Figure 1: Impulse-response Analysis
(a) Benchmark
(b) Scarring
Panel A: Benchmark Specification: Four-panel chart showing responses over 20 quarters to a one-time shock to the cycle that increases GDP by one percent on impact.
First panel (GDP): Shows contributions from cycle (blue bars). GDP rises to peak around 3% at quarter 8-10, then gradually declines. The cycle has a pronounced hump-shaped profile.
Second panel (Unemployment Rate): Shows negative response peaking around -2% at quarters 8-10, then slowly returning toward zero.
Third panel (Price Inflation): Shows gradual rise peaking around 1% at quarters 14-16. Response builds slowly and persists.
Fourth panel (Wage Inflation): Similar pattern to price inflation, peaking around 1% at quarters 14-16 with gradual buildup.
Panel B - Scarring Specification: Four-panel chart with same structure as 1a but showing markedly different dynamics.
First panel (GDP): Shows contributions from cycle (blue bars) and scarring effects on productive capacity (orange bars). Peak occurs earlier (around quarter 4-6) at approximately 3%, with faster decay. Orange bars show significant scarring contribution to the hump shape.
Second panel (Unemployment Rate): Sharper negative response peaking around -2% at quarters 4-6, with more rapid return to baseline compared to benchmark.
Third panel (Price Inflation): Much more front-loaded response, peaking around 1% by quarter 4-6, then declining more quickly than benchmark.
Fourth panel (Wage Inflation): Similarly front-loaded, peaking around 1% by quarter 4-6 with faster decay.
Figure 2: Trend Output
Line chart spanning 1987-2024 showing multiple trend output estimates:
Blue dotted line (Benchmark y*): Smooth, relatively steady upward trend from approximately 910 (log scale) in 1987 to 1050 in 2024. Shows minimal cyclical variation.
Orange dashed line (Scarring ỹ*): More volatile trend that tracks closer to actual log GDP data (black line). Shows notable dips during 2008-2009 (GFC) and 2020 (COVID). Exhibits more pronounced short-term dynamics than benchmark.
Green dash-dotted line (Scarring y*): The purely supply-driven component under scarring. Deviates from benchmark blue line particularly during 2008-2012, showing lower levels during this period.
Key observation: The orange dashed line (productive capacity with scarring) shows significantly more cyclical movement than the smooth benchmark trend, particularly visible during recession periods.
Figure 3: Trend Unemployment
Line chart spanning 1987-2024 showing unemployment trend estimates:
Blue dotted line (Benchmark u*): Relatively stable trend starting from about 5.5% in 1987 and ending at about 6% in 2024. Modest rise during GFC to about 5.5%.
Orange dashed line (Scarring ũ*): More cyclically responsive. Shows sharp rise to approximately 8% during 2008-2009, gradual decline through 2019, increases again during COVID-2020.
Green dash-dotted line (Scarring u*): Supply-driven component under scarring. Rises to approximately 7% during GFC (compared to 5% in benchmark), then declines more substantially through late 2010s, reaching about 4.5% by 2019.
Key observation: The scarring specification attributes more of the GFC unemployment rise to supply-side factors (green line rises sharply) compared to benchmark.
Figure 4: Output Gap
Line chart spanning 1987-2024 comparing cyclical components:
Blue dotted line (Benchmark c): Large, persistent swings. Shows deep negative gap of approximately -6% during 2008-2009. Very persistent movements.
Orange dashed line (Scarring c̃): Much less volatile. Peak-to-trough range approximately ±2%. Gaps close more quickly.
Green dashed line (Scarring y-y*): Deviation of output from its supply-driven trend under scarring. Shows less variability than the benchmark cycle. Gap of approximately -4% during 2008-2009, compared to -6% in the benchmark.
Key observation: The scarring specification produces less volatile cycles than the benchmark. The deviation of output from its purely supply-driven trend also differs meaningfully at times between the two specifications (notably during the GFC).
Figure 5: Stochastic Volatilities
Eleven-panel chart showing estimated time-varying volatilities of different shocks from 1987-2024. Blue-dotted lines for the benchmark specification. Orange-dashed line for the scarring specification
First panel (shock to the cycle ε): Blue-dotted line (benchmark) consistently higher than orange-dashed line (scarring). Both lines go up, showing an increase in volatility over time.
Second panel (shock to y*): Orange-dashed line (scarring) higher than blue-dotted line (benchmark) for most of the sample. Both lines go up, showing an increase in volatility over time.
Third panel (shock to trend output growth μ): Orange-dashed line (scarring) slightly and consistently higher than blue-dotted line (benchmark). Both lines go up, showing an increase in volatility over time.
Fourth panel (cost-push shock in price Phillips curve ε_π): Blue-dotted line (benchmark) consistently higher than orange-dashed line (scarring). Both lines go up, showing an increase in volatility over time.
Fifth panel (shock to inflation trend η_ π*): Blue-dotted line (benchmark) indistinguishable from orange-dashed line (scarring). Both lines go down, showing a decrease in volatility over time.
Sixth panel (shock to inflation expectations e): Blue-dotted line (benchmark) indistinguishable from orange-dashed line (scarring). Both lines go down, showing a decrease in volatility over time.
Seventh panel (cost-push shock in wage Phillips curve ε_π^w): Blue-dotted line (benchmark) consistently higher than orange-dashed line (scarring). Both lines go down, showing a decrease in volatility over time.
Eighth panel (shock to u*): Orange-dashed line (scarring) consistently higher than blue-dotted line (benchmark). Both lines go up, showing an increase in volatility over time.
Ninth panel (Okun’s law errors � ): Blue-dotted line (benchmark) and the orange-dashed line (scarring) are close to each other until the few years before the GFC; thereafter, the former line moves up and away from the latter line until the end of the sample. Both lines go up, showing an increase in volatility over time.
Tenth panel (shock to labor productivity growth v): Blue-dotted line (benchmark) indistinguishable from orange-dashed line (scarring). Both lines go down, showing a decrease in volatility over time.
Eleventh panel (shock to trend labor productivity growth η_g*): Blue-dotted line (benchmark) indistinguishable from orange-dashed line (scarring). Both lines go up over the first ten years over the sample, after which they go down for the remainder of the sample.
Key observation: Scarring specification exhibits a rebalancing in the volatility of shocks compared to the benchmark specification: higher volatility in the supply shocks driving the trends, lower cyclical and cost-push shock volatility.
Figure 6: Price Inflation Shock Decomposition
(a) Benchmark
(b) Scarring
Panel A - Benchmark: Stacked area chart 1987-2024 decomposing price inflation deviations from trend:
Black line: Actual inflation gap (deviation from trend)
Orange areas: Cycle shock contributions
Blue/Green areas: Cost-push shock contributions - dominate the decomposition during 2021-2022 surge (+3% contribution)
Panel B - Scarring: Stacked area chart with same structure:
Black line: Actual inflation gap (deviation from trend)
Orange areas: Cycle shock contributions - larger role than in the benchmark
Blue/Green areas: Cost-push shocks - smaller role than in the benchmark
Key pattern: Scarring specification attributes more inflation variation to cyclical shocks, less to cost-push factors.
Figure 7: Data decomposition of y*
(a) Benchmark
(b) Scarring
Panel A - Benchmark: Stacked bar chart 1987-2024 showing contributions of different observables to y* estimates:
Black line: debased (the contribution of the model’s initial conditions has been removed) estimate of y*
Blue bars (Real activity data): Negative contribution, dominate entirely, showing close to 100% of variation in y* estimates
Red bars (Inflation data): Almost no visible contribution
Key finding: Inflation data plays essentially no role in identifying y* in benchmark.
Panel B - Scarring: Stacked bar chart with same structure:
Black line: debased (the contribution of the model’s initial conditions has been removed) estimate of y*
Blue bars (Real activity component): Negative contribution, still primary driver but reduced role, accounting for approximately 60-80% of variation
Red bars (Inflation component): Substantial positive contribution
Key finding: Scarring specification gives inflation data meaningful weight in estimating y*.
Figure 8: Bayesian Model Comparison
Contour plot with joint posterior density of κ₂ (x-axis, 0 to 1) and γ₂ (y-axis, -2 to 0):
Density scale: Color gradient from blue (low, ~0.5) through yellow (high, ~5.5)
Peak location: Yellow region centered around κ₂ ≈ 0.2, γ₂ ≈ -0.3, with density approximately 5.5
Benchmark location: Upper-left corner at κ₂ = 0, γ₂ = 0, with density approximately 0.5
Text annotation: "BF_s:b=4.74, P(Ms|data)=0.83"
Interpretation: Posterior density at scarring parameter estimates is over 10 times higher than at benchmark location, yielding Bayes factor of 4.74 favoring scarring specification with 83% posterior probability.
Figure 9: Savage-Dickey Density Ratios
(a) Test of a model without scarring in output
(b) Test of a model without scarring in
unemployment
Panel A - Test of model without scarring in output: Overlaid density plots:
Blue solid line (Prior): Normal distribution centered at 0 for κ₂, density at κ₂=0 approximately 1.5
Orange dashed line (Posterior): Shifted right with mode around 0.2, density at κ₂=0 approximately 0.5
Ratio calculation indicates strong evidence favoring output scarring over benchmark.
Panel B - Test of model without scarring in unemployment: Overlaid density plots:
Blue solid line (Prior): Normal distribution centered at 0 for γ₂, density at 0 approximately 1.6
Orange dashed line (Posterior): Shifted left with mode around -0.4, density at γ₂=0 approximately 0.2
Similar pattern favoring unemployment scarring mechanism.
Figure B.1: Distribution of model parameters estimates (part 1)
Note: The graphs show the distribution of the percent
deviation of the estimator from the true parameter value.
Monte Carlo simulation results showing the distributions of 6 parameter estimates obtained from 1,000 simulations:
Each graph shows the distribution of the percent deviation of estimator from the true parameter value:
κ₁: Centered at 0, range ±20-25%
κ₂: Centered at 0, range ±20-20%
γ₁: Centered at 0, range ±30-30%
γ₂: Centered at 0, range ±30-30%
θ₁: Centered at 0, range ±20-20%
θ₂: Centered at 0, range ±30-30%
φ₁: Centered at 0, range ±10-10%
φ₂: Centered at 0, range ±25-25%
All distributions approximately normal, centered on true values, demonstrating parameter identification.
Figure B.2: Distribution of model parameters estimates (part 2)
Note: The graphs show the distribution of the
estimator minus the true parameter value.
Additional distributions show absolute deviations (not percent) of the estimator from the true parameter value:
μ: Centered at 0, range ±20-20
Variance parameters: All centered at 0 deviation, rather symmetric distributions
Combined with B.1, confirms all model parameters are well-identified and accurately estimated.
Figure C.1: Outliers
Six-panel time series chart 1987-2024 showing estimated additive outliers:
GDP: Single large negative outlier approximately -8% in 2020:Q2, otherwise near zero
Price inflation: Large negative outlier of approximately -0.8% in 2020:Q2, smaller positive outlier of the opposite sign a couple quarters later
Trend inflation: Negative outlier of about -0.08/-0.09 in 2022
Unemployment Rate: Large positive outlier of approximately +8% in 2020:Q3, smaller positive outliers in subsequent periods
Wage Inflation: A couple of small outliers over the sample, notably in 2021
Productivity growth: Small outlier in late 1996, early 1997
Demonstrates outlier treatment successfully isolates COVID-related extreme observations.
Figure C.2: Probability of Outliers
Six-panel chart showing estimated probability that observation is an outlier (0 to 1):
GDP: Probability =1.0 (certainty) at 2020:Q2, zero or close to zero elsewhere
Price inflation: Probability = 0.2 in 2020:Q2, zero or close to zero elsewhere
Trend inflation: Probability = 0.2 in 2022, zero elsewhere
Unemployment Rate: Probability =1.0 at 2020:Q2 and Q3, zero elsewhere
Wage Inflation: Probability = 0.2 in 2021, zero or close to zero elsewhere
Productivity growth: Probability close to zero everywhere
Confirms outlier detection is concentrated precisely at COVID pandemic quarters for real variables only.
Figure D.1: GDP Shock Decomposition
(a) Benchmark
(b) Scarring
Panel (a) - Benchmark: Stacked area chart 1987-2024 decomposing four-quarter GDP growth deviations from trend:
Black line: Actual detrended 4Q GDP growth
Orange (cycle shocks): Dominant driver, large negative contribution of -4% during GFC
Blue (level shocks to y*): Modest positive contributions throughout
Green (trend ouput growth shocks): Persistent negative contributions starting in the early 2000s
Panel (b) - Scarring: Stacked area chart with different composition:
Black line: Actual detrended 4Q GDP growth
Orange (cycle shocks): Reduced role compared to benchmark, -3% during GFC vs -4%
Blue (level shocks to y*): Slightly larger role than in the benchmark
Green (trend ouput growth shocks): Persistent negative contributions starting in the early 2000s
Pattern: The scarring specification rebalances volatility from cycle shocks toward shocks to y*
Figure D.2: Unemployment Shock Decomposition
(a) Benchmark
(b) Scarring
Panel (a) - Benchmark: Stacked area chart 1987-2024 showing unemployment rate decomposition:
Black line: Actual detrended unemployment rate
Orange (cycle shocks): Primary driver of deviations, +5% during GFC
Yellow (trend unemployment): Modest positive contributions throughout
Pattern: Cycle shocks dominate unemployment fluctuations.
Panel (b) - Scarring: Modified composition:
Black line: Actual detrended unemployment rate
Orange (cycle shocks): Reduced contributions to unemployment fluctuations compared to benchmark specification, contribution of about +3% during GFC
Yellow (trend unemployment): Larger contributions throughout compared to benchmark specification, contribution of about +2% during GFC
Pattern: Supply-driven unemployment trend explains larger share of unemployment variations in scarring specification
Figure D.3: Inflation and Expected Inflation
Time series chart 1987-2024 showing:
Black solid line (Actual Core PCE inflation): Four-quarter moving average, ranging 1-5%
Orange-dashed line (Trend inflation - scarring): Smooth trend declining from ~5% in 1987 to ~2% by mid-1990s, stable around 2% thereafter, notable uptick to above 3% in 2021-2022
Blue-dotted line (Trend inflation - benchmark): Nearly identical path to scarring specification
Key observation: Both specifications produce very similar inflation expectations estimates
Figure D.4: Wage Inflation Shock Decomposition
(a) Benchmark
(b) Scarring
Panel (a) - Benchmark: Stacked area chart 1987-2024:
Black line: Four-quarter moving average of the wage inflation gap (the sum of the contributions from the state of the Kalman filter’s initial conditions, the trend inflation process and the trend productivity growth rate)
Green (wage cost-push shocks): Large contributions, dominant factor
Orange (cycle shocks): More modest contributions
Pattern: Cost-push shocks heavily relied upon to explain wage dynamics.
Panel (b) - Scarring: Modified composition:
Black line: Four-quarter moving average of the wage inflation gap
Green (wage cost-push shocks): Substantially reduced role compared to the benchmark specification
Orange (cycle shocks): Larger contribution than in the benchmark specification
Pattern: Balanced contribution of cycle shocks and cost-push factors to explain wage inflation dynamics
Figure E.1: Stochastic Volatility Estimates
Eleven-panel comparison of volatility estimates across three specifications (benchmark-blue, scarring-green, hysteresis-red):
All panels span 1987-2024:
First panel (shock to the cycle ε): Blue line above green line, green line above red line, all lines going up
Second panel (shock to y*): Red line above green line, green line above blue line, all lines going up except for the red line in early part of the sample
Third panel (shock to trend output growth μ): Red line above green line, green line slightly above blue line, all lines going up
Fourth panel (cost-push shock in price Phillips curve ε_π): Blue line above green line, green line above red line, all lines going up
Fifth panel (shock to inflation trend η_ π*): Lines are indistinguishable, all going down
Sixth panel (shock to inflation expectations e): Lines are indistinguishable, all going down
Seventh panel (cost-push shock in wage Phillips curve ε_π^w): Blue line above green line, green line above red line, all lines going down
Eighth panel (shock to u*): Red line above green line, green line above blue line, all lines going up
Ninth panel (Okun’s law errors � ): Lines are indistinguishable in the earlier part of the sample, in the later part of the sample blue line above green line and green line above red line, all lines going ip
Tenth panel (shock to labor productivity growth v): Lines are indistinguishable, all going down
Eleventh panel (shock to trend labor productivity growth η_g*): Lines are indistinguishable, going up over the first ten years over the sample, after which they go down for the remainder of the sample.
Key finding: Hysteresis specification shows highest supply shock volatility, lowest cyclical and cost-push volatility - most extreme rebalancing.
Figure E.2: Productive Capacity with History Dependence
Line chart 1987-2024 comparing productive capacity estimates:
Yellow dashed (Scarring ỹ*): Shows cyclical movements, dips during GFC and 2020
Orange dashed-dotted (Hysteresis ỹ*): Very close to scarring ỹ but slightly more volatile
Blue dotted (Benchmark y*): Smooth upward trend
Key observation: Hysteresis specification produces slightly more volatile productive capacity estimates than the scarring specification
Figure E.3: Output Gap with History Dependence
Line chart 1987-2024 comparing cyclical components:
Blue dotted line (Benchmark c): Large, persistent swings. Shows deep negative gap of approximately -6% during 2008-2009. Very persistent movements.
Orange dashed-dotted line (Scarring c̃): Much less volatile. Peak-to-trough range approximately ±2%. Gaps close more quickly.
Yellow dashed line (Scarring c̃): Even shallower fluctuations around zero than the scarring c̃
Key observation: Under hysteresis, cycle is less volatile because more persistence is captured by permanent trend movements.
Figure E.4: Trend Unemployment with History Dependence
Line chart spanning 1987-2024 showing unemployment trend estimates:
Blue dotted line (Benchmark u*): Relatively stable trend starting from about 5.5% in 1987 and ending at about 6% in 2024. Modest rise during GFC to about 5.5%.
Orange dashed-dotted line (Scarring ũ*): More cyclically responsive. Shows sharp rise to approximately 8% during 2008-2009, gradual decline through 2019, increases again during COVID-2020.
Yellow dashed line (Hysteresis ũ*): Minimal differences with scarring ũ*
Key observation: Minimal difference between scarring and hysteresis specifications for unemployment trend estimates.
Table E.1
Extended version of Table 1 adding hysteresis column:
Cyclical dynamics parameters:
φ₁ (AR coefficient 1): Hysteresis: 0.73 [0.41,1.03]
φ₂ (AR coefficient 2): Hysteresis: 0.17 [-0.13,0.49]
Okun's law parameters:
θ₁: Hysteresis: -0.49 [-0.68,-0.28]
θ₂: Hysteresis: -0.55 [-0.73,-0.37]
Price Phillips curve:
β (lagged inflation): Hysteresis: 0.45 [0.33,0.57]
κ₁ (level effect): Hysteresis: 0 (restricted)
κ₂ (speed effect): Hysteresis: 1.01 [0.64,1.37]
Wage Phillips curve:
λ (constant): Hysteresis: -0.68 [-0.83,-0.53]
βw (lagged wage inflation): Hysteresis: 0.29 [0.18,0.40]
γ₁ (level effect): Hysteresis: -0.94 [-1.33,-0.54]
γ₂ (speed effect): Hysteresis: -0.29 [-0.52,-0.07]
Initial conditions:
y₀*: Prior N(911.9,5) | Similar across specifications
Other initial conditions for μ₀, π₀*, u₀*, g₀* are provided and are similar across specifications
Implied parameters:
κ (total price Phillips slope): Hysteresis: 1.01 [0.64,1.37]
γ (total wage Phillips slope): Hysteresis: -1.24 [-1.69, -0.76]
ρʸ=δʸ (output scarring coefficient): Hysteresis: 1 (restricted)
ρᵘ=δᵘ (unemployment scarring coefficient): Hysteresis: 0.24 [0.07,0.40]
Key observations: Shows hysteresis produces most extreme parameter estimates: steepest Phillips curves, lowest cycle persistence, complete permanent effects on output.
Figure F.1: Productive Capacity with Correlation
Line chart 1987-2024 comparing:
Orange dashed (Scarring ỹ*): Baseline scarring estimate
Light blue dotted (Scarring with correlation ỹ*): Very similar path, slightly wider uncertainty bands
Key message: Adding correlation between trend and cycle shocks produces minimal change in productive capacity estimates.
Figure F.2: Output Gap with Correlation
Line chart 1987-2024 comparing:
Orange dashed (Scarring c̃): Baseline scarring cycle
Light blue dotted (Scarring with correlation c̃): Nearly identical path, similar uncertainty
Key message: Cycle estimates robust to allowing correlated disturbances.
Figure F.3: Trend Unemployment with Correlation
Line chart 1987-2024 comparing:
Orange dashed (Scarring ũ*): Baseline estimate
Light blue dotted (Scarring with correlation ũ*): Nearly identical, overlapping uncertainty bands
Key message: Unemployment trend estimates unchanged by correlation specification.
Figure F.4: Savage-Dickey Density Ratio
Overlaid density plots for correlation coefficient ω:
Blue dashed (Prior): Beta distribution centered at 0, symmetric, density at ω=0 approximately 0.5
Orange solid (Posterior): Shifted right, mode around 0.6, density at ω=0 approximately 0.1
Posterior probability favors positive correlation between trend and cycle shocks. Data support correlation structure.