## APPENDIX B. Statistical Model

## Source details

**Canonical URL:** [APPENDIX B. Statistical Model](https://www.imf.org/-/media/files/publications/wp/2018/wp18115.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2018/wp18115.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2018/wp18115.pdf.json)

---

### A. Data construction, sample, and measurement
- Individual-level panel construction:
  - Annual earnings: sum of contributive earnings across jobs, adjusted using CPI; earnings growth Δy_it computed as difference in logs.
  - Employment time: number of weeks worked = minimum(52, sum of weeks worked across all jobs within a calendar year).
  - Changes in employment time Δx_it: difference in logs across every two consecutive years in which the worker is observed.
  - Weekly earnings growth Δw_it recovered as: Δw_it ≡ Δy_it − Δx_it.
- Data source and sample:
  - LoSai dataset from the Italian Ministry of Labor and Social Policy (MLPS); contributive earnings (imponibile previdenziale).
  - Final panel: unbalanced panel of 974,686 workers over 27 years with a total of 9,293,543 individual-year observations.
  - Average observations per two-year period: 335,000.
  - Average tenure in sample per worker: 9.7 years; median tenure: 8 years.
- Sample restrictions and treatment:
  - Restricted to males aged 25-60 with records in both yeart and t−1, earnings above the 2.5 percentile, and at least 3 weeks worked in a given year.
  - The 2.5 percentile is equivalent to an annual income of €800 in 2012.
  - Main work contract restricted to full-time (defined as the record associated with the highest share of overall earnings in a year). Including part-time workers does not change main results.
- Top-coding and robustness:
  - 0.5 percent of all matched observations are top-coded; these observations are excluded in the main analysis.
  - Sensitivity checks: including top-coded observations or assigning extreme values yields virtually unaffected series for annual earnings; weekly-earnings series show a bump in variance between 1998 and 2000 but are judged unlikely to alter main conclusions.
- Measurement error and overlapping jobs:
  - Weeks of work measured accurately in INPS; exclusion of very-low-weeks observations reduces measurement error concerns.
  - Subsample accounting for overlapping jobs yields quantitatively similar results.

### B. Key empirical findings (Italy, INPS 1985–2012) — patterns and moments
- Dominance of employment time in cross-sectional variation:
  - When measured in logs, changes in employment time generate more than four-fifths of the variance in annual earnings growth.
  - Share working 52 weeks: 77% of workers (77.2% if only include people with more than two weeks worked; 77.3% if also restrict to non-top-coded observations).
  - 80% work 50 weeks or more.
  - Comparison to CPS (US): 68.9% of prime-aged males work 52 weeks; rises to 82.90% when considering only individuals working at the time of interview.
- Tail behavior and group evidence (reference year t = 2002):
  - Workers employed 52 weeks in yeart−1 (Group A) are much less likely to appear in the right tail: probability of > 90th percentile in 2002 for Group A is less than 2.1 percent (never above 4.3 percent in other years).
  - Workers employed 52 weeks in both yeart−1 and yeart are unlikely to experience large positive or negative earnings growth.
  - Non-parametric calculation: complement group is about 19.8 times more likely to experience large earnings growth (calculation in text: 0.021×0.8+p×0.2=0.1⇒p=0.416 ⇒ 0.416/0.021=19.8).
- Variance and third-moment patterns:
  - Example variance decomposition (2002): var(Δy) 0.158 = var(Δx) 0.150 + 2cov(Δx,Δw) −0.014 + var(Δw) 0.023.
  - Standard deviations implied by 2002 variances: employment time standard deviation 40 percent; weekly earnings standard deviation 15 percent.
  - Across years: variance of changes in employment time is greater than four-fifths of the variance of annual earnings growth in all years; cross-term (twice the covariance) is small and typically negative.
  - Third central moment: third moment of employment time growth closely tracks that of annual earnings growth in sign and magnitude; weekly earnings third moments are generally one order of magnitude smaller (exceptions: 1998 and 1999).
- Time-series moments and cyclical behavior:
  - Mean change in employment time reaches a low of −0.07 at 2009.
  - Mean change in weekly earnings averages +0.02 during expansions and falls to −0.02 in 2012.
  - The third moment of annual earnings growth and of changes in employment time are pro-cyclical, dropping to about −0.1 in 2009.
  - Five-year horizons: five-year weekly earnings variance 0.05; one-year weekly earnings variance 0.025 — weekly-earnings shocks have a considerable permanent component; employment-time variation appears largely transitory.
- Job stayers vs switchers:
  - Raw differences: variance of switchers ≈ twice that of stayers; switchers’ third central moment more negative.
  - Restricting to workers employed 52 weeks in both years removes most of the switcher-stayer differences (example: variance difference 0.007, third moment difference 0.001 in 2009/2008 comparison).

### C. Statistical decomposition of moments and empirical magnitudes
- Decomposition identities:
  - Δy_it = Δx_it + Δw_it.
  - Mean: m_1(Δy) = m_1(Δx) + m_1(Δw).
  - Variance: m_2(Δy) = m_2(Δx) + 2 m_1,1(Δx,Δw) + m_2(Δw).
  - Third central moment: m_3(Δy) = m_3(Δx) + 3 m_2,1(Δx,Δw) + 3 m_1,2(Δx,Δw) + m_3(Δw).
- Selected year statistics and decompositions:
  - 2002 variance decomposition (preserved): var(Δy) 0.158 = var(Δx) 0.150 + 2cov(Δx,Δw) −0.014 + var(Δw) 0.023.
  - 2009 example (Table summary): m2(Δy) 0.203; m2(Δx) 0.168; m2(Δw) 0.028; cross-term 0.007.
  - Third-moment decomposition (selected years, Table A.3):
    - 1986: m3(Δy) −0.028 (0.002); m3(Δx) −0.040 (0.002); m3(Δw) 0.000 (0.000); cross-term 0.012 (0.001).
    - 2000: m3(Δy) 0.028 (0.002); m3(Δx) 0.026 (0.002); m3(Δw) 0.001 (0.001); cross-term 0.000 (0.001).
    - 2009: m3(Δy) −0.094 (0.002); m3(Δx) −0.092 (0.002); m3(Δw) −0.002 (0.000); cross-term 0.000 (0.001).
    - 2012: m3(Δy) −0.076 (0.003); m3(Δx) −0.071 (0.002); m3(Δw) −0.002 (0.000); cross-term −0.002 (0.001).
- Cyclicality regressions (Table 4 and Table A.7 highlights):
  - Third moment of annual earnings growth regressed on GDP growth: coefficient 0.659 (0.124) (column 1).
  - Adding third moment of changes in employment time: GDP coefficient −0.120 (0.086); third moment of employment time coefficient 1.020 (0.091) (column 2). R-squared rises from 0.43 to 0.86. Observations: 27 (table bottom states Observations 2727 — preserved as 2727 in source).
  - Time-series cyclicality tests (Italy, Table A.7 selected rows):
    - Mean: GDP Growth coefficient 0.722 (0.110); controlling for Mean of employment time growth GDP coefficient 0.063 (0.120); Mean of employment time growth coefficient 0.833 (0.150); R-squared 0.52 → 0.78; Observations 27.
    - Variance: GDP Growth coefficient −0.422 (0.176); controlling for Variance of employment time growth GDP coefficient −0.257 (0.096); Variance of employment time growth coefficient 0.626 (0.117); R-squared 0.18 → 0.54; Observations 27.
    - Skewness: GDP Growth coefficient 0.629 (0.160); controlling for Skewness of employment time growth GDP coefficient −0.091 (0.071); Skewness of employment time growth coefficient 0.976 (0.081); R-squared 0.40 → 0.83; Observations 27.
  - Interpretation: including employment-time moments largely accounts for the cyclicality of mean, variance, and especially the third moment of annual earnings growth.

### D. Statistical model structure and mechanism for procyclical skewness
- Conceptual decomposition:
  - Annual earnings change Δy = change in employment time Δx (extensive margin) + change in weekly earnings Δw (intensive margin).
- Employment process:
  - Random monthly transitions between discrete labor market states (Markov chain); separation and hiring rates vary over the cycle.
  - Mechanism for procyclical skewness: in recessions separation rate rises and hiring rate falls → more workers experience reductions in employment time and longer non-employment spells → negative skewness in cross-sectional Δx and thereby in Δy.
- Wage process:
  - log wage w_it = η_it + e_it.
  - Permanent component η_it follows a unit root: η_it = η_it−1 + u_it, with u_it ∼ N(μ_t, σ^2).
  - Transitory shock e_it ∼ Exponential(b); changes Δe_it follow Laplace distribution (producing straight tails for weekly-earnings changes).
  - Time variation allowed only in μ_t to reflect acyclical weekly-earnings distribution around its mean.
- Calibration / numerical example and external validation:
  - Simulation parameters (preserved):
    - Separation rate: 1 percent in expansion; 1.25 percent in recession.
    - Job finding (hiring) rate: 15 percent in expansion; 9 percent in recession.
    - Wage process parameters: σ = 0.07 (permanent shock standard deviation), b = 0.10 (transitory shock parameter → transitory stdev 10 percent).
    - Simulation: 100,000 worker histories over a 10 years period, with the 8th and 9th years as recession years.
  - Feeding matched monthly CPS labor-market flows from the US into the model produces model-implied cross-sectional distributions of Δx whose time series of the third moment follows timing and magnitude of observed third moment of earnings growth — presented as suggestive external validation.

### E. Markov-chain algorithm for annual employment distributions (CPS-based exercise)
- Objective: transform monthly labor flows (CPS) into cross-sectional distributions of annual changes in employment time.
- Definitions and forward/reverse counting processes:
  - Forward counting →V_n(t) = sum_{τ=0}^{n-1} 1(s_{t+τ} ∈ E) (months employed in next n months).
  - Reverse counting ←V_n(t) counts employment months in previous n months.
  - Forward PMF →Π_i(k,t) = sum_j →Π_{i,j}(12,k,t) — probability for worker in state i at t to have k months employed in next 12 months.
  - Unconditional joint distribution: P(←V_{12}(t)=k, →V_{12}(t)=l) = sum_{i,j} ←Π_i(k,t) P_{i,j}(t) →Π_j(l,t).
  - Final object: distribution of log difference log →V_{12}(t) − log ←V_{12}(t).
- Recursive computation (Proposition 1):
  1. For every n, k > 0:
     - [ →Π_EE(n,k,t) ; →Π_EN(n,k,t) ] = [ P_EE(t+1)  P_EN(t+1) ] · →Π(n−1,k−1,t+1)
     - [ →Π_NE(n,k,t) ; →Π_NN(n,k,t) ] = [ P_NE(t+1)  P_NN(t+1) ] · →Π(n−1,k,t+1)
  2. If k = 0 and n > 0:
     - →Π(n,0,t) = [ 0 0 ; 0 P_NN(t+1) ] · →Π(n−1,0,t+1)
  3. If n = 0 and k > 0:
     - →Π(0,k,t) = 0
  4. If n = 0 and k = 0:
     - →Π(0,0,t) = I_{S×S}
- TPM estimation:
  - TPMs are estimated directly from flows data by dividing each flow by the stock at the destination state (forward) or origin state (backward).

### F. Limitations, robustness, and interpretation
- Data limitations:
  - INPS observes annual earnings and weeks of employment but not hours per week or hourly wages; thus Δw aggregates Δh and Δẇ.
  - SSA MEF and similar administrative datasets often observe Δy but not Δx or Δw, complicating decomposition in those datasets.
  - CPS matched January-to-January weekly-wage changes suffer small sample sizes (≈ two thousands) and measurement error, producing large standard errors for third moments.
- Model limitations:
  - Markov assumption imposes ex-ante homogeneity and no history dependence (no duration dependence). Duration dependence and heterogeneity by occupation/age may matter.
  - Mitigation: expand state space or stratify TPMs (age-group specific TPMs) when data permit.
- Robustness checks:
  - Overlapping-jobs subsample, top-coding sensitivity, age-profile removal (Figures A.5 and A.6) — main results remain qualitatively unchanged.
- Comparisons with other studies:
  - Alternative interpretation to Guvenen, Ozkan, and Song (2014): procyclical skewness attributed largely to employment-time dynamics rather than only wage dynamics.
  - Differences with Busch et al. (2016) and others may reflect methodological choices (third central moment vs quantile-based skewness) and institutional/sample differences.

### G. Policy-relevant implications and recommendations
- Modeling guidance:
  - Explicitly model employment margin separately from wage margin in earnings processes; failing to do so risks overstating the magnitude and persistence of downside risk in recessions.
  - Distinguish transitory employment-time shocks (largely driving cyclical tails) from more persistent wage shocks (weekly earnings).
- Redistribution and insurance:
  - Tax and transfer system reduces variance and asymmetry of earnings growth, especially after the 2007 recession.
  - Example: adding unemployment benefits and tax adjustments reduces aggregate variance of earnings growth by 25%, from 0.2 to 0.15 (1997–2012 evidence).
  - Most insurance arises from unemployment benefits; income tax plays a marginal role.
- Empirical guidance for comparative studies:
  - When comparing one-year vs five-year distributions, account for persistence differences: employment-time variation appears mostly transitory; weekly-earnings shocks include a larger permanent component.
  - Use datasets that observe both Δx and Δw (where possible) for reliable decomposition; otherwise interpret higher-order moments cautiously.

*Source: APPENDIX B. Statistical Model — wp18115.*

### APPENDIX A. DATA APPENDIX ___________________________________________42

### APPENDIX A. DATA APPENDIX

### A. Italy: Additional Summary Statistics
- A. Italy: Additional Summary Statistics  ________________________________42

### B. Italy: Top-Coding
- B. Italy: Top-Coding  _______________________________________________43

### C. Italy: The Italian Income Tax
- C. Italy: The Italian Income Tax    _____________________________________43

### D. Italy: The Employment Time Statistics
- D. Italy: The Employment Time Statistics _______________________________43

### E. United States: Moments from GOS
- E. United States: Moments from GOS __________________________________43

### F. United States: CPS Data
- F. United States: CPS Data  __________________________________________50

*Source: wp18115 - APPENDIX A. DATA APPENDIX (wp18115 - APPENDIX A. DATA APPENDIX ___________________________________________42)*

### APPENDIX B. Statistical Model ______________________________________________53

### APPENDIX B. Statistical Model

### A. Setup — decomposition and motivation
- Earnings decomposition identity (logs, first differences): Δy_it = Δx_it + Δw_it, where
  - Δy_it is (log) annual earnings growth,
  - Δx_it is change in (log) employment time measured in weeks,
  - Δw_it is change in (log) weekly earnings (Δw = Δh + Δẇ when hours and hourly wage are separable).
- Rationale for decomposition:
  - Employment time (extensive margin) and weekly earnings (intensive margin + hourly wage) have different policy implications and persistence properties.
  - Unemployment insurance insures against drops in employment time but not wage declines; wage insurance targets wage declines.
  - Empirical datasets frequently observe Δy but not both Δx and Δw (e.g., SSA MEF lacks weeks employed), while INPS permits decomposition.

### B. Key empirical findings (Italy, INPS 1985–2012 and referenced US evidence)
- Data and sample:
  - INPS sample covers period 1985-2012, based on workers born on 24 randomly selected birth dates, representing a 6.6 percent sample of non-farm private sector employees.
  - Representative sample example: males 25–60 with 300,000 observations (approximately 6.5% of all male workers in the private sector in that age range) for year 2002.
- Dominance of employment time in cross-sectional variation:
  - When measured in logs, changes in employment time generate more than four-fifths of the variance in annual earnings growth.
  - Workers employed for 52 weeks in a given year are 12 times less likely than those employed for less than 52 weeks to experience an increase larger than the 90th percentile in the following year.
  - Workers employed for 52 weeks are 20 times less likely than those who are not to experience a decrease smaller than the 10th percentile from the previous year.
- Moments and cyclicality:
  - The third moment (skewness measure) of annual earnings growth is highly procyclical: correlation with GDP growth is 0.66 and significant.
  - Controlling for the third moment of changes in employment time reduces this correlation to a non-significant -0.12.
  - Time-series behavior: the third moment of changes in employment time closely tracks that of annual earnings growth; the third moment of changes in weekly earnings is relatively flat and close to zero.
- Persistence at different horizons:
  - Five-year variance of weekly earnings changes: 0.051.
  - One-year variance of weekly earnings changes: 0.024.
  - Five-year variance of employment time changes: about 0.11 (approximately the same as one-year changes).
  - Interpretation: weekly earnings have an important persistent component; changes in employment time are largely transitory.
- Role of taxes and transfers (Italy):
  - Tax and transfer system reduced variance and asymmetry of earnings growth, especially after the 2007 recession.
  - Example for 2009: post-tax earnings growth variance (third moment) was 0.15 (-0.04) compared to 0.20 (-0.11) pre-tax.
  - Most of the insurance effect arises from unemployment benefits; income tax plays a marginal role.
- Switchers vs. stayers:
  - In 2009, variance of earnings growth: switchers 0.40 vs. stayers 0.16.
  - Third central moment more negative for switchers in recessions (e.g., -0.13 vs. -0.08 in 2009).
  - Almost all the switcher-stayer difference is due to workers experiencing changes in employment time; among those employed 52 weeks in both years the difference is negligible (example: variance difference 0.007, third moment difference 0.001 in 2009/2008 comparison).

### C. Statistical decomposition of moments
- Notation:
  - m_j(·) denotes the jth central moment of the cross-sectional distribution.
  - m_k,l(a,b) ≡ E_i[(a−m_1(a))^k (b−m_1(b))^l] for cross-terms.
- Decompositions:
  - Mean: m_1(Δy) = m_1(Δx) + m_1(Δw).
  - Variance: m_2(Δy) = m_2(Δx) + 2 m_1,1(Δx,Δw) + m_2(Δw).
  - Third central moment: m_3(Δy) = m_3(Δx) + 3 m_2,1(Δx,Δw) + 3 m_1,2(Δx,Δw) + m_3(Δw).
- Interpretation:
  - If cross-terms are small (e.g., mean-independence), component shares of variance are approximately the component variances.
  - Third-moment decomposition isolates contribution of asymmetry from employment-time shocks, weekly-earnings shocks, and their interaction.

### D. Model structure (statistical model summary and mechanism)
- Model ingredients:
  - Employment process: random monthly transitions between discrete labor market states (in the spirit of Diamond-Mortensen-Pissarides / job search models); separation and hiring rates vary over the cycle.
  - Wage process: sum of a Gaussian permanent component and a transitory shock, generating a symmetric wage-growth distribution.
- Mechanism for procyclical skewness:
  - Recessions: separation rate rises, hiring rate falls → more workers experience reductions in employment time and longer non-employment spells → large negative Δx more common → negative skewness in cross-sectional Δx and thereby in Δy.
- Calibration / numerical example:
  - A numerical calibration to match key moments from Italy shows that the combination of this employment process and wage process can reproduce the shape and procyclical left skewness of the annual earnings growth distribution with few parameters.
- External validation using US flows:
  - Feeding matched monthly CPS labor market flows from the US into the model produces model-implied cross-sectional distributions of Δx whose time series of the third moment follows timing and magnitude of observed third moment of earnings growth — presented as suggestive evidence for the mechanism.

### E. Limitations and interpretation
- Data limitations:
  - INPS observes annual earnings and weeks of employment but not hours per week or hourly wages; thus Δw aggregates Δh and Δẇ.
  - SSA MEF and similar administrative datasets observe Δy but often not Δx or Δw, complicating decomposition in those datasets.
- Modeling implications:
  - Failure to model employment time and weekly earnings separately can lead to misleading conclusions about dynamics of annual earnings, higher-order moments, and nonlinearities in richer consumption/wealth models.
- Comparisons with other studies:
  - The paper provides an alternative interpretation to Guvenen, Ozkan, and Song (2014) by attributing procyclical skewness largely to employment-time dynamics.
  - Differences with other studies (e.g., Busch et al. (2016) on German data) may stem from methodological choices (third central moment vs. quantile-based skewness) and sample/institutional differences.

*Source: APPENDIX B. Statistical Model (content unit from wp18115).*

### 0.5 percent of all matched observations.  In our main analysis we exclude these observations.

### wp18115 - 0.5 percent of all matched observations.  In our main analysis we exclude these observations.

### Data construction and variable definitions
- Individual-level panel created by combining multiple job records for the same individual in a given year by summing over all records associated with the same worker:
  - Annual earnings: sum of contributive earnings across jobs, adjusted using CPI; earnings growth Δy_it computed as difference in logs.
  - Employment time: number of weeks worked = minimum(52, sum of weeks worked across all jobs within a calendar year).
  - Changes in employment time Δx_it: difference in logs across every two consecutive years in which the worker is observed.
  - Weekly earnings growth Δw_it recovered as: Δw_it ≡ Δy_it − Δx_it.
- Data source: LoSai dataset from the Italian Ministry of Labor and Social Policy (MLPS); contributive earnings (imponibile previdenziale) differ from taxable earnings (imponibile fiscale).
- Top-coding: 0.5 percent of all matched observations are top-coded; these observations are excluded in the main analysis. Appendix A shows adding them back does not change the analysis. An exercise assigning extreme values to top-coded observations indicates effects are likely small.
- Overlapping jobs: If a worker held two jobs at the same time the main construction may not capture overlap. A subsample with exact months worked that accounts for overlapping jobs yields quantitatively very similar and qualitatively unchanged results.

### Sample restrictions and composition
- Restricted to males aged 25-60 with records in both yeart and t−1, earnings above the 2.5 percentile, and at least 3 weeks worked in a given year.
  - The 2.5 percentile is equivalent to an annual income of €800 in 2012.
  - Removing workers who worked less than 3 weeks corresponds to 0.25% of the full sample; 60% of these removed workers have 0 weeks recorded, leaving 0.1% working 1 or 2 weeks.
- Main work contract restricted to full-time (defined as the record associated with the highest share of overall earnings in a year). Including part-time workers does not change main results.
- Final panel: unbalanced panel of 974,686 workers over 27 years with a total of 9,293,543 individual-year observations.
  - Average observations per two-year period: 335,000.
  - Average tenure in sample per worker: 9.7 years; median tenure: 8 years.

### Key summary statistics (all years pooled; earnings in 2013 euros)
- Observations: 9,293,543
- Age:
  - Mean: 40.7
  - Std. Dev.: 19.0
  - P10: 29.00
  - P50: 40.00
  - P90: 54.00
- Annual Earnings:
  - Mean: 27,988
  - Std. Dev.: 18,919
  - P10: 12,181
  - P50: 23,905
  - P90: 45,565
- Employment Time (Weeks):
  - Mean: 48.20
  - Std. Dev.: 9.77
  - P10: 36.00
  - P50: 52.00
  - P90: 52.00
- Weekly Earnings:
  - Mean: 572
  - Std. Dev.: 360
  - P10: 321
  - P50: 474
  - P90: 896
- Δ Earnings (Δy):
  - Mean: 0.01
  - Std. Dev.: 0.40
  - P10: -0.20
  - P50: 0.01
  - P90: 0.22
- Δ Employment Time (Δx):
  - Mean: -0.01
  - Std. Dev.: 0.38
  - P10: -0.10
  - P50: 0.00
  - P90: 0.08
- Δ Weekly Earnings (Δw):
  - Mean: 0.02
  - Std. Dev.: 0.17
  - P10: -0.10
  - P50: 0.01
  - P90: 0.15

### Employment time patterns and variance decomposition
- Share working 52 weeks:
  - 77% of workers in the sample work for 52 weeks (77.2% if only include people with more than two weeks worked; 77.3% if also restrict to non-top-coded observations).
  - 80% work 50 weeks or more.
- Comparison to CPS (US): 68.9% of prime-aged males work 52 weeks; rises to 82.90% when considering only individuals working at the time of interview.
- Variance decomposition of changes in employment:
  - Within-worker component accounts for 86 percent of the variance.
  - Between-worker component accounts for 14 percent of the variance.

### Measurement error considerations
- Division bias concern: measurement errors in employment time could mechanically induce negative correlation with weekly earnings.
  - Weeks of work are measured accurately in INPS data.
  - Exclusion of observations with very low weeks worked reduces measurement error concerns.
  - Standard deviation of weekly earnings is 17 percent, suggesting measurement error in employment time is unlikely to materially bias the measurement exercise.

### Cross-sectional evidence on the role of employment time
- Distributions:
  - Annual earnings growth Δy_it has heavier tails than weekly earnings growth Δw_it; changes in employment time Δx_it match the tails of the distribution of annual earnings growth.
  - Standard deviation: Δy Std. Dev. = 0.40; Δw Std. Dev. = 0.17.
- Group analysis for reference year t = 2002 (comparison to full-sample distribution):
  - Group A: employed 52 weeks in yeart−1 (cannot have positive Δx): right tail of earnings growth distribution much thinner than full sample; left tail remains similar. Implies right tail largely generated by workers not employed 52 weeks in yeart−1.
  - Group B: employed 52 weeks in yeart (mirror image of Group A): members unlikely to experience a drop in earnings larger than the 10th percentile of the full sample distribution.
  - Group C: employed 52 weeks in both yeart−1 and yeart (full 104 weeks): unlikely to experience large positive or negative earnings growth.
- Quantification (Table 2 reference):
  - Probability that a worker in Group A experiences earnings growth > 90th percentile in 2002 is less than 2.1 percent (never above 4.3 percent in other years).
  - Group A includes more than 80 percent of workers in that year, implying the complement (those not working 52 weeks in yeart−1) account for most large positive earnings growth observations.

### Comparability with U.S. evidence
- Figure 2 (Italy vs United States, 1996) shows similar heavy left and right tails in the log-density of annual earnings growth Δy_it.
  - Italian distribution (1996) is less dispersed and more symmetric.
  - Similarities suggest forces shaping the earning-growth distribution are not unique to the United States.

*Source: INPS panel data and associated analysis described in the source content.*

### Appendix TableA.1for the median and other statistics).

### wp18115 - Appendix TableA.1for the median and other statistics).

### Major empirical findings: Italy 1986-2012
- Changes in employment time generate the tails of the annual earnings growth distribution and drive its cyclical asymmetry (third moment).
- Non-parametric evidence: workers not employed for 52 weeks are far more likely to experience extreme changes — the complementary group to Group A are about 19.8 times more likely to experience large earnings growth (calculation in text: 0.021×0.8+p×0.2=0.1⇒p=0.416 ⇒ 0.416/0.021=19.8).
- Variance decomposition (year 2002 example):
  - var(Δy) 0.158 = var(Δx) 0.150 + 2cov(Δx,Δw) −0.014 + var(Δw) 0.023
  - Estimated values shown beneath each term in the paper.
- Standard deviations implied by 2002 variances:
  - employment time: standard deviation of 40 percent (from variance 0.158)
  - weekly earnings: standard deviation of 15 percent (from variance 0.022)
- Across all years in the sample (Table 3): the variance of changes in employment time is greater than four-fifths of the variance of annual earnings growth in all years; the cross-term (twice the covariance) is small and typically negative.
- Third-moment decomposition: the third moment of employment time growth closely tracks that of annual earnings growth in sign and magnitude; weekly earnings third moments are generally one order of magnitude smaller (exceptions: 1998 and 1999).
- Time-series moments:
  - Mean change in employment time reaches a low of −0.07 at 2009.
  - Mean change in weekly earnings averages +0.02 during expansions and falls to −0.02 in 2012.
  - The third moment of annual earnings growth and of changes in employment time are pro-cyclical, dropping to about −0.1 in 2009.
- Five-year horizon:
  - Variance of five-year weekly earnings growth: 0.05 (text)
  - Variance of one-year weekly earnings growth: 0.025 (text)
  - Interpretation: weekly-earnings shocks have a considerable permanent component; most variation in employment time appears transitory (variance of five-year employment-time changes roughly the same as one-year).
- Job stayers vs switchers:
  - Raw differences: variance of switchers ≈ twice that of stayers; switchers’ third central moment more negative.
  - Once restricted to workers employed 52 weeks in both years, these differences largely disappear — composition (spells of non-employment) explains much of the gap.
- Tax and unemployment benefits (1997–2012 evidence):
  - Adding unemployment benefits and tax adjustments reduces the aggregate variance of earnings growth by 25%, from 0.2 to 0.15 (text).
  - Unemployment benefits thin the left tail of the earnings growth distribution (illustrated for 2009), reducing extreme negative earnings changes caused by drops in employment time.

### Key statistics and illustrative numbers (preserved exactly)
- var(Δy) 0.158 = var(Δx) 0.150 + 2cov(Δx,Δw) −0.014 + var(Δw) 0.023 (2002 decomposition, equation (8)).
- Employment-time variance 0.158 → standard deviation 40 percent; weekly-earnings variance 0.022 → standard deviation 15 percent (2002).
- Five-year weekly earnings variance 0.05; one-year weekly earnings variance 0.025.
- Mean change in employment time: −0.07 at 2009.
- Mean change in weekly earnings: +0.02 during expansions; −0.02 in 2012.
- Third moment reaches −0.1 in 2009.
- Reduction in variance due to unemployment benefits and taxes after 2007: 25%, from 0.2 to 0.15.
- Simulation parameters for the proposed earnings process:
  - Separation rate: 1 percent in expansion; 1.25 percent in recession.
  - Job finding (hiring) rate: 15 percent in expansion; 9 percent in recession.
  - Wage process parameters: σ = 0.07 (permanent shock standard deviation), b = 0.10 (transitory shock parameter → transitory stdev 10 percent).
  - Simulation: 100,000 worker histories over a 10 years period, in which the 8th and 9th years are recession years.
- US labor-market illustrative figures:
  - Male unemployment rate: 6.27 percent in 2005 and 12.13 percent in 2014 (text).
  - Transition flows documented (monthly): notable E→N > 1 percent exits per month; N→E ≈ 8 percent per month (textual magnitudes preserved).

### Variance and third-moment decomposition evidence (summary of patterns)
- Variance decomposition (Table 3, selected moments):
  - Example 2009: m2(Δy) 0.203; m2(Δx) 0.168; m2(Δw) 0.028; cross-term 0.007.
  - Across years, m2(Δx) explains the bulk of m2(Δy); m2(Δw) is relatively small and flatter over time.
- Third central moment:
  - Annual earnings growth and changes in employment time trace almost identical third-moment time series; weekly earnings third moment is near zero and flat.
- Cyclicality test (Table 4):
  - Third moment of annual earnings growth regressed on GDP growth: coefficient 0.659 (0.124) (column 1).
  - Adding third moment of changes in employment time: GDP coefficient −0.120 (0.086); third moment of employment time coefficient 1.020 (0.091) (column 2).
  - R-squared: 0.43 (column 1) and 0.86 (column 2). Observations: 27 27 (text lists 2727 but table bottom states Observations 2727 — preserved as 2727).

### Mechanism and proposed earnings process
- Conceptual decomposition: annual earnings = employment process (employment time) + wage process (weekly earnings).
- Employment process:
  - Discrete-time Markov chain with states E (employment) and U (unemployment) (and augmented N, not-in-labor-force, in US evidence).
  - Transition probability matrix P(m): P_E in expansions, P_R in recessions; transition probabilities common across workers at time m.
  - Employment time X_it: number of months in E over months {m, m+1, ..., m+11}.
- Wage process:
  - log wage w_it = η_it + e_it.
  - Permanent component η_it follows a unit root: η_it = η_it−1 + u_it, with u_it ∼ N(μ_t, σ^2).
  - Transitory shock e_it ∼ Exponential(b); changes Δe_it follow Laplace distribution (captures straight tails of weekly earnings growth).
  - Time variation allowed only in μ_t (mean wage growth) to reflect acyclical weekly-earnings distribution around its mean.
- Numerical example calibrated to match moments and labor-market facts reproduces:
  - Heavy tails in changes in employment time.
  - Straight tails in weekly-earnings changes.
  - Negative skewness (left tail rise) in recessions and recovery patterns.

### US labor-market flows: suggestive external evidence
- Augmented employment process with N (not-in-labor-force) necessary to match observed flows in CPS.
- Transition-rate stylized facts (IPUMS-CPS, males 25-60, Feb 1976–Dec 2015):
  - E→U (separation) rises in recessions; U→E (hiring) falls in recessions.
  - E→N trend increases over long run (participation declines among prime-age men).
  - N→E flows substantial and countercyclical.
- Model fed with observed monthly transition rates generates an implied third moment of changes in employment time that matches timing and magnitude of the third moment of annual earnings growth reported by GOS (figure comparison described in text).

### Policy-relevant implications and recommendations
- Modeling implication: explicitly model employment margin separately from wage margin in earnings processes; failing to do so risks overstating the magnitude and persistence of downside risk in recessions.
- Redistribution/insurance effects:
  - Unemployment benefits and taxes materially reduce variance and left-tail risk of annual earnings growth (example: 25% reduction in variance after 2007 when accounting for transfers and taxes).
- Empirical guidance:
  - Distinguish transitory employment-time shocks (largely driving cyclical tails) from more persistent wage shocks (weekly earnings).
  - When comparing one-year vs five-year distributions, account for persistence differences: employment-time variation appears mostly transitory; weekly-earnings shocks include a larger permanent component.

*Source: INPS data provided by MLPS, figures and tables as reproduced in the supplied content of the IMF working paper appendix.*

### References

### References

### Key citations
- Seminal and related works cited include (selection from the list): Abowd and Card (1989); Abowd and Zellner (1985); Arellano, Blundell, and Bonhomme (2017); Bagger et al. (2014); Berger et al. (2016); Blanchard and Diamond (1989); Blundell, Bozio, and Laroque (2011); Blundell, Graber, and Mogstad (2015); Busch et al. (2016); Chetty (2012); Constantinides and Duffie (1996); Constantinides and Ghosh (2014); Davis and von Wachter (2011); DeNardi, Fella, and Pardo (2016); Elsby, Hobijn, and Şahin (2015); Flood et al. (2015); French and Song (2014); Guvenen (2009); Guvenen et al. (2014, 2015); Hall (2014); Hall and Schulhofer-Wohl (2015); Hansen (1985); Heckman and Borjas (1980); Hobijn and Şahin (2009); Holzheu (2018); Hubmer (2018); Kiefer (1988); Kopczuk, Saez, and Song (2010); Krebs (2007); LaLonde (2007); Lise (2012); Low, Meghir, and Pistaferri (2010); Low and Pistaferri (2015); MaCurdy (1982); McKay (2017); McKay and Reis (2016); Meghir and Pistaferri (2004); Meyer (1990); Postel-Vinay and Robin (2002); Rogerson (1988); Rothstein (2011); Saporta-Eksten (2014); Schmidt (2016); Sericola (2000); Shimer (2012); Song and Manchester (2007); Storesletten, Telmer, and Yaron (2004a, 2004b, 2007); Topel and Ward (1992); von Wachter, Song, and Manchester (2011).

### Italy: Data Appendix — sample coverage and key summary statistics
- Sample restriction: males 25-60 years old who ever had a record at the Italian social security administration between 1984 and 2012.
- Table A.1 selected-year highlights:
  - Year 1992: Obs 339,296; Age mean 40.62; Annual Earnings mean 29,116.23; Employment Time (Weeks) mean 48.10; Earnings Rate (Weekly Earnings) mean 598.15; Δ Earnings (Δy) mean -0.01; Δ Employment Time (Δx) mean -0.02; Δ Earnings Rate (Δw) mean 0.01.
  - Year 2002: Obs 347,323; Age mean 39.99; Annual Earnings mean 27,841.43; Employment Time (Weeks) mean 48.33; Earnings Rate mean 568.56; Δ Earnings mean 0.02; Δ Employment Time mean 0.00; Δ Earnings Rate mean 0.02.
  - Year 2012: Obs 374,901; Age mean 42.41; Annual Earnings mean 27,218.92; Employment Time (Weeks) mean 46.98; Earnings Rate mean 563.79; Δ Earnings mean -0.05; Δ Employment Time mean -0.04; Δ Earnings Rate mean -0.02.
- Table A.2 (annual time series moments): Annual Earnings Growth, Employment Time Growth, and Weekly Earnings Growth reported for 1986–2012 with year-by-year Mean, Sd, p10, p50, p90 (table preserved in source).

### Italy: Top-coding and sensitivity analysis
- INPS top-coding:
  - In 2012, income is top coded at e645 per day.
  - Approximately 0.5 percent of the worker-year observations are affected by top-coding each year.
  - Main analysis: top-coded observations are dropped.
- Sensitivity checks (described and plotted in Figure A.1):
  - Columns reproduced: (1) baseline (drop top-coded); (2) include top-coded without correction; (3) assign to each top-coded observation in yeart and yeart−1 two times the maximum observed weekly earnings and recompute moments; (4) assign maximum observed weekly earnings to observations top-coded in yeart and twice the maximum to those top-coded in yeart−1; (5) assign twice the maximum in yeart and maximum in yeart−1.
  - Findings: series relative to annual earnings are virtually unaffected; weekly earnings series remain close to the baseline with a bump in variance between 1998 and 2000 and some third-moment fluctuations in the same years. The movements in the weekly earnings growth distribution are judged unlikely to play a relevant quantitative role in driving the evolution of annual earnings growth moments over time.

### Italy: Employment time persistence and variance decomposition
- Autocovariance in log employment time:
  - Positive and significant up to the tenth lag for levels; changes in employment time display negative autocovariance.
- Decomposition of variance of changes in employment time:
  - Within-worker effects account for 86 percent of the variance of changes in employment time.
  - Between-worker effects account for the remaining 14 percent.
- Decomposition of variance in levels of employment time:
  - Between-worker effect accounts for 40 percent of the overall variance.
- Table A.5 (Employment Time – Autocovariance Structure) selected values:
  - (a) Levels: Order 0 autocov. 0.1258; Order 1 autocov. 0.0330; Order 2 autocov. 0.0198; Order 10 autocov. 0.0111. Observations and Workers: Order 0 row shows Observations 25059,291 and Workers 974 (thousands formatting in source).
  - (b) Growth: Order 0 autocov. 0.1421; Order 1 autocov. -0.0228; Order 2 autocov. -0.0049; Order 10 autocov. -0.00000. Observations and Workers: Order 0 row shows Observations 35489,291 and Workers 974 (thousands formatting in source).
  - Notes: standard errors and t-statistics are computed clustering at the individual level. Number of observations and workers are in thousands.

### Italy: Marginal Tax Rates (IRPEF) — selected years and rates
- Table A.4 collects brackets and Marginal Tax Rates (MTR) for IRPEF. Income brackets expressed in million Italian Liras up to 2001, and in Euros thereafter. 1 Euros is 1936.27 Liras.
- Selected bracket and MTR examples from the table:
  - 1984-1985: bracket [0-11) MTR 18%; [11-24) MTR 27%; ... >500 MTR 65%.
  - 1986-1988: bracket [0-6) MTR 12%; ... >600 MTR 62%.
  - 1998-1999: bracket [0-15) MTR 18.5%; [15-30) MTR 26.5%; [30-60) MTR 33.5%; [60-135) MTR 39.5%; >135 MTR 45.5%.
  - 2002: bracket [0-10329.14) MTR 18%; [10329.14-15493.71) MTR 26%; [15493.71-30987.41) MTR 33%; [30987.41-69721.68) MTR 39%; >69721.68 MTR 45%.
  - 2007-2012: bracket [0-15000) MTR 23%; [15000-28000) MTR 27%; [28000-55000) MTR 38%; [55000-75000) MTR 41%; >75000 MTR 43%.

### Third central moment decomposition for Italy (annual earnings growth)
- Table A.3 reports annual time series decomposition of the third moment of earnings growth into components:
  - Columns: (1) m3(Δy) total; (2) m3(Δx) third moment of changes in employment time; (3) m3(Δw) third moment of changes in weekly earnings; (4) Cross-term 3m2,1(Δx,Δw)+3m1,2(Δx,Δw).
  - Selected year entries (value with standard deviation in parentheses as in source):
    - 1986: (1) -0.028 (0.002); (2) -0.040 (0.002); (3) 0.000 (0.000); (4) 0.012 (0.001).
    - 2000: (1) 0.028 (0.002); (2) 0.026 (0.002); (3) 0.001 (0.001); (4) 0.000 (0.001).
    - 2009: (1) -0.094 (0.002); (2) -0.092 (0.002); (3) -0.002 (0.000); (4) 0.000 (0.001).
    - 2012: (1) -0.076 (0.003); (2) -0.071 (0.002); (3) -0.002 (0.000); (4) -0.002 (0.001).
  - Notes: standard deviations of each component are reported in parentheses. Sample includes all 25-60 years old males who appear in the data for two consecutive years.

### United States: data sources used in analysis
- Master Earnings File (MEF) moments:
  - Guvenen, Ozkan, and Song (2014) provide time series of moments used in the analysis based on a 10 percent sample of males, 25-60 years old, covering the period 1978-2011 from the SSA Master Earnings File (MEF). These moments (first three) are taken from their appendix tables A.1 and A.8.
- CPS matched monthly files:
  - Data source: CPS matched monthly files collected by the Bureau of Labor Statistics and made available by IPUMS-CPS (Flood et al., 2015).
  - Sample coverage: period 1976-2015.
  - Sample restriction: males 25-60 years old for consistency across samples.
  - Method: recover monthly transition probabilities from person-month records (employment state in previous month and current period). Statistics weighted using the “final weight” provided by the BLS. Standard errors computed using 500 bootstrapped samples.
  - Labor market states distinguished: Employed (E), Unemployed (U), Not-in-labor-force (N).
  - Notable statistics:
    - Great Recession peak: unemployment share increased to 9.2% in January 2010.
    - Long-term trend: share not-in-labor-force (N) increased from around 7.5% at the beginning of the sample to 14.2% at the end of the sample.

*Source: wp18115 - References (IMF Working Paper PDF content).*

### 2015. This rise has been noticed most clearly during the great recession, and has typically been

### wp18115 - 2015. This rise has been noticed most clearly during the great recession, and has typically been

### Data, sample, and measurement issues
- Sample includes males 25-60 years old that appear in two consecutive months, and are matched using CPSIDP. Each row in Table A.6 refers to January of that year. Age and employment state shares are computed using the final weights provided by the CPS. Sources: IPUMS-CPS (Flood et al., 2015).
- Unmatched months excluded due to technical issues with household identifiers: all months of 1977, July to October 1985, and May to September 1995; these periods and the following and previous 12 months are excluded from analysis.
- Note: unemployment share does not correspond to the unemployment rate — the unemployment rate denominator is the civilian labor force (employed plus unemployed) while the share is denominated by the total working age population of males.
- Seasonality: series are not seasonally adjusted because within-year seasonality of employment may be quantitatively important as a source of variation.
- Weekly wages: January-to-January matched CPS used to construct distribution of changes in weekly earnings, but limitations include:
  - Matched observations ≈ two thousands (about 2,000), which is insufficient to reliably measure third moments compared to 300,000 in INPS and five million in GOS data.
  - Self-reported weekly wages may suffer from misreporting and measurement error, producing large standard errors; point estimates from this data are not relied upon in the analysis.
- Table A.6 (Summary Statistics – January Matched CPS) selected rows (preserve format and values as presented):
  - 1989: Observations 21,713; Age 39.8; Employment 0.878; State EUN 0.043; (last column) 0.079
  - 1990: Observations 23,103; Age 39.8; Employment 0.873; State EUN 0.045; 0.082
  - 1991: Observations 23,253; Age 39.9; Employment 0.858; State EUN 0.055; 0.086
  - 1992: Observations 23,303; Age 40.0; Employment 0.843; State EUN 0.070; 0.087
  - 1993: Observations 22,902; Age 40.2; Employment 0.846; State EUN 0.064; 0.090
  - 1994: Observations 22,757; Age 40.2; Employment 0.847; State EUN 0.058; 0.096
  - 1995: Observations 23,008; Age 40.4; Employment 0.858; State EUN 0.046; 0.095
  - 1996: Observations 19,667; Age 40.6; Employment 0.854; State EUN 0.046; 0.100
  - 1997: Observations 20,208; Age 40.7; Employment 0.858; State EUN 0.043; 0.098
  - 1998: Observations 20,508; Age 41.0; Employment 0.866; State EUN 0.037; 0.097
  - 1999: Observations 20,498; Age 41.2; Employment 0.873; State EUN 0.032; 0.096
  - 2000: Observations 20,870; Age 41.4; Employment 0.873; State EUN 0.030; 0.096
  - 2001: Observations 20,064; Age 41.6; Employment 0.866; State EUN 0.034; 0.100
  - 2002: Observations 24,008; Age 41.8; Employment 0.842; State EUN 0.052; 0.106
  - 2003: Observations 24,181; Age 41.9; Employment 0.833; State EUN 0.053; 0.114
  - 2004: Observations 23,576; Age 42.0; Employment 0.841; State EUN 0.049; 0.111
  - 2005: Observations 23,139; Age 42.1; Employment 0.838; State EUN 0.043; 0.119
  - 2006: Observations 22,781; Age 42.2; Employment 0.843; State EUN 0.039; 0.118
  - 2007: Observations 22,424; Age 42.3; Employment 0.851; State EUN 0.039; 0.110
  - 2008: Observations 22,698; Age 42.4; Employment 0.847; State EUN 0.041; 0.112
  - 2009: Observations 22,760; Age 42.4; Employment 0.808; State EUN 0.076; 0.116
  - 2010: Observations 22,960; Age 42.4; Employment 0.783; State EUN 0.093; 0.124
  - 2011: Observations 22,388; Age 42.4; Employment 0.789; State EUN 0.083; 0.128
  - 2012: Observations 21,996; Age 42.6; Employment 0.803; State EUN 0.069; 0.129
  - 2013: Observations 21,898; Age 42.7; Employment 0.807; State EUN 0.065; 0.128
- Additional information about IPUMS-CPS available at cps.ipums.org.

### Extended employment process: model setup and objectives
- Time is monthly; labor market state follows a first order Markov chain.
- Individual state at period t is s_t ∈ S = {1,2,...,S}. Transition probabilities:
  - P_i,j(t) = P(s_t = j | s_{t-1} = i); P(t) is the time t transition probability matrix (TPM).
  - Reverse transition probability matrix (rTPM) defined as:
    - ̃P_i,j(t) = P(s_{t-1}=i | s_t = j) = P_i,j(t) P(s_{t-1}=i) / P(s_t=j)
- Objective: transform monthly labor flows (CPS) into cross-sectional distributions of annual changes in employment time for every period in the sample.
- Define E ⊂ S as employment states, N = S \ E as non-employment states.
- Forward counting process (number of months employed in next n months):
  - →V_n(t) = sum_{τ=0}^{n-1} 1(s_{t+τ} ∈ E)
- Reverse counting process ←V_n(t) counts employment months between t-n and t-1.
- Joint probability matrices:
  - →Π_{i,j}(n,k,t) = P(→V_n(t)=k, s_{t+n}=j | s_t = i)
  - ←Π_{i,j}(n,k,t) = P(←V_n(t)=k, s_{t-n}=j | s_t = i)
- Forward probability mass function of annual employment:
  - →Π_i(k,t) = sum_{j=1}^S →Π_{i,j}(12,k,t) — probability for worker in state i at t to have k months of employment in next 12 months.
- Unconditional joint distribution of annual employment time:
  - P(←V_{12}(t)=k, →V_{12}(t)=l) = sum_{i,j} ←Π_i(k,t) P_{i,j}(t) →Π_j(l,t)
- Final object of interest: distribution of log difference log →V_{12}(t) − log ←V_{12}(t).

### Computing the forward probability mass function (algorithm and proposition)
- Partition TPM P(t) consistent with employment/non-employment ordering:
  - P(t) = [ P_EE(t)  P_EN(t) ; P_NE(t)  P_NN(t) ]
- Similarly partition →Π(n,k,t) into blocks →Π_EE, →Π_EN, →Π_NE, →Π_NN.
- Proposition 1 (recursive computation):
  1. For every n, k > 0:
     - [ →Π_EE(n,k,t) ; →Π_EN(n,k,t) ] = [ P_EE(t+1)  P_EN(t+1) ] · →Π(n−1,k−1,t+1)
     - [ →Π_NE(n,k,t) ; →Π_NN(n,k,t) ] = [ P_NE(t+1)  P_NN(t+1) ] · →Π(n−1,k,t+1)
  2. If k = 0 and n > 0:
     - →Π(n,0,t) = [ 0 0 ; 0 P_NN(t+1) ] · →Π(n−1,0,t+1)
  3. If n = 0 and k > 0:
     - →Π(0,k,t) = 0
  4. If n = 0 and k = 0:
     - →Π(0,0,t) = I_{S×S}
- Interpretation: recursion provides initial conditions and exact computation of cross-sectional distribution of changes in the extensive margin given sequence of TPMs {P(t+τ)}_{τ=1...12}.
- TPMs are estimated directly from flows data by dividing each flow by the stock at the destination state (forward) or origin state (backward).

### Limitations of the Markov assumption
- Two main restrictions imposed:
  - Ex-ante homogeneity: future distribution for two workers in same state follows same probability distribution — concern when pooling across occupations, education, age.
  - No history dependence: future realization depends only on current state and not on path (duration dependence is precluded).
- Evidence of declining exit rates with unemployment duration noted (references cited in source).
- Mitigation: expand state space or stratify sample (e.g., age-group specific TPMs) when data permit; longer panels could allow conditioning on paths.

### Time series evidence (US and comparison to Italy) and cyclicality accounting
- Figures and moments:
  - Figure A.3 (Moment Decomposition of Annual Earnings Growth and its Components, US 1976-2013) shows:
    - Panel (a): mean change in annual earnings and two components are procyclical and move together.
    - Panel (b): variance of implied changes in employment time and weekly earnings growth exhibit an upward trend, similar to Italy, contrasting with a downward trend in variance of earnings growth reported by GOS.
    - Panel (c): third moment (skewness proxy) — third moment of weekly earnings growth has large confidence intervals due to low sample size; point estimates are relatively flat and close to zero, consistent with Italian results.
  - Interpretation: employment time is the primary source of asymmetry in cross-section of earnings growth in the US, as in Italy.
- Quantitative exercise to assess cyclicality:
  - Define cyclicality of each moment as contemporaneous correlation with GDP growth (after removal of trend).
  - Let standardized, detrended series be: moments of earnings growth ˆm^n_t(Δy), moments of employment time ˆm^n_t(Δx), and GDP growth ˆg_t.
  - Unconditional regression:
    - ˆm^n_t(Δy_it) = β_n ˆg_t + ε_t
  - Controlled regression:
    - ˆm^n_t(Δy_it) = β_{n,x} ˆg_t + γ_{n,x} ˆm^n_t(Δx_it) + ε_{x,t}
  - If β_n ≠ 0 and β_{n,x} = 0 after controlling for employment time moments, interpret employment time as accounting for cyclicality of that moment.
- Results for Italy (Table A.7):
  - Mean of annual earnings growth:
    - GDP Growth coefficient 0.722 (standard error 0.110) — significant at 5%
    - When controlling for Mean of employment time growth, GDP Growth coefficient becomes 0.063 (standard error 0.120) — no longer significant.
    - Mean of employment time growth coefficient 0.833 (standard error 0.150).
    - R-squared rises from 0.52 to 0.78; Observations 27.
  - Variance of annual earnings growth:
    - GDP Growth coefficient −0.422 (standard error 0.176) — not significant at 5% in presented table (bold indicates significance at 5% only when shown).
    - When controlling for Variance of employment time growth, GDP Growth coefficient −0.257 (standard error 0.096) — halves in magnitude and stays non-significant.
    - Variance of employment time growth coefficient 0.626 (standard error 0.117).
    - R-squared rises from 0.18 to 0.54; Observations 27.
  - Skewness of annual earnings growth:
    - GDP Growth coefficient 0.629 (standard error 0.160) — significant at 5%
    - When controlling for Skewness of employment time growth, GDP Growth coefficient −0.091 (standard error 0.071) — becomes insignificant.
    - Skewness of employment time growth coefficient 0.976 (standard error 0.081).
    - R-squared rises from 0.40 to 0.83; Observations 27.
  - Notes: time series are detrended with a linear trend and standardized. Standard errors are Newey-West with two lags. Coefficients in bold significant at 5 percent.
- Conclusion from cyclicality exercise: changes in employment time can account for the cyclicality of the third moment (skewness) and substantially reduce or eliminate the cyclicality of mean and variance of annual earnings growth when included as controls.

### Earnings growth over the life cycle (Italy evidence) and robustness
- Age profiles (Figure A.4; pooled 1986-2012, males 26-60):
  - Mean annual earnings growth declines from 7.5 percent at age 26 to −20 percent at age 60.
  - Employment time growth declines over the life cycle: initially 4 percent as workers join labor force, declines until age 35 and near zero until about age 45, then declines more sharply reaching double digits mean decline at age 57 (retirement effects).
  - Weekly earnings growth increases initially by 3 percent, then declines and flattens, and increases again after age 50 — possibly composition effects from retirement.
- Robustness: removing time-age fixed effects (to remove age profile and correlated movements across cohorts and age groups), residuals are used to recreate decomposition figures. Figures A.5 and A.6 (age profiles removed) show no major changes in the main results other than the mean of distributions being set to zero.
  - Figure A.5: log densities of one-year growth of annual earnings and components after removing age fixed effects (representative sample ~300,000 observations, year 2002).
  - Figure A.6: moment decomposition of annual earnings, employment time, and weekly earnings growth with age profiles removed — patterns remain consistent with main findings.

*Source: wp18115 (PDF chapter/section) content provided.*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18115.pdf_