## wpiea2021289-print-pdf - Section 3 quantifies the contribution of idiosyncratic shocks to aggregate fluctuations

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### Data and sample construction
- Data sources:
  - VAT registry from the tax authority for total firm sales.
  - National Customs Service for firms’ export sales (free on board) by country destination.
  - Annual statements for number of employees.
  - Sales converted to Unidad de Fomento (CPI-indexed unit of account).
- Sample selection:
  - Period: 2007–2019 (analysis focuses on 2008–2019 sample for most computations).
  - Drop discontinued firm-destination sale series over 2008–2019.
  - Drop firm-destination sale series with sales more than doubling in a year or contracting by more than a half.
  - Final sample: 40,150 firms and 529,884 firm-destinations, accounting for 67.3 percent of total sales by Chilean firms.
- Data quality checks:
  - Correlation between firms’ aggregate sales and GDP: 83.2 percent.
  - Correlation between firms’ export sales and BOP export data: 93 percent.
- Summary statistics (sample period averages and measures):
  - Average aggregate growth rate (Chile whole economy): 0.027.
  - Average of individual growth rates (Chile whole economy): 0.012.
  - SD of individual growth rates (Chile whole economy): 0.486.
  - SD by size percentiles (Chile whole economy):
    - 0 - 20 size percentile: 0.581
    - 21 - 40 size percentile: 0.475
    - 41 - 60 size percentile: 0.467
    - 61 - 80 size percentile: 0.457
    - 81 - 100 size percentile: 0.433
  - Top firm volatilities (Chile whole economy):
    - Top 1000: 0.398
    - Top 100: 0.220
  - Average √Herf(f,n) (Chile whole economy): 0.057
  - Average √Herf(f) (Chile whole economy): 0.076
- Comparative context:
  - Statistics for France and the US are reported (from Di Giovanni, Levchenko and Mejean (2014) and Yeh (2017)) and highlight that Chilean firms exhibit higher sales volatility and higher market concentration.

### Identification of idiosyncratic and macro-sectoral shocks
- Decomposition of firm-destination sales growth γ_fnt:
  - γ_fnt = δ_jnt + ε_fnt
    - δ_jnt: macro-sectoral shock (sector j selling to destination n, includes macro and sectoral common shocks).
    - ε_fnt: firm idiosyncratic shock.
- Estimation:
  - δ_jnt captured by averaging sale growth rates of all firms in sector j selling to destination n.
  - ε_fnt is the residual: actual sales growth minus δ_jnt.
- Empirical properties (Table 2 highlights):
  - Total sales (Whole economy):
    - Actual γ_fnt: Obs 529,884; Mean 0.012; SD 0.486; Corr with actual 1.000.
    - Macro-sectoral δ_jnt: Obs 9,264; Mean 0.011; SD 0.643; Corr with actual 0.266.
    - Idiosyncratic ε_fnt: Obs 529,884; Mean 0.000; SD 0.468; Corr with actual 0.964.
  - Domestic sales (Whole economy):
    - Actual γ_fnt: Obs 481,044; Mean 0.012; SD 0.438; Corr 1.000.
    - Macro-sectoral δ_jnt: Obs 504; Mean 0.012; SD 0.085; Corr 0.137.
    - Idiosyncratic ε_fnt: Obs 481,044; Mean 0.000; SD 0.434; Corr 0.991.
  - Export sales (Whole economy):
    - Actual γ_fnt: Obs 48,840; Mean 0.017; SD 0.818; Corr 1.000.
    - Macro-sectoral δ_jnt: Obs 8,760; Mean 0.011; SD 0.661; Corr 0.467.
    - Idiosyncratic ε_fnt: Obs 48,840; Mean 0.000; SD 0.723; Corr 0.884.
- Key empirical patterns:
  - Idiosyncratic shocks are more volatile than macro-sectoral shocks, especially for domestic sales.
  - Idiosyncratic shocks are much more correlated with actual firm sales than macro-sectoral shocks.
  - Results replicate for manufacturing firms.

### Quantifying contributions to aggregate fluctuations
- Aggregate growth decomposition (with fixed weights at year τ to avoid time-varying weights bias):
  - γ_At|τ = Σ_{j,n} ω_jn,τ-1 δ_jnt + Σ_{f,n} ω_f n,τ-1 ε_fnt
- Variance decomposition for each τ:
  - σ^2_Aτ = σ^2_MSτ + σ^2_Iτ + COV_τ
    - σ_MSτ: variance due to macro-sectoral shocks.
    - σ_Iτ: variance due to idiosyncratic shocks.
    - COV_τ: covariance between macro-sectoral and idiosyncratic contributions.
- Average results over τ ∈ [2008,2019] (Table 3 averages):
  - Total sales (Whole economy, Chile):
    - Actual σ_A: 0.082
    - Macro-sectoral σ_MS: 0.058
    - Idiosyncratic σ_I: 0.036
    - σ_I / σ_A: 0.434
    - σ_I / σ_MS: 0.609
  - Manufacturing (Chile):
    - Actual σ_A: 0.087
    - Macro-sectoral σ_MS: 0.055
    - Idiosyncratic σ_I: 0.046
    - σ_I / σ_A: 0.522
    - σ_I / σ_MS: 0.826
  - Domestic sales (Chile):
    - Actual σ_A: 0.076
    - Macro-sectoral σ_MS: 0.053
    - Idiosyncratic σ_I: 0.037
    - σ_I / σ_A: 0.489
    - σ_I / σ_MS: 0.700
  - Export sales (Chile):
    - Actual σ_A: 0.135
    - Macro-sectoral σ_MS: 0.129
    - Idiosyncratic σ_I: 0.070
    - σ_I / σ_A: 0.514
    - σ_I / σ_MS: 0.538
- Comparative statements:
  - Idiosyncratic shocks explain about 43 percent of aggregate volatility in total sales for Chile (whole economy) and 52 percent in manufacturing.
  - These contributions are smaller than in France and the US, where idiosyncratic shocks account for 80 and 52 percent of aggregate volatility in whole-economy sales, respectively.

### Disentangling granularity and linkage components
- Decomposition of idiosyncratic variance (for year τ):
  - σ^2_Iτ = σ^2_Gτ + σ^2_Lτ
    - σ^2_Gτ (granular): Σ_{f,n} w^2_{f n,τ-1} Var(ε_fnt) — captures contribution absent propagation across firms; nonzero if firm size distribution is fat-tailed.
    - σ^2_Lτ (linkage): Σ_{g,f,m,n} w_{gm,τ-1} w_{f n,τ-1} Cov(ε_gmt, ε_fnt) — captures contribution from firm linkages / covariances across firms.
- Empirical contributions (Table 4 averages):
  - Total sales (Whole economy, Chile):
    - Actual σ_A: 0.082
    - Granular σ_G: 0.017
    - Link σ_L: 0.031
    - σ_G / σ_A: 0.203
    - σ_L / σ_A: 0.381
  - Manufacturing (Chile):
    - Actual σ_A: 0.087
    - Granular σ_G: 0.029
    - Link σ_L: 0.035
    - σ_G / σ_A: 0.336
    - σ_L / σ_A: 0.397
- Interpretation:
  - The granular component accounts for about one fifth of aggregate sales volatility in Chile (σ_G / σ_A ≈ 0.203 for whole economy).
  - The linkage component accounts for about 38 percent of aggregate fluctuations in Chile (σ_L / σ_A ≈ 0.381 for whole economy).
  - Compared to France and the US, Chile’s granular component is similar or slightly larger, but its linkage component is considerably lower (France link ≈ 0.721 of σ_A; US link ≈ 0.489 of σ_A).
- Counterfactual exercises on granularity:
  - If all firms had equal size (for a given τ), σ_G would drop to 0.0022 (about 7 times smaller than baseline σ_G = 0.0166), highlighting the importance of market concentration.
  - If all firms had equal variance (set to the sample average), σ_G would increase to 0.0276 (almost twice the baseline), highlighting the role of the negative relationship between firm size and volatility.
- Firm size–volatility relationship:
  - Regression estimated: ln σ_f t = α ln size_f t + λ_f + η_j + υ_t + ε_f t
    - σ_f t: average volatility of firm f during a four-year non-overlapping period t.
    - size_f t: average size of firm f measured using firm sales or employment.
    - λ_f, η_j, υ_t: firm, sector, and period fixed effects.
  - Result: highly statistically significant negative association between firm volatility and size (larger firms tend to be less volatile), which reduces the granular contribution.
- Network asymmetry and degree distributions:
  - First- and second-degree connections (suppliers and buyers) follow highly skewed (Pareto-like) distributions.
  - Estimated Pareto parameters for Chile (first-degree):
    - suppliers: -0.28
    - buyers: -0.40
  - Estimated Pareto parameters for Chile (second-degree):
    - suppliers: -0.10
    - buyers: -0.07
  - Comparative evidence: parameters in Chile are less skewed than in advanced economies (e.g., Japan first-degree buyers and sellers: -1.32 and -1.50), which helps explain lower linkage-driven aggregate volatility in Chile.

### Conclusions and implications
- Main quantitative findings:
  - Idiosyncratic shocks explain about 43 percent of aggregate volatility in total sales for Chile (whole economy) and about 52 percent in manufacturing.
  - Granularity accounts for about one fifth of aggregate sales volatility (σ_G / σ_A ≈ 0.203).
  - Firm linkages account for about 38 percent of aggregate sales volatility (σ_L / σ_A ≈ 0.381).
- Drivers of Chile’s lower contribution of idiosyncratic shocks compared with advanced economies:
  - Amplifying force: higher market concentration in Chile increases potential impact of shocks to large firms.
  - Dampening force: large firms in Chile are considerably less volatile than smaller firms, reducing the granular effect.
  - Dampening force: more symmetric firm network structure in Chile (firms have similar numbers of connections) reduces propagation via input-output linkages and lowers linkage-driven volatility.
- Suggested directions for future research:
  - Better understanding and monitoring of individual firm dynamics and firm-to-firm linkages given sizable aggregate effects of idiosyncratic shocks in emerging markets.
  - Systematic investigation of differences in firms’ network structures between emerging and advanced economies as firm-to-firm transaction data become available for more countries.

*Source: wpiea2021289-print-pdf - Section 3 quantifies the contribution of idiosyncratic shocks to aggregate fluctuations.*

### Section 3 quantifies the contribution of idiosyncratic shocks to aggregate fluctuations.

### wpiea2021289-print-pdf - Section 3 quantifies the contribution of idiosyncratic shocks to aggregate fluctuations

### Data and sample construction
- Data sources:
  - VAT registry from the tax authority for total firm sales.
  - National Customs Service for firms’ export sales (free on board) by country destination.
  - Annual statements for number of employees.
  - Sales converted to Unidad de Fomento (CPI-indexed unit of account).
- Sample selection:
  - Period: 2007–2019 (analysis focuses on 2008–2019 sample for most computations).
  - Drop discontinued firm-destination sale series over 2008–2019.
  - Drop firm-destination sale series with sales more than doubling in a year or contracting by more than a half.
  - Final sample: 40,150 firms and 529,884 firm-destinations, accounting for 67.3 percent of total sales by Chilean firms.
- Data quality checks:
  - Correlation between firms’ aggregate sales and GDP: 83.2 percent.
  - Correlation between firms’ export sales and BOP export data: 93 percent.
- Summary statistics (sample period averages and measures):
  - Average aggregate growth rate (Chile whole economy): 0.027.
  - Average of individual growth rates (Chile whole economy): 0.012.
  - SD of individual growth rates (Chile whole economy): 0.486.
  - SD by size percentiles (Chile whole economy): 
    - 0 - 20 size percentile: 0.581
    - 21 - 40 size percentile: 0.475
    - 41 - 60 size percentile: 0.467
    - 61 - 80 size percentile: 0.457
    - 81 - 100 size percentile: 0.433
  - Top firm volatilities (Chile whole economy):
    - Top 1000: 0.398
    - Top 100: 0.220
  - Average √Herf(f,n) (Chile whole economy): 0.057
  - Average √Herf(f) (Chile whole economy): 0.076
- Comparative context: statistics for France and the US are reported (from Di Giovanni, Levchenko and Mejean (2014) and Yeh (2017)) and highlight that Chilean firms exhibit higher sales volatility and higher market concentration.

### Identification of idiosyncratic and macro-sectoral shocks
- Decomposition of firm-destination sales growth γ_fnt:
  - γ_fnt = δ_jnt + ε_fnt
    - δ_jnt: macro-sectoral shock (sector j selling to destination n, includes macro and sectoral common shocks).
    - ε_fnt: firm idiosyncratic shock.
- Estimation:
  - δ_jnt captured by averaging sale growth rates of all firms in sector j selling to destination n.
  - ε_fnt is the residual: actual sales growth minus δ_jnt.
- Empirical properties (Table 2 highlights):
  - Total sales (Whole economy):
    - Actual γ_fnt: Obs 529,884; Mean 0.012; SD 0.486; Corr with actual 1.000.
    - Macro-sectoral δ_jnt: Obs 9,264; Mean 0.011; SD 0.643; Corr with actual 0.266.
    - Idiosyncratic ε_fnt: Obs 529,884; Mean 0.000; SD 0.468; Corr with actual 0.964.
  - Domestic sales (Whole economy):
    - Actual γ_fnt: Obs 481,044; Mean 0.012; SD 0.438; Corr 1.000.
    - Macro-sectoral δ_jnt: Obs 504; Mean 0.012; SD 0.085; Corr 0.137.
    - Idiosyncratic ε_fnt: Obs 481,044; Mean 0.000; SD 0.434; Corr 0.991.
  - Export sales (Whole economy):
    - Actual γ_fnt: Obs 48,840; Mean 0.017; SD 0.818; Corr 1.000.
    - Macro-sectoral δ_jnt: Obs 8,760; Mean 0.011; SD 0.661; Corr 0.467.
    - Idiosyncratic ε_fnt: Obs 48,840; Mean 0.000; SD 0.723; Corr 0.884.
- Key empirical patterns:
  - Idiosyncratic shocks are more volatile than macro-sectoral shocks, especially for domestic sales.
  - Idiosyncratic shocks are much more correlated with actual firm sales than macro-sectoral shocks.
  - Results replicate for manufacturing firms.

### Quantifying contributions to aggregate fluctuations
- Aggregate growth decomposition (with fixed weights at year τ to avoid time-varying weights bias):
  - γ_At|τ = Σ_{j,n} ω_jn,τ-1 δ_jnt + Σ_{f,n} ω_f n,τ-1 ε_fnt
- Variance decomposition for each τ:
  - σ^2_Aτ = σ^2_MSτ + σ^2_Iτ + COV_τ
    - σ_MSτ: variance due to macro-sectoral shocks.
    - σ_Iτ: variance due to idiosyncratic shocks.
    - COV_τ: covariance between macro-sectoral and idiosyncratic contributions.
- Average results over τ ∈ [2008,2019] (Table 3 averages):
  - Total sales (Whole economy, Chile):
    - Actual σ_A: 0.082
    - Macro-sectoral σ_MS: 0.058
    - Idiosyncratic σ_I: 0.036
    - σ_I / σ_A: 0.434
    - σ_I / σ_MS: 0.609
  - Manufacturing (Chile):
    - Actual σ_A: 0.087
    - Macro-sectoral σ_MS: 0.055
    - Idiosyncratic σ_I: 0.046
    - σ_I / σ_A: 0.522
    - σ_I / σ_MS: 0.826
  - Domestic sales (Chile):
    - Actual σ_A: 0.076
    - Macro-sectoral σ_MS: 0.053
    - Idiosyncratic σ_I: 0.037
    - σ_I / σ_A: 0.489
    - σ_I / σ_MS: 0.700
  - Export sales (Chile):
    - Actual σ_A: 0.135
    - Macro-sectoral σ_MS: 0.129
    - Idiosyncratic σ_I: 0.070
    - σ_I / σ_A: 0.514
    - σ_I / σ_MS: 0.538
- Comparative statements:
  - Idiosyncratic shocks explain about 43 percent of aggregate volatility in total sales for Chile (whole economy) and 52 percent in manufacturing.
  - These contributions are smaller than in France and the US, where idiosyncratic shocks account for 80 and 52 percent of aggregate volatility in whole-economy sales, respectively.

### Disentangling granularity and linkage components
- Decomposition of idiosyncratic variance (for year τ):
  - σ^2_Iτ = σ^2_Gτ + σ^2_Lτ
    - σ^2_Gτ (granular): Σ_{f,n} w^2_{f n,τ-1} Var(ε_fnt) — captures contribution absent propagation across firms; nonzero if firm size distribution is fat-tailed.
    - σ^2_Lτ (linkage): Σ_{g,f,m,n} w_{gm,τ-1} w_{f n,τ-1} Cov(ε_gmt, ε_fnt) — captures contribution from firm linkages / covariances across firms.
- Empirical contributions (Table 4 averages):
  - Total sales (Whole economy, Chile):
    - Actual σ_A: 0.082
    - Granular σ_G: 0.017
    - Link σ_L: 0.031
    - σ_G / σ_A: 0.203
    - σ_L / σ_A: 0.381
  - Manufacturing (Chile):
    - Actual σ_A: 0.087
    - Granular σ_G: 0.029
    - Link σ_L: 0.035
    - σ_G / σ_A: 0.336
    - σ_L / σ_A: 0.397
- Interpretation:
  - The granular component accounts for about one fifth of aggregate sales volatility in Chile (σ_G / σ_A ≈ 0.203 for whole economy).
  - The linkage component accounts for about 38 percent of aggregate fluctuations in Chile (σ_L / σ_A ≈ 0.381 for whole economy).
  - Compared to France and the US, Chile’s granular component is similar or slightly larger, but its linkage component is considerably lower (France link ≈ 0.721 of σ_A; US link ≈ 0.489 of σ_A).
- Counterfactual exercises on granularity:
  - If all firms had equal size (for a given τ), σ_G would drop to 0.0022 (about 7 times smaller than baseline σ_G = 0.0166), highlighting the importance of market concentration.
  - If all firms had equal variance (set to the sample average), σ_G would increase to 0.0276 (almost twice the baseline), highlighting the role of the negative relationship between firm size and volatility.
- Firm size–volatility relationship:
  - Regression estimated: ln σ_f t = α ln size_f t + λ_f + η_j + υ_t + ε_f t
    - σ_f t: average volatility of firm f during a four-year non-overlapping period t.
    - size_f t: average size of firm f measured using firm sales or employment.
    - λ_f, η_j, υ_t: firm, sector, and period fixed effects.
  - Result: highly statistically significant negative association between firm volatility and size (larger firms tend to be less volatile), which reduces the granular contribution.
- Network asymmetry and degree distributions:
  - First- and second-degree connections (suppliers and buyers) follow highly skewed (Pareto-like) distributions.
  - Estimated Pareto parameters for Chile (first-degree):
    - suppliers: -0.28
    - buyers: -0.40
  - Estimated Pareto parameters for Chile (second-degree):
    - suppliers: -0.10
    - buyers: -0.07
  - Comparative evidence: parameters in Chile are less skewed than in advanced economies (e.g., Japan first-degree buyers and sellers: -1.32 and -1.50), which helps explain lower linkage-driven aggregate volatility in Chile.

### Conclusions and implications
- Main quantitative findings:
  - Idiosyncratic shocks explain about 43 percent of aggregate volatility in total sales for Chile (whole economy) and about 52 percent in manufacturing.
  - Granularity accounts for about one fifth of aggregate sales volatility (σ_G / σ_A ≈ 0.203).
  - Firm linkages account for about 38 percent of aggregate sales volatility (σ_L / σ_A ≈ 0.381).
- Drivers of Chile’s lower contribution of idiosyncratic shocks compared with advanced economies:
  - Amplifying force: higher market concentration in Chile increases potential impact of shocks to large firms.
  - Dampening force: large firms in Chile are considerably less volatile than smaller firms, reducing the granular effect.
  - Dampening force: more symmetric firm network structure in Chile (firms have similar numbers of connections) reduces propagation via input-output linkages and lowers linkage-driven volatility.
- Suggested directions for future research:
  - Better understanding and monitoring of individual firm dynamics and firm-to-firm linkages given sizable aggregate effects of idiosyncratic shocks in emerging markets.
  - Systematic investigation of differences in firms’ network structures between emerging and advanced economies as firm-to-firm transaction data become available for more countries.

*Source: wpiea2021289-print-pdf - Section 3 quantifies the contribution of idiosyncratic shocks to aggregate fluctuations.*

### References

### References

### Bibliographic citations
- Acemoglu, Daron, Vasco M Carvalho, Asuman Ozdaglar, and Alireza Tahbaz-Salehi. 2012. “The network origins of aggregate fluctuations.” Econometrica, 80(5): 1977–2016.
- Alfaro-Urena, Alonso, Mariany Fuentes Fuentes, Isabela Manelici, and José P Vásquez. 2018. “Costa Rican Production Network: Stylized Facts.” Documento de Investigación N. 002.
- Atalay, Enghin. 2017. “How important are sectoral shocks?” American Economic Journal: Macroeconomics, 9(4): 254–80.
- Baqaee, David Rezza. 2018. “Cascading failures in production networks.” Econometrica, 86(5): 1819–1838.
- Baqaee, David Rezza, and Emmanuel Farhi. 2019. “The macroeconomic impact of microeconomic shocks: beyond Hulten’s Theorem.” Econometrica, 87(4): 1155–1203.
- Bernard, Andrew B, Andreas Moxnes, and Yukiko U Saito. 2019. “Production Networks, Geography, and Firm Performance.” Journal of Political Economy, 127(2): 639–688.
- Cardoza, Marvin, Francesco Grigoli, Nicola Pierri, Cian Ruane, Maria Soledad, and Martinez Peria. 2020. “Worker mobility and domestic production networks.” IMF Working Papers (205).
- Carvalho, Vasco, and Xavier Gabaix. 2013. “The great diversification and its undoing.” American Economic Review, 103(5): 1697–1727.
- Carvalho, Vasco M, and Basile Grassi. 2019. “Large firm dynamics and the business cycle.” American Economic Review, 109(4): 1375–1425.
- Di Giovanni, Julian, and Andrei A Levchenko. 2012. “Country size, international trade, and aggregate fluctuations in granular economies.” Journal of Political Economy, 120(6): 1083–1132.
- Di Giovanni, Julian, Andrei A Levchenko, and Isabelle Mejean. 2014. “Firms, destinations, and aggregate fluctuations.” Econometrica, 82(4): 1303–1340.
- Di Giovanni, Julian, Andrei A Levchenko, and Isabelle Mejean. 2018. “The micro origins of international business-cycle comovement.” American Economic Review, 108(1): 82–108.
- Eaton, Jonathan, Samuel Kortum, and Francis Kramarz. 2011. “An anatomy of international trade: Evidence from French firms.” Econometrica, 79(5): 1453–1498.
- Eaton, Jonathan, Samuel Kortum, Brent Neiman, and John Romalis. 2016. “Trade and the global recession.” American Economic Review, 106(11): 3401–38.
- Foerster, Andrew T, Pierre-Daniel G Sarte, and Mark W Watson. 2011. “Sectoral versus aggregate shocks: A structural factor analysis of industrial production.” Journal of Political Economy, 119(1): 1–38.
- Gabaix, Xavier. 2011. “The granular origins of aggregate fluctuations.” Econometrica, 79(3): 733–772.
- Koren, Miklós, and Silvana Tenreyro. 2013. “Technological diversification.” American Economic Review, 103(1): 378–414.
- Lucas, Robert E. 1977. “Understanding business cycles.” Vol. 5, 7–29, North-Holland.
- Melitz, Marc J. 2003. “The impact of trade on intra-industry reallocations and aggregate industry productivity.” Econometrica, 71(6): 1695–1725.
- Stanley, Michael HR, Luis AN Amaral, Sergey V Buldyrev, Shlomo Havlin, Heiko Leschhorn, Philipp Maass, Michael A Salinger, and H Eugene Stanley. 1996. “Scaling behaviour in the growth of companies.” Nature, 379(6568): 804–806.
- Yeh, Chen. 2017. “Are firm-level idiosyncratic shocks important for US aggregate volatility?” US Census Bureau Center for Economic Studies Paper No. CES-WP-17-23.

### Appendix

### A Correlation between micro and macro data
- Sample: 40,150 firms representing 67.3 percent of sales of the universe of Chilean firms.
- Correlation between growth rate of aggregate sales (domestic and export sales) of the sample and GDP growth: 83.2 percent.
- Correlation between growth rate of firm-level export data from customs and growth rate of exports from balance of payments data: 93 percent.
- Notes from figures:
  - Panel (a) shows the growth rate of the sum of firms’ sales and the growth rate of GDP; the correlation is 83.2 percent.
  - Panel (b) shows the growth rate of the sum of firms’ export sales and the growth rate of exports from the national accounts; the correlation is 93.0 percent.
- Footnote: The customs data provide the free on board value of each firm’s exports to each of the foreign destinations.

### B Firms’ size and volatility
- Empirical facts:
  - Probability plot of the log of firms’ sales shows a fat right tail: firms on the right tail are larger than implied by a normal distribution.
  - Anderson-Darling test rejects the hypothesis that the data follows a normal distribution.
  - Larger firms (those with a larger share of total sales) observe a less volatile growth rate of sales.
  - The negative relationship between firm size and volatility is consistent whether size is measured by sales or employment.
- Visualization note:
  - In Figure B.1(b) the maximum of the horizontal axis is set to 0.2 percent of total sales to ease visualization.
- Regression evidence (Table B.1: Estimation of the power law between firm size and variance):
  - Dependent variables: Ln SD of sales growth; Ln SD of employment growth.
  - Observations: 120,450 in all reported columns.
  - Coefficients and statistics:
    - Ln sales
      - Column (1): -0.039*** (0.007)
      - Column (2): -0.174*** (0.015)
      - Column (3): -0.041*** (0.005)
      - Column (4): -0.174*** (0.018)
    - Ln employment
      - Column (5): -0.032*** (0.003)
      - Column (6): -0.162*** (0.009)
      - Column (7): -0.035*** (0.003)
      - Column (8): -0.162*** (0.011)
    - R2
      - Columns (1)–(4): 0.047, 0.642, 0.147, 0.642 respectively.
      - Columns (5)–(8): 0.032, 0.529, 0.058, 0.529 respectively.
  - Fixed effects:
    - Columns with firm fixed effects and/or sector fixed effects are indicated in the table; preferred specification is column (4) which includes both firm and sector fixed effects and uses sales as measure of size.
  - Notes: The regressions are run on four-year non-overlapping periods. All regressions include time fixed effects. Columns (1) and (5) include a constant. Clustered standard errors at the industry level in parentheses. ***p <0.01, **p <0.05, *p <0.1.
- Figure B.2 presents firm-destination variances and weights; axes range shown include Weight up to 0.0010 and Individual variance up to 14 in plotted units.

### C Firm-specific volatility: a decomposition
- Decomposition described: idiosyncratic shocks (firm-specific volatility) into granular and link components over τ as defined in equation (5).
- Key findings:
  - For the whole economy and for the manufacturing sector, the contribution of the link component is generally larger than the contribution of the granular component.
  - The link component is significantly smaller than for France or the US.
  - The link component varies the most over time and effectively drives fluctuations in the idiosyncratic shocks.
- Figure notes and ranges:
  - Panel C.1(a) (Whole economy): plotted values of Idiosyncratic, Granular, Link roughly span 0.02 to 0.05 over 2008–2019.
  - Panel C.1(b) (Manufacturing): plotted values of Idiosyncratic, Granular, Link roughly span 0.02 to 0.07 over 2008–2019.
  - Notes: Figures present the contribution of firm-specific variances and cross-firm covariances to firm-specific fluctuations.

*Content derived from the PDF chapter "wpiea2021289-print-pdf - References".*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021289-print-pdf.pdf_