## Annex 2.1 Data Sources, Sample Coverage, and Variable Definitions

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### Data sources, frequency, aggregation, and seasonal adjustment
- Primary analysis uses data in quarterly frequency.
- Primary sources on wages, employment, unemployment, and inflation are combined by taking one source as primary and extending backwards and forwards using growth rates from other available sources.
- Where available, OECD data is taken first, followed by ILO, and other sources listed in Annex Table 2.1.1.
- For quarterly frequency, all original source data that was not seasonally adjusted by the source was seasonally adjusted by the authors using X-13ARIMA-SEATS procedure from the U.S. Census Bureau.
- Inflation expectations are sourced from Consensus Forecasts (CF).
  - Monthly CF surveys provide expected current- and next-year inflation (fixed-event forecasts); the twelve-month-ahead (fixed-horizon) inflation expectations are constructed as the weighted sum of monthly vintages, following the standard approach in the literature (see Buono and Formai 2018, Methodological Appendix).
- Note on indicators and data providers (sources preserved verbatim):
  - Inflation: Haver Analytics; International Monetary Fund, World Economic Outlook database; Organisation for Economic Co-operation and Development
  - Wage (per worker, per hour; by sector): Eurostat; Haver Analytics; International Labour Organization; Organisation for Economic Co-operation and Development; US Bureau of Labor Statistics
  - Employment (number of people, hours worked; by sector): Eurostat; Haver Analytics; International Labour Organization; Organisation for Economic Co-operation and Development; US Bureau of Labor Statistics
  - Unemployment: Haver Analytics; International Labour Organization; International Monetary Fund, International Financial Statistics and World Economic Outlook databases; Organisation for Economic Co-operation and Development
  - Unemployment-to-Vacancy Ratio: Duval and others (2022); Barnichon (2010); and national authorities (Australian Bureau of Statistics; Eurostat; Statistics Canada; UK Office for National Statistics; US Bureau of Labor Statistics, Job Openings and Labor Turnover Survey)
  - Output-Side Real GDP in Chained Purchasing-Power-Parity Dollars (mil. 2017US$) per Worker: Penn World Table 10.0
  - GDP in Purchasing-Power-Parity Dollars: International Monetary Fund, World Economic Outlook database
  - Markups: Diez, Leigh, and Tambunlertchai (2018) based on the Industry Classification Benchmark by FTSE Russell
  - Stringency of Contract Regulation: Indicators of Employment Protection, Organisation for Economic Co-operation and Development
  - Inflation Expectations: Consensus Economics Inc.
  - Government Long-Term Rates: Organisation for Economic Co-operation and Development
  - Particpation in GVCs: Organisation for Economic Co-operation and Development, Trade in Value Added (TiVA)
  - Global Supply Chain Pressure Index: Benigno and others (2022); Federal Reserve Bank of New York, Global Supply Chain Pressure Index (GSCPI)
  - Monetary Policy Shocks: Jarociński and Karadi (2020), updated version by Jarociński as of July 2022.
  - Index of Inflation Expectations Anchoring: Bems and others (2021)
  - Inter-Country Input-Output Tables: Organisation for Economic Co-operation and Development, Inter-Country Input-Output Database (ICIO); Organisation for Economic Co-operation and Development, Trade in Employment (TiM)
  - Household Consumption Composition: United States Census Bureau; USA Trade Online
  - Commodity Prices: International Monetary Fund, Primary Commodity Price System
  - International Trade Costs: United States Census Bureau; USA Trade Online
  - Fiscal Policy: Organisation for Economic Co-operation and Development, Quarterly and Annual National Accounts; United States Bureau of Economic Analysis
  - Monetary Policy: Haver Analytics; Wu-Xia Shadow Rates (Federal Funds Rate; European Central Bank Policy Rates)
  - Household Savings Rate: Organisation for Economic Co-operation and Development, Annual National Accounts
  - Real GDP per Capita: Haver Analytics; Brazilian Institute of Geography and Statistics (IBGE); Federal Reserve Economic Data (FRED)
  - IPCA-15 (Consumer Price Index): Brazilian Institute of Geography and Statistics (IBGE)
  - Personal Consumption Expenditures: Federal Reserve Economic Data (FRED)
  - Federal Reserve Funds Rate: Federal Reserve Economic Data (FRED)
  - Selic Interest Rate: Central Bank of Brazil
  - Constant Composition Real Wages: Dizioli and Wang (2022); Howard, Rich, and Tracy (2022)

### Constructed wage and employment series and aggregation
- Wage series constructed (four series):
  - wage per hour in local currency
  - wage per worker in local currency
  - wage per hour index
  - wage per worker index
- For quarterly frequency, wage data in local currency was annualized.
- Employment series constructed (four series):
  - number of people employed
  - number of employees
  - total number of hours worked
  - number of hours worked per employee
- Sector aggregation:
  - Sectors defined based on ISIC, revision 4.
  - Wages and employment aggregated separately by economy into two broad sectors: industry and services.
  - Industry includes manufacturing; construction; mining and quarrying; electricity, gas, and water supply.
  - Services include market services (trade; transportation; accommodation and food; and business and administrative services); and non-market services (public administration; community, social, and other services and activities).
  - Aggregate employment in industry and services is calculated as the sum of employment in each of their respective subsectors.
  - Average wages in industry and services are the employment-weighted average of wages in each subsector.

### Sample coverage and exercises
- Samples for analytical exercises are provided in Annex Table 2.1.2 (economy lists preserved in the original).
- Economy groupings and counts:
  - AEs (33): Australia; Austria; Belgium; Canada; Czech Republic; Denmark; Estonia; Finland; France; Germany; Greece; Hong Kong SAR; Ireland; Israel; Italy; Japan; Korea; Latvia; Lithuania; Luxembourg; Netherlands; New Zealand; Norway; Portugal; Singapore; Slovak Republic; Slovenia; Spain; Sweden; Switzerland; Taiwan Province of China; United Kingdom; United States
  - EMDEs (22): Argentina; Belarus; Brazil; Bulgaria; Colombia; Croatia; Hungary; Kazakhstan; Mexico; Moldova; Peru; Philippines; Poland; Romania; Russia; Saudi Arabia; Serbia; South Africa; Thailand; Türkiye; Ukraine; Vietnam
- For the United States, a more granular sectoral decomposition used data for 17 sectors; broader advanced economies used 9 sectors. The annex notes that using the restrictive sample with 17 sectors does not overturn conclusions on the limited role of sectoral reallocation.

### Key empirical findings on wage and employment dynamics
- Wages per worker vs wages per hour:
  - The chapter focused on average wages per worker when describing dynamics of nominal and real wages.
  - During the acute COVID-19 shock:
    - wages per worker spiked down, reflecting the negative impact of the pandemic on nominal wages.
    - wages per hour spiked up, as the adjustment in hours was more severe than the adjustment on wages.
  - Wages per hour quickly returned to their previous trend and there are no clear signs of severe, above average wage inflation by the end of 2021.
- Sectoral dynamics:
  - Employment in advanced economies is back to its pre-pandemic level in both industry and service sectors on average, while the recovery in emerging market and developing economies has been skewed towards industry.
  - Both nominal and real wages have displayed a consistent dynamic for both advanced and emerging market and developing economies—wages across sectors appear to return to (or, in one case, fall short of) the same common, aggregate trend.
  - Any wage pressures are currently broad-based, reflecting wider economic pressures rather than sectoral composition changes.
- Role of sectoral reallocation:
  - Between 2019 and 2021, less than 5 percent of the change in the average nominal wages per worker can be accounted for by sectoral reallocation of workers within a sample of advanced economies (Annex Figure 2.2.3, panel 1).
  - For the United States, the contribution from sectoral reallocation is somewhat larger at 10 percent on average during the period, with an even larger contribution during the acute pandemic phase (Annex Figure 2.2.3, panel 2).
  - The exact contribution from sectoral reallocation varies across economies and depends on how broadly sectors are defined.
- Real wages:
  - In advanced economies, average real wage rose after the acute phase of the pandemic due to a pickup in nominal wage growth; growth in inflation during the second half of 2021 undid a large portion of those gains.
    - As a result, real wages in services are slightly above pre-pandemic levels, while real wages in industry have mostly returned to their levels in the last quarter of 2019.
  - In emerging market and developing economies, real wages across sectors have been generally flat through the whole period.

### Sectoral composition of wage growth — decomposition formula
- Change in wages per worker in economy c between time t and time t+k is written as:
  - 푤푤_{ccc+kk} − 푤푤_{ccc} = [ (푊푊_{ccc, s, t+kk} / 퐸퐸_{ccc, s, t+kk} ) − (푊푊_{ccc, s, t} / 퐸퐸_{ccc, s, t}) ] * (퐸퐸_{ccc, s, t+kk} / 퐸퐸_{ccc, t+kk}) + [ (퐸퐸_{ccc, s, t+kk} / 퐸퐸_{ccc, t+kk}) − (퐸퐸_{ccc, s, t} / 퐸퐸_{ccc, t}) ] * (푊푊_{ccc, s, t} / 퐸퐸_{ccc, s, t})
  - 푊푊_{ccc, s, t} are total wages in national currencies paid in economy c at sector s and time t, and 퐸퐸_{ccc, s, t} is total number of employees in each sector.
  - The first term captures the contribution from within-sector wage change; the second term captures the contribution from sectoral reallocation of labor to the overall change in wage levels.
  - To get the contribution to relative wage growth (from period t), the equation is divided through by 푤푤_{ccc} and simplified to the expression shown in the annex.

*Source: IMF staff compilation.*

### Annex Figure 2.2.2 — Sectoral Contributions to Recent Wage Dynamics
- Figure measures cumulative change in nominal wages per employee from 2019:Q4–21:Q4, decomposed into:
  - Within sector wage changes
  - Sectoral reallocation
  - Total change
- Units and baseline: Percentage points; 2019:Q4 = 0.
- Sectoral definitions:
  - Panel 1 (aggregated sectors): B-E, F, G-I, J, K, L, M-N, O-Q (8 aggregated sectors).
  - Panel 2 (disaggregated for United States): A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q (17 sectors).
- Samples and coverage:
  - Panel 1 sample consists of 22 economies.
  - Time period covered: 2019:Q4–21:Q4.
- Figure key observations:
  - Decomposes per-worker nominal wage growth into within-sector wage changes and sectoral reallocation.
  - Time path shown across quarterly points from 2019:Q4 through 21:Q4 with y-axis tick marks including: –6; –4; –2; 0; 2; 4; 6; 8; 10.
  - Quarters explicitly shown along the x-axis include: 2019:Q4, 20:Q1, 20:Q2, 20:Q3, 20:Q4, 21:Q1, 21:Q2, 21:Q3, 21:Q4.
- Data sources: International Labour Organization; Organisation for Economic Co-operation and Development; US Bureau of Economic Analysis; and IMF staff calculations.

### Relevant numeric and sample facts (preserved exactly)
- Time window: 2019:Q4–21:Q4.
- Panel 1 sample size: 22 economies.
- Aggregated sectors in panel 1: B-E, F, G-I, J, K, L, M-N, and O-Q.
- Disaggregated sectors in panel 2: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, and Q.

---

### Annex 2.5 Contributions of supply and demand shocks to wages and prices

### Model structure and key features
- Multi-economy, multi-sector general equilibrium model based on Baqaee and Farhi (2022a; 2022b) and Gourinchas and others (2021).
- Nested constant elasticity of substitution (CES) structure across all decision points.
- Economy composition:
  - Two economies (domestic and foreign/ROW aggregate).
  - Five sectoral categories: agriculture and food manufacturing; energy; other industry; construction; services.
  - Three types of final demand: household consumption, government consumption, other final demand.
- Time structure: model features 2 periods, with the second period thought of as a return to steady state after shocks resorb.
- Frictions and heterogeneity:
  - Nominal downward wage rigidities (wages cannot decline from steady state).
  - Credit constraints generating hand-to-mouth households and endogenous aggregate demand fluctuations.
  - Trade links via Armington aggregation and inter-economy input-output (ICIO) linkages.
- International trade: partner economies combined into a single rest of the world (ROW) aggregate.

### Calibration of structural parameters
- Main building blocks derived from inter-economy input-output (ICIO) tables published by the OECD; 2018 data used for pre- and post-COVID steady states.
- Labor and capital shares by economy and sector calculated from the Trade in Employment (TiM) database using the same 44 ISIC category classification.
- Household parameters:
  - Intertemporal CES coefficient of 0.95, which implies a marginal propensity to consume for Ricardian households of 5 percent.
  - Consumption composition CES parameter across 5 goods: 0.8.
- Firm production parameters:
  - Highest nest (value-added and intermediate inputs) CES coefficient: 0.6 (constant returns to scale).
  - Value-added nest (labor and capital) CES parameter: 0.5.
  - Intermediate input bundle CES parameter: 0.2.
- Labor market assumptions:
  - Downward nominal wage rigidities lead to Keynesian unemployment when sectoral labor demand falls.
  - Unemployed households cannot borrow or consume without government transfers, reducing aggregate demand and creating a role for aggregate demand management and monetary policy.

### Calibration of supply and demand shocks (overview)
- Seven types of shocks are used to decompose wage and price dynamics in response to the COVID pandemic.
- Shocks calibrated using cumulative changes between the last year of pre-pandemic data and observed values at end-2020 and end-2021; the model is solved separately for each year.
- Supply-side shocks:
  - Production capacity and labor supply shocks calibrated to match changes in total hours worked by economy and sector between 2019 and 2020 and 2021.
  - International trade cost shocks measured by log difference between CIF and FOB values of imports by detailed goods classification (United States Census Bureau); assumed exogenous and identical increase across goods in all other economies.
    - By the 4th quarter of 2021, trade costs had increased by 7 percent compared to 2019 for the manufacturing sector on average.
    - Trade costs assumed to be zero for services.
  - Commodity price shocks calibrated by adjusting total-factor productivity for the food and energy sectors so that model-based sectoral price changes match IMF Primary Commodity Prices indexes.
    - Published indexes show energy and food prices had increased by 85 and 20 percent year-on-year respectively in 2021.
- Demand-side shocks:
  - Consumption composition shocks modeled by changing consumption weights in household CES utility to match changes in expenditure shares over time.
  - Fiscal policy support calibrated by multiplying government consumption in input-output tables by changes in the nominal value of government final consumption in 2020 and 2021; government transfers to households included by using the ratio of transfers to government consumption over time.
  - Monetary policy support calibrated using changes in central bank policy rates; for the United States and the euro area (at their effective lower bound), shadow policy rates from Wu and Xia (2016, 2017) are used instead.
  - Changes in consumers’ saving behavior calibrated by adjusting the discount rate in the consumption Euler equation so the model matches aggregate savings rates by economy over time as measured by the OECD’s Annual National Accounts database.

### Empirical estimation: local projections and shock identification
- Local Projection (LP) framework estimated on a quarterly panel of sixteen euro area economies over 1999:Q4–2019:Q4.
- Outcome variables (y): nominal and real wage growth, unemployment rate, realized inflation, and 12-month ahead expected inflation, considered at horizons h = 0,...,8 quarters.
- Main explanatory variables (s): inflationary shock (first exercise) and monetary shocks (second exercise).
- Controls and identification:
  - VIX used as proxy for global financial market uncertainty; monetary policy shocks control for policy reaction.
  - For monetary shock exercise, central bank communication shocks included (Jarociński and Karadi 2020).
  - Standard errors clustered at economy-level and corrected for heteroscedasticity and autocorrelation up to the eighth lag.
- Supply-side shock proxy:
  - Fed Global Supply Chain Pressure Index (GSCPI) used as proxy for inflationary (supply) shocks; constructed as principal components of twenty-seven variables purged for demand factors.
  - GSCPI enters lagged and is multiplied by economy-level openness (exports + imports as share of GDP, lagged) to address endogeneity.
- Monetary policy shocks:
  - Sourced from Jarociński and Karadi (2020) identified via Bayesian structural VAR with high-frequency data and sign restrictions.
  - Time series aggregated to quarterly frequency using Ottonello and Winberry (2020) weighting scheme.

### Key empirical results and robustness
- Figures referenced (2.7, 2.6.1, 2.8) show cumulative effects at different horizons of a one standard deviation shock on real and nominal wage growth, expected and realized inflation, government long-term rate, and unemployment.
- Annex Figure 2.6.1 displays impulse responses (with 90 percent confidence intervals) of unemployment and government long-term rate to:
  - A one standard deviation supply chain pressure shock (GSCPI weighted by trade openness).
  - A one standard deviation monetary policy shock (Jarociński and Karadi 2020).
- Robustness:
  - Results robust to alternative samples (dropping one economy at a time; considering only economies in the euro area since 1999).
  - Robust to alternative (i) orders of autocorrelation in residuals, and (ii) measures of economy exposure (using OECD participation in GVCs).
- State-dependent analysis:
  - A state-dependent LP is estimated to test if inflation expectations are less sensitive to shocks where expectations are better anchored.
  - Anchoring indicator: Bems and others (2021) index (simple average of three metrics; reference horizon 5-year ahead).
  - Dummy D = 1 if index in economy i at t-1 is above cross-economy/time median.
  - Findings: impact of inflationary shocks on 12-month ahead inflation expectations is less persistent in economies with better-anchored inflation expectations.

### Notable exact numeric calibrations and empirical sample notes
- CES and substitution parameters:
  - Intertemporal household CES: 0.95.
  - Marginal propensity to consume for Ricardian households implied: 5 percent.
  - Household consumption CES across 5 goods: 0.8.
  - Production nests: CES coefficients 0.6 (value-added vs intermediates), 0.5 (labor vs capital), 0.2 (intermediate input bundle).
- Time and data references:
  - Model uses ICIO 2018 as last available year for pre- and post-pandemic steady states.
  - Shocks calibrated using cumulative changes between 2019 and end-2020 and end-2021.
  - Trade cost increase by 4th quarter of 2021 vs 2019 for manufacturing: 7 percent.
  - Energy and food price increases year-on-year in 2021: 85 percent and 20 percent, respectively.
  - Sample periods referenced for other exercises: 2000:Q1–19:Q4 (wage Phillips curve regressions), and LP sample 1999:Q4–2019:Q4 for euro area panel.
- Sectoral aggregation and model dimensions:
  - Two economies (domestic and ROW), five sectoral categories, three final demand types.
  - Seven types of shocks used for decomposition.

*Source: Annex 2.5, "Contributions of supply and demand shocks to wages and prices," World Economic Outlook, October 2022.*

---

### Annex 2.7 Role of Wage and Price Expectations: Scenarios from a Small DSGE Model

### Model setup and equilibrium equations
- Workhorse model based on Galí, Smets, and Wouters (2012) and Berg and others (2006) — a New Keynesian model that includes wage and price Phillips curves (PC).
- Key variables:
  - y: the output gap.
  - π: quarter-on-quarter, annualized core inflation rate.
  - r: the nominal monetary policy interest rate.
  - w: the constant composition real wage gap.
  - π_w: real wage inflation.
- Representative linearized equilibrium equations (labels preserved):
  - IS Curve: y_c = α_y y_{c−1} + α_y y_{c+1} + γ(π_{c+1} − r_c) + s_{yc}
  - Shock process: s_{yc} = ρ_ε s_{y t−1} + ε_{y t}
  - Price PC: π_c = α_πy π_{c−1} + α_πy π_{c+1} + k π_{w c} + ε_{π c}
  - Nominal wage definition: π_{w c} = w_c − w_{c−1} + π_c
  - Wage PC: π_{w c} = −α_wy w_{c−1} + α_wy π_{w c+1} + K_w y_c + ε_{w c}
  - Policy reaction function: r_c = ρ_r r_{c−1} + (1−ρ_r)(ρ_π π_{c+1} + ρ_y y_c) + ε_{r c}

### Adaptive learning (AL) expectations specification
- In Rational Expectations (RE): E_c[x_{c+1}] = x_{c+1} given ε_{c+1} = 0.
- AL: agents form expectations via a small forecasting model updated as new data arrive.
- Forecasting equation (AR(2)):
  - E_c[x_{c+1}] = α_c + β_{c1} x_c + β_{c2} x_{c−1}
- Fully adaptive expectations: α_c = 0, β_{c1} = 1, β_{c2} = 0.
- Kalman filter updating of learning coefficients:
  - B_{c|c} = B_{c|c−1} + P_{c|c−1} X_{c−1} [Σ_c + X'_{c−1} P_{c|c−1} X_{c−1}]^{-1} * (forecast errors)
  - B_{c|c} stacks AR(2) coefficients; P_{c|c−1} is the covariance matrix; Σ_c is variance-covariance of AR(2) residuals.

### Estimation sample, data, and filters
- Estimation method: Bayesian.
- Sample period: 2000:Q1 to 2019:Q4.
- Countries estimated: Brazil and the USA.
- Variables in estimation: output gap, real wage gap, annualized quarterly price inflation deviation from target, policy rate.
- Real wage construction:
  - Composition-constant real wage used: Howard, Rich, and Tracy (2022) for the USA; series used in (Dizioli and Wang, forthcoming) for Brazil.
- Output and real wage gaps calculated with both HP and linear filters.
  - Linear filter chosen for reported results due to better in-sample and out-of-sample performance.
- Note on sample reduction: Due to limited data availability of the Bems and others (2021) index, the sample is reduced to 10 euro area economies.

### In-sample model evidence (log marginal likelihood)
- Log Marginal Likelihood:
  - REAL Linear Filter: -391.1
  - AL Linear Filter: -341.4
  - REAL HP Filter: -342.9
  - AL HP Filter: -338.5

### Out-of-sample forecast performance (RMSEs, selected entries)
- Linear Filter RMSEs (selected entries):
  - 1-Quarter Ahead RMSE: REAL 0.15, RE 0.22, AL 0.05, 0.11, 0.06, 0.01, 1.41, 0.47
  - 4-Quarters Ahead RMSE: REAL 0.95, RE 1.44, AL 0.63, 1.11, 0.56, 0.27, 0.66, 0.31
  - 8-Quarters Ahead RMSE: REAL 1.39, RE 1.13, AL 1.52, 1.52, 0.64, 0.48, 0.31, 0.19
- HP Filter RMSEs (selected entries):
  - 1-Quarter Ahead RMSE: REAL 0.09, RE 0.53, AL 0.02, 0.11, 0.05, 0.03, 0.20, 0.28
  - 4-Quarters Ahead RMSE: REAL 1.43, RE 1.27, AL 0.5, 0.88, 0.73, 0.44, 0.80, 0.64
  - 8-Quarters Ahead RMSE: REAL 1.18, RE 1.62, AL 1.02, 1.05, 1.52, 0.25, 0.49, 0.41
- Note: Table lists RMSEs across variables "Real Wage Gap", "Output Gap", "Policy Rate", "Inflation" and across horizons; numbers above transcribed directly as in source.

### Comparison of estimation results: Brazil versus the USA
- Findings:
  - Expectations in Brazil depend more on past outcomes than in the USA — seen by adding coefficients on the first two lags of both inflation and wages.
  - Real wage expectations are substantially more persistent in Brazil than in the USA.
  - Coefficient stability: coefficients were stable over the last ten years before the pandemic; the mean expected inflation coefficient was zero as households expected inflation to be at the central bank target.
  - Pandemic disruption: during the pandemic both economies experienced inflation outcomes above target, challenging prior stability.
  - As inflation expectations respond more to past inflation in Brazil, there is feedback from inflation to inflation expectations that keeps inflation higher for longer across shocks, despite a stronger monetary policy response in Brazil.
  - Implication: Monetary policy has to do more to lower inflation in an EM economy like Brazil, even if hit with the same shocks as the US.

### Optimal monetary policy and channels in the AL model
- Optimal policy objective: choose interest rate path {H_c} for t = 1 to ∞ minimizing:
  - Σ_{c=j}^{∞} β_c (0.75(H_c − H_{c−1}) + (y_c − 0)^2 + (π̃_c − 0)^2)
- Assumptions:
  - Equal weights for output gap and inflation deviations from target.
  - Interest rate smoothing weight: 0.75 on interest rate change.
  - Central bank assumed to have full knowledge of current shocks, all future shocks, and how actions impact expectations.
- Three channels through which central bank influences inflation:
  1. Direct channel: tighter policy cools demand, lowering output gap and hence inflation.
  2. Expectations channel via current inflation: tightening lowers current inflation, which enters forecasting equations and lowers next-period expectations.
  3. Learning-coefficient channel: policy can affect households’ forecasting coefficients—if households observe less inflation than expected this period, they update their model of how past inflation matters for future inflation.

### Key implications and contributions
- Modeling shows a price Phillips curve that includes only the output gap would predict that, under adaptive expectations, the only way to lower inflation is via a negative output gap.
- This model shows that a negative real wage gap could enable anchoring of inflation even with fully adaptive expectations.
- Policy implication: In economies where expectations are more backward-looking (Brazil in the estimation), monetary policy needs to act more forcefully to lower inflation because of stronger feedback from realized inflation to expectations.

*Source: Annex 2.7 "Role of Wage and Price Expectations: Scenarios from a Small DSGE Model", IMF staff estimates, World Economic Outlook (October 2022).*

### Annex 2.1 Data Sources, Sample Coverage, and Variable Definitions

### Annex 2.1 Data Sources, Sample Coverage, and Variable Definitions

### Data sources and frequency, aggregation, and seasonal adjustment
- Primary analysis uses data in quarterly frequency.
- Primary sources on wages, employment, unemployment, and inflation are combined by taking one source as primary and extending backwards and forwards using growth rates from other available sources.
- Where available, OECD data is taken first, followed by ILO, and other sources listed in Annex Table 2.1.1.
- For quarterly frequency, all original source data that was not seasonally adjusted by the source was seasonally adjusted by the authors using X-13ARIMA-SEATS procedure from the U.S. Census Bureau.
- Inflation expectations are sourced from Consensus Forecasts (CF).
  - Monthly CF surveys provide expected current- and next-year inflation (fixed-event forecasts); the twelve-month-ahead (fixed-horizon) inflation expectations are constructed as the weighted sum of monthly vintages, following the standard approach in the literature (see Buono and Formai 2018, Methodological Appendix).
- Note on indicators and data providers (as listed in Annex Table 2.1.1; sources preserved verbatim):
  - Inflation: Haver Analytics; International Monetary Fund, World Economic Outlook database; Organisation for Economic Co-operation and Development
  - Wage (per worker, per hour; by sector): Eurostat; Haver Analytics; International Labour Organization; Organisation for Economic Co-operation and Development; US Bureau of Labor Statistics
  - Employment (number of people, hours worked; by sector): Eurostat; Haver Analytics; International Labour Organization; Organisation for Economic Co-operation and Development; US Bureau of Labor Statistics
  - Unemployment: Haver Analytics; International Labour Organization; International Monetary Fund, International Financial Statistics and World Economic Outlook databases; Organisation for Economic Co-operation and Development
  - Unemployment-to-Vacancy Ratio: Duval and others (2022); Barnichon (2010); and national authorities (Australian Bureau of Statistics; Eurostat; Statistics Canada; UK Office for National Statistics; US Bureau of Labor Statistics, Job Openings and Labor Turnover Survey)
  - Output-Side Real GDP in Chained Purchasing-Power-Parity Dollars (mil. 2017US$) per Worker: Penn World Table 10.0
  - GDP in Purchasing-Power-Parity Dollars: International Monetary Fund, World Economic Outlook database
  - Markups: Diez, Leigh, and Tambunlertchai (2018) based on the Industry Classification Benchmark by FTSE Russell
  - Stringency of Contract Regulation: Indicators of Employment Protection, Organisation for Economic Co-operation and Development
  - Inflation Expectations: Consensus Economics Inc.
  - Government Long-Term Rates: Organisation for Economic Co-operation and Development
  - Particpation in GVCs: Organisation for Economic Co-operation and Development, Trade in Value Added (TiVA)
  - Global Supply Chain Pressure Index: Benigno and others (2022); Federal Reserve Bank of New York, Global Supply Chain Pressure Index (GSCPI)
  - Monetary Policy Shocks: Jarociński and Karadi (2020), updated version by Jarociński as of July 2022.
  - Index of Inflation Expectations Anchoring: Bems and others (2021)
  - Inter-Country Input-Output Tables: Organisation for Economic Co-operation and Development, Inter-Country Input-Output Database (ICIO); Organisation for Economic Co-operation and Development, Trade in Employment (TiM)
  - Household Consumption Composition: United States Census Bureau; USA Trade Online
  - Commodity Prices: International Monetary Fund, Primary Commodity Price System
  - International Trade Costs: United States Census Bureau; USA Trade Online
  - Fiscal Policy: Organisation for Economic Co-operation and Development, Quarterly and Annual National Accounts; United States Bureau of Economic Analysis
  - Monetary Policy: Haver Analytics; Wu-Xia Shadow Rates (Federal Funds Rate; European Central Bank Policy Rates)
  - Household Savings Rate: Organisation for Economic Co-operation and Development, Annual National Accounts
  - Real GDP per Capita: Haver Analytics; Brazilian Institute of Geography and Statistics (IBGE); Federal Reserve Economic Data (FRED)
  - IPCA-15 (Consumer Price Index): Brazilian Institute of Geography and Statistics (IBGE)
  - Personal Consumption Expenditures: Federal Reserve Economic Data (FRED)
  - Federal Reserve Funds Rate: Federal Reserve Economic Data (FRED)
  - Selic Interest Rate: Central Bank of Brazil
  - Constant Composition Real Wages: Dizioli and Wang (2022); Howard, Rich, and Tracy (2022)

### Constructed wage and employment series and aggregation
- Wage series constructed (four series):
  1. wage per hour in local currency
  2. wage per worker in local currency
  3. wage per hour index
  4. wage per worker index
- For quarterly frequency, wage data in local currency was annualized.
- Employment series constructed (four series):
  1. number of people employed
  2. number of employees
  3. total number of hours worked
  4. number of hours worked per employee
- Sector aggregation:
  - Sectors defined based on ISIC, revision 4.
  - Wages and employment aggregated separately by economy into two broad sectors: industry and services.
  - Industry includes manufacturing; construction; mining and quarrying; electricity, gas, and water supply.
  - Services include market services (trade; transportation; accommodation and food; and business and administrative services); and non-market services (public administration; community, social, and other services and activities).
  - Aggregate employment in industry and services is calculated as the sum of employment in each of their respective subsectors.
  - Average wages in industry and services are the employment-weighted average of wages in each subsector.

### Sample coverage and exercises
- Samples for analytical exercises are provided in Annex Table 2.1.2 (economy lists preserved in the original).
- Economy groupings and counts (as presented):
  - AEs (33): Australia; Austria; Belgium; Canada; Czech Republic; Denmark; Estonia; Finland; France; Germany; Greece; Hong Kong SAR; Ireland; Israel; Italy; Japan; Korea; Latvia; Lithuania; Luxembourg; Netherlands; New Zealand; Norway; Portugal; Singapore; Slovak Republic; Slovenia; Spain; Sweden; Switzerland; Taiwan Province of China; United Kingdom; United States
  - EMDEs (22): Argentina; Belarus; Brazil; Bulgaria; Colombia; Croatia; Hungary; Kazakhstan; Mexico; Moldova; Peru; Philippines; Poland; Romania; Russia; Saudi Arabia; Serbia; South Africa; Thailand; Türkiye; Ukraine; Vietnam
  - Additional AE and EMDE subsets are listed for specific figures and exercises in the annex (counts and country lists preserved verbatim in the source).
- For the United States, a more granular sectoral decomposition used data for 17 sectors; broader advanced economies used 9 sectors. The annex notes that using the restrictive sample with 17 sectors does not overturn conclusions on the limited role of sectoral reallocation.

### Key empirical findings on wage and employment dynamics
- Wages per worker vs wages per hour:
  - The chapter focused on average wages per worker when describing dynamics of nominal and real wages.
  - During the acute COVID-19 shock:
    - wages per worker spiked down, reflecting the negative impact of the pandemic on nominal wages.
    - wages per hour spiked up, as the adjustment in hours was more severe than the adjustment on wages.
  - Wages per hour quickly returned to their previous trend and there are no clear signs of severe, above average wage inflation by the end of 2021.
- Sectoral dynamics:
  - Employment in advanced economies is back to its pre-pandemic level in both industry and service sectors on average, while the recovery in emerging market and developing economies has been skewed towards industry (Figure references preserved in source).
  - Both nominal and real wages have displayed a consistent dynamic for both advanced and emerging market and developing economies—wages across sectors appear to return to (or, in one case, fall short of) the same common, aggregate trend.
  - Any wage pressures are currently broad-based, reflecting wider economic pressures rather than sectoral composition changes.
- Role of sectoral reallocation:
  - Between 2019 and 2021, less than 5 percent of the change in the average nominal wages per worker can be accounted for by sectoral reallocation of workers within a sample of advanced economies (Annex Figure 2.2.3, panel 1).
  - For the United States, the contribution from sectoral reallocation is somewhat larger at 10 percent on average during the period, with an even larger contribution during the acute pandemic phase (Annex Figure 2.2.3, panel 2).
  - The exact contribution from sectoral reallocation varies across economies and depends on how broadly sectors are defined.
- Real wages:
  - In advanced economies, average real wage rose after the acute phase of the pandemic due to a pickup in nominal wage growth; growth in inflation during the second half of 2021 undid a large portion of those gains.
    - As a result, real wages in services are slightly above pre-pandemic levels, while real wages in industry have mostly returned to their levels in the last quarter of 2019.
  - In emerging market and developing economies, real wages across sectors have been generally flat through the whole period.

### Sectoral composition of wage growth — decomposition formula
- The change in wages per worker in economy c (푤푤_{ccc}), between time t and time t+k, is written as:
  - 푤푤_{ccc+kk} − 푤푤_{ccc} = [ (푊푊_{ccc, s, t+kk} / 퐸퐸_{ccc, s, t+kk} ) − (푊푊_{ccc, s, t} / 퐸퐸_{ccc, s, t}) ] * (퐸퐸_{ccc, s, t+kk} / 퐸퐸_{ccc, t+kk}) + [ (퐸퐸_{ccc, s, t+kk} / 퐸퐸_{ccc, t+kk}) − (퐸퐸_{ccc, s, t} / 퐸퐸_{ccc, t}) ] * (푊푊_{ccc, s, t} / 퐸퐸_{ccc, s, t})
  - where 푊푊_{ccc, s, t} are total wages in national currencies paid in economy c at sector s and time t, and 퐸퐸_{ccc, s, t} is total number of employees in each sector.
  - The first term captures the contribution from within-sector wage change; the second term captures the contribution from sectoral reallocation of labor to the overall change in wage levels.
  - To get the contribution to relative wage growth (from period t), the equation is divided through by 푤푤_{ccc} and simplified to the expression shown in the annex.

*Source: IMF staff compilation.*

### Annex Figure 2.2.2.  Sectoral Contributions to Recent Wage

### Annex Figure 2.2.2.  Sectoral Contributions to Recent Wage Dynamics

### Figure description and methodology
- Measures: Cumulative change in nominal wages per employee from 2019:Q4–21:Q4, decomposed into:
  - Within sector wage changes
  - Sectoral reallocation
  - Total change
- Units and baseline: Percentage points; 2019:Q4 = 0.
- Sectoral definitions:
  - Panel 1 (aggregated sectors): B-E, F, G-I, J, K, L, M-N, O-Q (8 aggregated sectors).
  - Panel 2 (disaggregated for United States): A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q (17 sectors).
- Samples and coverage:
  - Panel 1 sample consists of 22 economies.
  - Time period covered: 2019:Q4–21:Q4.
  - See Online Annex 2.2 for methodology details and Online Annex 2.1 for sample coverage.
- Data sources: International Labour Organization; Organisation for Economic Co-operation and Development; US Bureau of Economic Analysis; and IMF staff calculations.

### Key observations conveyed by the figure
- The figure decomposes per-worker nominal wage growth into contributions from:
  - Within sector wage changes (how wages changed inside sectors).
  - Sectoral reallocation (shifts in employment shares across sectors affecting aggregate wages).
- Time path shown across quarterly points from 2019:Q4 through 21:Q4 with scale spanning at least from –6 to 10 percentage points on the vertical axis (figure axes labels include –6, –4, –2, 0, 2, 4, 6, 8, 10).
- Two panels illustrate:
  1. Advanced Economies: Change in nominal wage per worker, cumulative — separate series for "Within sector wage changes", "Sectoral reallocation", and "Total change".
  2. United States: Change in nominal wage per worker, cumulative — separate series for "Within sector wage changes", "Sectoral reallocation", and "Total change".

### Related analytic context in the annex (how this decomposition is used)
- The decomposition algorithm is applied:
  - To data for eight aggregated sectors covering 22 advanced economies over 2019:Q4–2021:Q4.
  - To a more disaggregated set of 17 sectors for the United States covering the same period.
- The decomposition is part of broader analyses in the annex that:
  - Link sectoral wage dynamics to aggregate wage movements during the pandemic.
  - Feed into event studies and wage Phillips curve analysis elsewhere in the annex (e.g., Figures 2.2.3, 2.3.1, and the wage Phillips curve estimations in Annex 2.4).

### Relevant numeric and sample facts (preserved exactly)
- Time window: 2019:Q4–21:Q4.
- Aggregated sectors in panel 1: B-E, F, G-I, J, K, L, M-N, and O-Q.
- Disaggregated sectors in panel 2: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, and Q.
- Panel 1 sample size: 22 economies.
- Figure y-axis tick marks shown (example values in the figure): –6; –4; –2; 0; 2; 4; 6; 8; 10.
- Quarters explicitly shown along the x-axis include: 2019:Q4, 20:Q1, 20:Q2, 20:Q3, 20:Q4, 21:Q1, 21:Q2, 21:Q3, 21:Q4.

*Sources: International Labour Organization; Organisation for Economic Co-operation and Development; US Bureau of Economic Analysis; and IMF staff calculations.*

### Annex 2.5. Contributions of supply

### Annex 2.5. Contributions of supply and demand shocks to wages and prices

### Model structure and key features
- Multi-economy, multi-sector general equilibrium model based on Baqaee and Farhi (2022a; 2022b) and Gourinchas and others (2021).
- Nested constant elasticity of substitution (CES) structure across all decision points.
- Economy composition:
  - Two economies (domestic and foreign/ROW aggregate) for numerical tractability.
  - Five sectoral categories: agriculture and food manufacturing; energy; other industry; construction; services.
  - Three types of final demand: household consumption, government consumption, other final demand (investment, changes in inventories, non-profits, direct purchases abroad by residents).
- Time structure: model features 2 periods, with the second period thought of as a return to steady state after shocks resorb.
- Frictions and heterogeneity:
  - Nominal downward wage rigidities (wages cannot decline from steady state).
  - Credit constraints generating hand-to-mouth households and endogenous aggregate demand fluctuations.
  - Trade links via Armington aggregation and inter-economy input-output (ICIO) linkages.
- International trade: partner economies combined into a single rest of the world (ROW) aggregate.

### Calibration of structural parameters
- Data sources and aggregation:
  - Main building blocks derived from inter-economy input-output (ICIO) tables published by the OECD; 2018 data used for pre- and post-COVID steady states.
  - Labor and capital shares by economy and sector calculated from the Trade in Employment (TiM) database using the same 44 ISIC category classification.
  - Intermediate and final uses aggregated into two economies and five categories (see model composition above).
- Household parameters:
  - Intertemporal CES coefficient of 0.95, which implies a marginal propensity to consume for Ricardian households of 5 percent.
  - Consumption composition CES parameter across 5 goods: 0.8.
  - Each consumption good consists of an Armington aggregate over domestic and foreign varieties (trade elasticities reported in Annex Table 2.5.1).
- Firm production parameters:
  - Highest nest (value-added and intermediate inputs) CES coefficient: 0.6 (constant returns to scale).
  - Value-added nest (labor and capital) CES parameter: 0.5.
  - Intermediate input bundle (aggregating sectoral output) CES parameter: 0.2.
  - Sectoral output for domestic use combined using Armington trade elasticities and economy- and sector-specific input shares from the ICIO.
- Labor market assumptions:
  - Downward nominal wage rigidities lead to Keynesian unemployment when sectoral labor demand falls.
  - Unemployed households cannot borrow or consume without government transfers, reducing aggregate demand and creating a role for aggregate demand management and monetary policy.

### Calibration of supply and demand shocks (overview)
- Seven types of shocks are used to decompose wage and price dynamics in response to the COVID pandemic.
- Due to the 2-period model, shocks are calibrated using cumulative changes between the last year of pre-pandemic data and observed values at end-2020 and end-2021; the model is solved separately for each year.

- Supply-side shocks:
  - Production capacity and labor supply shocks:
    - Calibrated to match changes in total hours worked by economy and sector between 2019 and 2020 and 2021.
    - Assumption: only labor supply was affected by lockdowns and social distancing (as in Baqaee and Farhi 2022b); declines in hours had to be accompanied by increases in hourly wages in the data for all sectors except construction in the United States for 2020.
  - International trade cost shocks:
    - Measured by log difference between CIF and FOB values of imports by detailed goods classification (United States Census Bureau).
    - Assumed exogenous and identical increase across goods in all other economies.
    - By the 4th quarter of 2021, trade costs had increased by 7 percent compared to 2019 for the manufacturing sector on average.
    - Trade costs assumed to be zero for services.
  - Commodity price shocks:
    - Calibrated by adjusting total-factor productivity for the food and energy sectors so that model-based sectoral price changes match IMF Primary Commodity Prices indexes.
    - Published indexes show energy and food prices had increased by 85 and 20 percent year-on-year respectively in 2021.
    - In the model, these are results of underlying TFP shocks plus endogenous supply and demand responses.

- Demand-side shocks:
  - Consumption composition shocks:
    - Modeled by changing consumption weights in household CES utility to match changes in expenditure shares over time.
    - For the United States, detailed quarterly BEA consumption categories are aggregated to the model’s 5-goods classification.
    - For OECD and selected other economies, household consumption by durability is used where available: food and energy matched to non-durables; other industries and construction considered durables; services tracked separately.
  - Fiscal policy support:
    - Calibrated by multiplying government consumption in input-output tables by changes in the nominal value of government final consumption in 2020 and 2021.
    - Government transfers to households (not in standard IO tables) included by using the ratio of transfers to government consumption over time; transfers included directly in hand-to-mouth household budgets. Ricardian households are assumed not to respond to transfer disbursements because the government intertemporal budget constraint must hold.
  - Monetary policy support:
    - Calibrated using changes in central bank policy rates.
    - For the United States and the euro area (at their effective lower bound), shadow policy rates from Wu and Xia (2016, 2017) are used instead.
  - Changes in consumers’ saving behavior:
    - Calibrated by adjusting the discount rate in the consumption Euler equation so the model matches aggregate savings rates by economy over time as measured by the OECD’s Annual National Accounts database.

### Empirical estimation: local projections and shock identification
- Local Projection (LP) framework estimated on a quarterly panel of sixteen euro area economies over 1999:Q4–2019:Q4.
- Outcome variables (y): nominal and real wage growth, unemployment rate, realized inflation, and 12-month ahead expected inflation, considered at horizons h = 0,...,8 quarters.
- Main explanatory variables (s): inflationary shock (first exercise) and monetary shocks (second exercise).
- Controls and identification:
  - VIX used as proxy for global financial market uncertainty; monetary policy shocks control for policy reaction.
  - For monetary shock exercise, central bank communication shocks included to capture outlook surprises (Jarociński and Karadi 2020).
  - Standard errors clustered at economy-level and corrected for heteroscedasticity and autocorrelation up to the eighth lag.
- Supply-side shock proxy:
  - Fed Global Supply Chain Pressure Index (GSCPI) used as proxy for inflationary (supply) shocks; index encapsulates manufacturing backlogs/delays and shipping/airfreight price indices, constructed as principal components of twenty-seven variables purged for demand factors.
  - To address endogeneity, GSCPI enters lagged and is multiplied by economy-level openness (exports + imports as share of GDP, lagged).
- Monetary policy shocks:
  - Sourced from Jarociński and Karadi (2020) identified via Bayesian structural VAR with high-frequency data and sign restrictions.
  - Time series aggregated to quarterly frequency using Ottonello and Winberry (2020) weighting scheme (triangular weights to reflect timing within quarter).

### Key empirical results and robustness
- Figures referenced (2.7, 2.6.1, 2.8) show cumulative effects at different horizons of a one standard deviation shock on real and nominal wage growth, expected and realized inflation, government long-term rate, and unemployment.
- Annex Figure 2.6.1 displays impulse responses (with 90 percent confidence intervals) of unemployment and government long-term rate to:
  - A one standard deviation supply chain pressure shock (GSCPI weighted by trade openness).
  - A one standard deviation monetary policy shock (Jarociński and Karadi 2020).
- Robustness:
  - Results robust to alternative samples (dropping one economy at a time; considering only economies in the euro area since 1999).
  - Robust to alternative (i) orders of autocorrelation in residuals, and (ii) measures of economy exposure (using OECD participation in GVCs).
- State-dependent analysis:
  - A state-dependent LP is estimated to test if inflation expectations are less sensitive to shocks where expectations are better anchored.
  - Anchoring indicator: Bems and others (2021) index (simple average of three metrics: deviation of long-term mean inflation forecasts from target; variability of mean long-term inflation forecasts; dispersion of long-term inflation forecasts), reference horizon 5-year ahead.
  - Dummy D = 1 if index in economy i at t-1 is above cross-economy/time median.
  - Findings: impact of inflationary shocks on 12-month ahead inflation expectations is less persistent in economies with better-anchored inflation expectations.

### Notable exact numeric calibrations and empirical sample notes
- CES and substitution parameters:
  - Intertemporal household CES: 0.95.
  - Marginal propensity to consume for Ricardian households implied: 5 percent.
  - Household consumption CES across 5 goods: 0.8.
  - Production nests: CES coefficients 0.6 (value-added vs intermediates), 0.5 (labor vs capital), 0.2 (intermediate input bundle).
- Time and data references:
  - Model uses ICIO 2018 as last available year for pre- and post-pandemic steady states.
  - Shocks calibrated using cumulative changes between 2019 and end-2020 and end-2021.
  - Trade cost increase by 4th quarter of 2021 vs 2019 for manufacturing: 7 percent.
  - Energy and food price increases year-on-year in 2021: 85 percent and 20 percent, respectively.
  - Sample periods referenced for other exercises: 2000:Q1–19:Q4 (wage Phillips curve regressions), and LP sample 1999:Q4–2019:Q4 for euro area panel.
- Sectoral aggregation and model dimensions:
  - Two economies (domestic and ROW), five sectoral categories, three final demand types.
  - Seven types of shocks used for decomposition.

*Source: Annex 2.5, "Contributions of supply and demand shocks to wages and prices," World Economic Outlook, October 2022.*

### Annex 2.7 Role of Wage and Price Expectations: Scenarios from a Small DSGE

### Annex 2.7 Role of Wage and Price Expectations: Scenarios from a Small DSGE Model

### Model setup and equilibrium equations
- Workhorse model based on Galí, Smets, and Wouters (2012) and Berg and others (2006) — a New Keynesian model that includes wage and price Phillips curves (PC).
- Key model variables and definitions:
  - y: the output gap (measure of slack).
  - π: quarter-on-quarter, annualized core inflation rate.
  - r: the nominal monetary policy interest rate.
  - w: the constant composition real wage gap (real wage deviations from labor productivity growth).
  - π_w: real wage inflation.
- Representative linearized equilibrium equations (labels preserved as in source):
  - IS Curve: y_c = α_y y_{c−1} + α_y y_{c+1} + γ(π_{c+1} − r_c) + s_{yc}
  - Shock process: s_{yc} = ρ_ε s_{y t−1} + ε_{y t}
  - Price PC: π_c = α_πy π_{c−1} + α_πy π_{c+1} + k π_{w c} + ε_{π c}
  - Nominal wage definition: π_{w c} = w_c − w_{c−1} + π_c
  - Wage PC: π_{w c} = −α_wy w_{c−1} + α_wy π_{w c+1} + K_w y_c + ε_{w c}
  - Policy reaction function: r_c = ρ_r r_{c−1} + (1−ρ_r)(ρ_π π_{c+1} + ρ_y y_c) + ε_{r c}

### Adaptive learning (AL) expectations specification
- In Rational Expectations (RE): E_c[x_{c+1}] = x_{c+1} given ε_{c+1} = 0.
- AL takes agents forming expectations via a simple statistical model (small forecasting model) that is updated as new data arrive.
- Forecasting equation used (AR(2) form):
  - E_c[x_{c+1}] = α_c + β_{c1} x_c + β_{c2} x_{c−1}
- Special cases:
  - Fully adaptive expectations corresponds to α_c = 0, β_{c1} = 1, β_{c2} = 0.
- Time variation:
  - Coefficients α_c, β_{c1}, β_{c2} vary over time depending on forecast accuracy.
- Kalman filter updating of learning coefficients:
  - B_{c|c} = B_{c|c−1} + P_{c|c−1} X_{c−1} [Σ_c + X'_{c−1} P_{c|c−1} X_{c−1}]^{-1} * (forecast errors)
  - B_{c|c} stacks AR(2) coefficients; P_{c|c−1} is the covariance matrix; Σ_c is variance-covariance of AR(2) residuals.

### Estimation sample, data, and filters
- Estimation method: Bayesian.
- Sample period: 2000:Q1 to 2019:Q4.
- Countries estimated: Brazil and the USA.
- Variables included in estimation: output gap, real wage gap, annualized quarterly price inflation deviation from target, policy rate.
- Real wage construction:
  - Composition-constant real wage used: Howard, Rich, and Tracy (2022) for the USA; the series used in (Dizioli and Wang, forthcoming) for Brazil.
- Output and real wage gaps were calculated with both HP and linear filters.
  - Linear filter chosen for the results reported because the model has better in-sample (Annex Table 2.7.1) and out-of-sample forecast performance for wages and prices (Annex Table 2.7.2).
- Note on sample reduction: Due to limited data availability of the Bems and others (2021) index, the sample is reduced to 10 euro area economies.

### In-sample model evidence (log marginal likelihood)
- Log Marginal Likelihood (comparison across filters and models):
  - REAL Linear Filter: -391.1
  - AL Linear Filter: -341.4
  - REAL HP Filter: -342.9
  - AL HP Filter: -338.5

### Out-of-sample forecast performance (Annex Table 2.7.2 RMSEs)
- Table reports RMSEs (root-mean-square error) for horizons and variables; numbers preserved exactly as in source. Bold numbers in source indicate best-performing models at that horizon.
- Linear Filter RMSEs (selected entries preserved in source order):
  - 1-Quarter Ahead RMSE: REAL 0.15, RE 0.22, AL 0.05, (additional juxtaposed numbers in table: 0.11, 0.06, 0.01, 1.41, 0.47) [table entries layout in source is compact; these numbers are all preserved as presented]
  - 4-Quarters Ahead RMSE: REAL 0.95, RE 1.44, AL 0.63, 1.11, 0.56, 0.27, 0.66, 0.31
  - 8-Quarters Ahead RMSE: REAL 1.39, RE 1.13, AL 1.52, 1.52, 0.64, 0.48, 0.31, 0.19
- HP Filter RMSEs (selected entries preserved in source order):
  - 1-Quarter Ahead RMSE: REAL 0.09, RE 0.53, AL 0.02, 0.11, 0.05, 0.03, 0.20, 0.28
  - 4-Quarters Ahead RMSE: REAL 1.43, RE 1.27, AL 0.5, 0.88, 0.73, 0.44, 0.80, 0.64
  - 8-Quarters Ahead RMSE: REAL 1.18, RE 1.62, AL 1.02, 1.05, 1.52, 0.25, 0.49, 0.41
- Note: The table in the source lists RMSEs across variables "Real Wage Gap", "Output Gap", "Policy Rate", "Inflation" and across horizons; the numbers above are transcribed directly as they appear.

### Comparison of estimation results: Brazil versus the USA
- Main empirical findings:
  - Expectations in Brazil depend more on past outcomes than in the USA — seen by adding coefficients on the first two lags of both inflation and wages.
  - Real wage expectations are substantially more persistent in Brazil than in the USA.
  - Coefficient stability: coefficients were stable over the last ten years before the pandemic; the mean expected inflation coefficient was zero as households expected inflation to be at the central bank target.
  - Pandemic disruption: during the pandemic both economies experienced inflation outcomes above target, which challenged prior stability.
  - As inflation expectations respond more to past inflation in Brazil, there is feedback from inflation to inflation expectations that keeps inflation higher for longer across shocks, despite a stronger monetary policy response in Brazil.
  - Implication: Monetary policy has to do more to lower inflation in an EM economy like Brazil, even if hit with the same shocks as the US.

### Optimal monetary policy and channels in the AL model
- Optimal policy objective defined as the interest rate path {H_c} for t = 1 to ∞ that minimizes the welfare function:
  - Σ_{c=j}^{∞} β_c (0.75(H_c − H_{c−1}) + (y_c − 0)^2 + (π̃_c − 0)^2)
- Assumptions in optimization:
  - Equal weights for output gap (y_c) and inflation deviations from target (π̃_c).
  - Role for interest rate smoothing included (0.75 weight on interest rate change).
  - Central bank assumed to have full knowledge of current shocks, all future shocks, and full knowledge of how their actions impact expectations.
- Three channels through which the central bank influences inflation (in the estimated AL model):
  1. Standard direct channel: tighter policy cools demand, lowering output gap and hence inflation.
  2. Expectations channel via current inflation: tightening lowers current inflation, which enters forecasting equations and lowers next-period expectations.
  3. Learning-coefficient channel: policy can affect the coefficients in the households’ forecasting equation—if households observe less inflation than expected this period, they update their model of how past inflation matters for future inflation.

### Key implications and contributions
- Modeling strategy contributes to debate on adaptive expectations by showing:
  - A price Phillips curve that includes only the output gap (and not marginal cost / real wage gap) would predict that, under adaptive expectations, the only way to lower inflation is via a negative output gap.
  - This model shows that a negative real wage gap could enable anchoring of inflation even with fully adaptive expectations.
- Policy implication highlighted:
  - In economies where expectations are more backward-looking (Brazil in the estimation), monetary policy needs to act more forcefully to lower inflation because of the stronger feedback from realized inflation to expectations.

*Source: Annex 2.7 "Role of Wage and Price Expectations: Scenarios from a Small DSGE Model", IMF staff estimates, World Economic Outlook (October 2022).*

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_Source: https://www.imf.org/-/media/files/publications/weo/2022/october/english/ch2annex.pdf_