - Figures plot first and second stage estimates across horizons 0 to 12 months with 90 percent confidence intervals.
- Graphical and inferential notes:
  - Dashed lines in figures indicate 90 percent confidence intervals.
  - Observations weighted by average labor force size in each MSA at 2019.
  - Standard errors clustered at the MSA level.

### Appendix highlights
- Appendix A.1 (MSA-level LP-IV with official statistics):
  - Quarterly LP-IV: sample Q1 2019 to Q4 2022; K = 1 lags; wage growth from QCEW; Bartik shock converted to quarterly by averaging.
  - Figures 11 and 12 present first and second stage estimates and a decomposition at the 1-quarter horizon.
- Appendix A.2 (national-level LP):
  - LP model: y^h_t = α^h + β^h ln θ_t + Σ_{k=1}^K γ^h_{θ,k} ln θ_t−k + Σ_{k=1}^K γ^h_{y,k} y_t−k + γ^h_x X_t + u^h_t, with monthly t and horizons 0–36 months; K = 3.
  - y_t: year-on-year services (excluding housing) inflation; ln θ_t: log vacancy-to-unemployment ratio (JOLTS-based).
  - Sample period: January 1986 to December 2023; standard errors: Newey-West.
  - Figure 13: β^h across horizons 0–36 months; Figure 14: 12-month horizon rolling estimates (3-year window) with estimate for 2021–2023 around 4.0.
  - Counterfactual: Figure 15 shows fitted predictions with and without ln(V/U), illustrating labor market tightness role in post-2021 services inflation surge.
- Appendix A.3 (Constructing nationally representative vacancies):
  - Procedure rescales Indeed vacancies to produce nationally representative 3-digit NAICS vacancies via regressions of 3-digit log V/U on 2-digit log V/U and conversion steps:
    - exp( \ ln(V^k3_t / U^k3_t) + ln(U^k3_t) ).

*Source: Inflation and Labor Markets: A Bottom-Up View — Working Paper No. WP/2024/220*

### 80.3 percent, with an average of 63.5 percent (see Figure 7).

### wpiea2024220-print-pdf - 80.3 percent, with an average of 63.5 percent (see Figure 7).

### Robustness: non-linearity and controls
- Recent theoretical and empirical work emphasizes non-linearity in Phillips curves (Ball et al., 2022; Benigno and Eggertsson, 2023; Dao et al., 2024).
- Labor shortage definition: vacancy-to-unemployment ratio exceeds 1 (Benigno and Eggertsson, 2023).
- Subsample analysis (High V/U vs Low V/U):
  - First stage results are similar across both subsamples.
  - Second stage results are larger in the presence of a labor shortage, particularly over the 6- to 12-month horizon.
- Polynomial terms:
  - Quadratic term of log vacancy-to-unemployment ratio estimated negative.
  - Cubic term estimated positive.
  - Neither quadratic nor cubic terms statistically significant (results not reported here).
- Additional controls:
  - Controlling for household income and housing wealth at the MSA level does not change results; findings are robust to these controls.
  - Controlling for time-varying economic conditions at the regional level using region×time fixed effects also leaves results robust and mitigates spatial correlation and spillovers.

### Comparison with recent literature
- Benigno and Eggertsson (2023):
  - Pass-through from log vacancy-to-unemployment ratio to core inflation over a 1-quarter horizon:
    - 4.7 percentage points with labor shortage (when vacancy-to-unemployment ratio is less than 1).
    - 0.5 percentage points without labor shortage.
  - Current study comparison:
    - Pass-through of 5.2 percentage points with labor shortage.
    - Pass-through of 3.6 percentage points without labor shortage.
- Ball et al. (2022):
  - Their results imply an increase in vacancy-to-unemployment ratio from 0.5 to 1.5 leads to a 3.5 percentage point rise in core inflation.
  - Current study: a similar change implies a 5.0 percentage point increase in services inflation (excluding housing).
- Interpretation:
  - Results are comparable to, although slightly larger than, recent aggregate-data estimates.
  - Possible reasons for differences: variations in data and methods; services are more labor-intensive and less tradable than goods, implying stronger pass-through for services inflation.
  - Hazell et al. (2022) theoretical result: slope of the regional Phillips curve is steeper than the aggregate Phillips curve when labor market conditions are persistent.
- Note on aggregate Phillips curve estimation:
  - In a monetary union regional Phillips curve, long-run inflation expectations are captured by time fixed effects.

### External validity: data and alternative specifications
- Data source advantages:
  - Homebase wage data provide granular information needed for MSA-level Phillips curve estimation and high-frequency capture of post-pandemic wage movements.
  - National Homebase wage trends closely align with CPS and CES (Dvorkin and Isaacson, 2022; Chen and Lee, 2024).
  - At the state level, monthly wage changes from Homebase and CES are highly correlated (Chen and Lee, 2024).
- Re-estimation with official data (QCEW):
  - MSA-quarter level QCEW sample period: Q1 2019 to Q4 2022 (baseline).
  - First stage coefficient ranges from 26.3 to 38.2 over the 1- to 4-quarter horizon.
  - Second stage coefficient ranges from 0.13 to 0.26 over the same horizon.
  - Combined pass-through peaks at 7.8 percentage points at the 3-quarter horizon.
  - Magnitudes similar to baseline but coefficients are less precisely estimated with QCEW.
  - Figure 12 at 1-quarter horizon: model closely predicts actual inflation; local labor market tightness identified as key driver since mid-2021.
- Aggregate Phillips curve exercises (details in Appendix A.2):
  - Log vacancy-to-unemployment ratio coefficient is significantly positive, implying an upward-sloping Phillips curve.
  - Coefficient time profile:
    - 0.6 at the 3-month horizon.
    - Between 1.5 and 2.0 at the 12-month horizon and beyond.
  - Rolling regressions with a 3-year window show substantial time variation in the coefficient.
  - Fitted values from the rolling window regression closely track actual inflation surge and decline during 2021-2023.
  - A model that imposes V/U = 1 (“Fitted w/o ln(V/U)”) predicts:
    - Peak inflation 1 percentage point lower.
    - Inflation falling below 2 percent by early 2023, contrary to observed data.

### Key empirical findings and implications
- Main mechanism:
  - Service sector wage growth is an important channel for services inflation through local labor market tightness.
- Strength of links:
  - Pass-through from tight labor market to service sector wage growth is strong.
  - Pass-through from service sector wage growth to services inflation is strong.
- Time period of prominence:
  - Local labor market tightness emerged as a key driver of inflation between Q3 2022 and Q1 2023.
- Policy-relevant implication:
  - Effects of overheated labor markets can be persistent when labor market tightness endures.
  - Substantial wage growth may impede efforts to curb inflation.

### Summary statistics (sample characteristics)
- Sample: monthly panel of MSAs over January 2019 to December 2022; observations weighted by average size of the labor force in each MSA at 2019.
- Table 1 reported variables (Obs, Mean, Std. Dev., Min, Median, Max):
  - Services inflation (excl. housing): 906; 3.30; 2.41; -3.18; 2.87; 10.03
  - Wage growth: 906; 0.95; 17.22; -66.33; 3.40; 63.33
  - Bartik shock: 906; 0.41; 0.60; -1.27; 0.52; 1.23
  - Vacancy-to-unemployment ratio: 906; 1.07; 0.54; 0.15; 1.04; 3.53
  - Labor productivity: 906; 1.94; 3.10; -4.50; 2.10; 7.70
  - Headline shock: 906; 0.54; 1.03; -1.81; 0.35; 3.89
  - Log household income per capita: 906; 11.15; 0.19; 10.65; 11.13; 11.73
  - Log housing wealth: 906; 4.97; 0.23; 4.40; 4.97; 5.55
- Measurement notes:
  - Inflation measured as year-on-year logarithmic difference ln(prices)i,t − ln(prices)i,t−12 for MSA i and month t.
  - Wage growth denotes year-on-year wage growth in the service sector from Homebase measured by logarithmic difference ln(wage)i,t − ln(wage)i,t−12.
  - Instrument: Bartik shock for the log vacancy-to-unemployment ratio, defined in equation (2).
  - Controls: labor productivity (BLS) proxies local labor demand; headline inflation shocks defined as the difference between year-on-year headline and core excluding food and energy inflation; housing wealth defined as product of Census homeownership rates and the Freddie Mac House Price Index.

*Source: wpiea2024220-print-pdf - 80.3 percent, with an average of 63.5 percent (see Figure 7).*

### 2019.  We cluster the standard errorsu

### wpiea2024220-print-pdf - 2019.

### Model specification and estimation details
- Estimation uses an LP-IV model estimated at the MSA level. Key equations (as presented):
  - y_h_i,t = α_h_2,t + η_h_2,i + β_h_2 b w_i,t + Σ_{k=1}^K γ_h_2,k w_i,t−k + Σ_{k=0}^K δ_h_2,k y_i,t−k + γ_h_2 X_i,s,t + u_h_2,i,t
  - w_i,t = α_h_1,t + η_h_1,i + β_h_1 Shock_i,t + Σ_{k=1}^K γ_h_1,k w_i,t−k + Σ_{k=0}^K δ_h_1,k y_i,t−k + γ_h_1 X_i,s,t + u_h_1,i,t
- Notation and settings:
  - i, s, and t denote MSA, state, and month respectively.
  - h denotes the estimation horizon; specific figures use h = 3 (3-month horizon) and h = 1,...,H for horizon plots.
  - K = 3 (controls include lags up to 3).
  - y_i,t is year-on-year inflation measured as ln(prices)_i,t − ln(prices)_i,t−12 for services excluding housing.
  - w_i,t is year-on-year wage growth in the service sector from Homebase measured as ln(wage)_i,t − ln(wage)_i,t−12.
  - Wage growth is instrumented using the Bartik shock for the log vacancy-to-unemployment ratio (ln(V/U)) defined in equation (2).
  - Controls X_i,s,t include labor productivity (from BLS), headline inflation shocks (difference between year-on-year headline and core excluding food and energy inflation), and other state-year level controls (varies by specification: log household income per capita, log housing wealth defined as product between Census homeownership rates and Freddie Mac House Price Index, and residuals from regressions for previous horizons).
  - Observations are weighted by the average size of the labor force in each MSA at 2019.
  - Standard errors u_h_i,t are clustered at the MSA level.
  - The solid line in figures plots point estimates; dashed lines show the 90 percent confidence interval.
- Sample periods reported:
  - Main monthly panel: January 2019 to December 2022.
  - Share-of-inflation calculation window: January 2021 to March 2023.
  - Subsample average for share calculation: average fitted value for ln(V/U) over January 2021 to March 2023, as a share of average actual services inflation over same period.

### Decomposition and fitted-value construction
- Fitted values and bar contributions in figures:
  - The solid line: actual inflation.
  - The dashed line: fitted values from the LP-IV model.
  - Bars show contribution of each independent variable where contribution captures combined effect from first and second stages.
  - “ln(V/U)” plots the fitted value from the Bartik shock Shock_i,t for the log vacancy-to-unemployment ratio.
  - “Lag Wage Growth” uses fitted values from w_i,t−k.
  - “Lag Inflation” uses fitted values from y_i,t−k.
  - “MSA FE” from α_h_1,t and α_h_2,t.
  - “Time FE” from η_h_1,t and η_h_2,t.
  - “Headline Shock” and “Productivity” come from controls in X_i,s,t.

### MSA-level predictions and heterogeneity
- Figures present predicted and actual services inflation (excluding housing) for multiple MSAs (examples shown explicitly):
  - Boston-Cambridge-Newton, MA-NH
  - New York-Newark-Jersey City, NY-NJ-PA
  - Philadelphia-Camden-Wilmington, PA-NJ-DE-MD
  - Chicago-Naperville-Elgin, IL-IN-WI
  - Detroit-Warren-Dearborn, MI
  - Minneapolis-St.Paul-Bloomington, MN-WI
  - St. Louis, MO-IL
  - Washington-Arlington-Alexandria, DC-VA-MD-WV
  - Miami-Fort Lauderdale-West Palm Beach, FL
  - Atlanta-Sandy Springs-Roswell, GA
  - Tampa-St. Petersburg-Clearwater, FL
  - Baltimore-Columbia-Towson, MD
  - Dallas-Fort Worth-Arlington, TX
  - Houston-The Woodlands-Sugar Land, TX
  - Phoenix-Mesa-Scottsdale, AZ
  - Denver-Aurora-Lakewood, CO
  - Los Angeles-Long Beach-Anaheim, CA
  - San Francisco-Oakland-Hayward, CA
  - Riverside-San Bernardino-Ontario, CA
  - Seattle-Tacoma-Bellevue, WA
  - San Diego-Carlsbad, CA
- Each MSA figure plots Actual vs Fitted and bars for ln(V/U), Lag Wage Growth, Lag Inflation, MSA FE, Time FE, Headline Shock, Productivity.

### Share of services inflation explained by ln(V/U)
- Calculation:
  - Share for each MSA = average fitted value for ln(V/U) over January 2021 to March 2023 divided by average actual services inflation (excluding housing) over same period.
- Figure 7 lists MSAs (ordered visually) with their shares; MSAs named explicitly in the figure notes include:
  - Philadelphia, Detroit, Chicago, Baltimore, Boston, Denver, Phoenix, Washington, St. Louis, New York, San Francisco, Miami, Houston, Minneapolis, Seattle, Tampa, Los Angeles, Dallas, Atlanta, San Diego, Riverside.
- Black dashed line in figure denotes the average across MSAs.

### Subsample and robustness analyses
- Labor-shortage subsample (Figure 8):
  - Subsamples split by whether MSA faced a labor shortage defined as V_i,t / U_i,t > 1.
  - LP-IV specification for subsample: y_h_i,t − y_0_i,t = α_h_t + η_h_i + β_h w_i,t + Σ_{k=1}^K γ_h_k w_i,t−k + Σ_{k=0}^K δ_h_k y_i,t−k + γ_h X_i,s,t + u_h_i,t
  - Sample: monthly panel January 2019 to December 2022.
  - Plotted outputs show: services inflation response (second stage) over horizons 0 to 12 months and first-stage regression coefficients over same horizons for Low V/U and High V/U groups.
- MSA-level models with region-time fixed effects and additional controls (Figures 9 and 10):
  - Specification adds α_h_r,t region-time fixed effects and includes controls at state-year level.
  - Additional controls in one specification include: log household income per capita, log housing wealth (Census homeownership rate × Freddie Mac House Price Index).
  - Observations: MSAs mapped to states based on principal state; states categorized into four Census Bureau designated regions.
  - Figures plot first and second stage estimates across horizons 0 to 12 months; first-stage regression coefficients are shown alongside second-stage responses.
  - All specifications cluster standard errors at the MSA level and show 90 percent confidence intervals.

### Graphical and inferential notes
- Confidence reporting:
  - Dashed lines in figures indicate 90 percent confidence intervals.
- Weighting and clustering:
  - Observations weighted by average labor force size in each MSA at 2019.
  - Standard errors clustered at the MSA level.

*Source: BLS, Homebase, and authors’ calculations.*

### References

### Inflation and Labor Markets: A Bottom-Up View — References and Appendix (Working Paper No. WP/2024/220)

### References
- The source includes an extensive references list (authors and working papers, journal articles, and institutional notes) cited throughout the paper, including recent NBER working papers, IMF Working Papers, BLS working papers, and articles in Journal of Monetary Economics, Econometrica, Quarterly Journal of Economics, and others.

### Appendix A.1 — Cross-sectional evidence from official statistics (MSA-level LP-IV)
- Estimation target and specification:
  - LP-IV model: y^h_i,t − y^0_i,t = α^h_t + η^h_i + β^h w_i,t + Σ_{k=1}^K γ^h_k w_i,t−k + Σ_{k=0}^K δ^h_k y_i,t−k + γ^h X_i,s,t + u^h_i,t
  - i, s, t denote MSA, state, and quarter; h = 1, ..., H denotes estimation horizon.
  - Controls for lags up to K = 1.
- Variable definitions:
  - y_i,t: year-on-year inflation of MSA i in quarter t measured as ln(prices)_i,t − ln(prices)_i,t−4 for services excluding housing.
  - Dependent variable: y^h_i,t − y^0_i,t = difference in inflation over horizon of h quarters starting at t.
  - w_i,t: year-on-year wage growth in the service sector from QCEW measured as ln(wage)_i,t − ln(wage)_i,t−4.
- Instrument and identification:
  - Wage growth instrumented using the Bartik shock for the log vacancy-to-unemployment ratio (monthly shock converted to quarterly by averaging within each calendar quarter).
- Controls and estimation details:
  - X_i,s,t includes state-year level controls: labor productivity (from BLS), headline inflation shocks (difference between year-on-year headline and core excluding food and energy inflation), and residuals from regressions for previous horizons.
  - Observations weighted by average size of the labor force in each MSA at 2019.
  - Standard errors clustered at the MSA level.
- Sample period and data notes:
  - Sample period: Q1 2019 to Q4 2022.
  - Wage data start from Q1 1990 but the Bartik shock data only starts in Q1 2019.
- Figures referenced:
  - Figure 11: plots estimated coefficients (first and second stage) from the LP-IV model.
  - Figure 12: decomposes estimation results at the 1-quarter horizon and plots fitted values and contributions of each independent variable.

### Appendix A.2 — Time-series evidence from official statistics (national-level LP)
- Estimation target and specification:
  - LP model: y^h_t = α^h + β^h ln θ_t + Σ_{k=1}^K γ^h_{θ,k} ln θ_t−k + Σ_{k=1}^K γ^h_{y,k} y_t−k + γ^h_x X_t + u^h_t
  - Monthly time t and horizon h = 0,1,...,36 months.
  - K = 3 lags of the dependent variable and each independent variable.
- Variable definitions:
  - y_t: year-on-year services (excluding housing) inflation in month t, computed as logarithmic changes.
  - ln θ_t: log vacancy-to-unemployment ratio (from Barnichon (2010) updated from JOLTS).
  - X_t: vector of controls including labor productivity (from BLS), headline inflation shocks, and inflation expectations.
  - Headline inflation shock: difference between year-on-year headline and core excluding food and energy inflation.
  - Inflation expectations: 2-year expectations from the Federal Reserve Bank of Cleveland, combined with 12-month Livingston survey expectations before 1982.
- Sample period and estimation:
  - Sample period: January 1986 to December 2023.
  - Standard errors: Newey-West.
  - Figure 13: plots estimated β^h across horizons 0–36 months.
- Time-variation and rolling regressions:
  - Rolling regressions: horizon fixed at h = 12 months (peak of the estimation) with a 3-year rolling window.
  - Figure 14: plots rolling estimates; the estimate for the 2021-2023 period is around 4.0.
  - Comparison: this magnitude is quantitatively similar to estimates in Benigno and Eggertsson (2023) during periods with a high vacancy-to-unemployment ratio.
- Counterfactual illustration:
  - Figure 15: plots fitted values from the rolling regression (“Fitted” line) and fitted values from the same regression with V/U imposed equal to 1 (“Fitted w/o ln(V/U)” line).
  - Key illustration: without a tight labor market, services (excluding housing) inflation would have peaked in early 2021 instead of early 2022 and would have fallen under 2 percent, instead of the actual rate of 4 percent, in early 2023.

### Appendix A.3 — Constructing nationally representative vacancies (Indeed → JOLTS/CPS bridging)
- Motivation:
  - Indeed vacancies may not be nationally representative because some job openings are not posted online; procedure rescales Indeed vacancies to obtain nationally representative 3-digit NAICS vacancies.
- Procedure (stepwise):
  - Compute log vacancy-to-unemployment ratio ln(V^k3_t / U^k3_t) at the 3-digit NAICS level using vacancies from Indeed and unemployment from CPS.
  - Regress the 3-digit ln(V^k3_t / U^k3_t) onto the corresponding 2-digit NAICS ln(V^k2_t / U^k2_t), where 2-digit values use vacancies from Indeed and unemployment from CPS.
  - Obtain fitted values \ ln(V^k3_t / U^k3_t) from that regression to produce a nationally representative 3-digit NAICS log vacancy-to-unemployment ratio.
  - Add back the 3-digit log unemployment from the denominator and exponentiate to convert fitted log ratios into nationally representative 3-digit vacancies:
    - exp( \ ln(V^k3_t / U^k3_t) + ln(U^k3_t) ).

### Figures and visual evidence (as described)
- Figure 11: MSA-level LP-IV first and second stage estimates (solid line point estimates, dashed 90 percent confidence interval).
- Figure 12: MSA-level fitted values and decomposition at 1-quarter horizon; bars show contributions from ln(V/U), lag wage growth, lag inflation, MSA FE, Time FE, Headline Shock, Productivity.
- Figure 13: National-level LP estimates of services inflation response to ln(V/U) for horizons 0–36 months.
- Figure 14: 12-month horizon rolling-sample estimates (3-year backward-looking window) showing time variation and peak ~4.0 in 2021–2023.
- Figure 15: National-level fitted predictions with and without ln(V/U) (imposing V/U = 1) showing the important role of labor market tightness in explaining the post-2021 services inflation surge.

*Source: Inflation and Labor Markets: A Bottom-Up View — Working Paper No. WP/2024/220*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024220-print-pdf.pdf_