## wpiea2024121-print-pdf - Appendix C.I

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---

### MRPK and financial frictions (summary)
- MRPK is evaluated as a proxy for heterogeneous sensitivity of firm investment to monetary policy shocks, compared to traditional proxies of financial constraints (age, leverage, liquidity).
- Empirical specification (equation (4)) interacts a high-MRPK dummy (MRPK above industry average) with indicators of financial constraints:
  - firm age (below/above industry median, which range between 5 to 16 years old),
  - leverage (below/above industry median, which range between 0 to 0.5),
  - liquidity/cash holdings (below/above industry median, which range between 0.02 to 0.2),
  - and the interaction of highly-leveraged and cash-poor firms.
- Main empirical finding: independently of the measure of financial constraints considered, only high-MRPK firms show higher sensitivity to monetary policy shocks.

### Intensive margin — effects on firm investment and debt
- Cash holdings:
  - Re-estimating the specification behind Figure 4 using net debt on the left-hand side yields average and differential effects similar to those in Figure 4, pointing at cash holdings not changing significantly after an expansionary monetary policy shock.
- Heterogeneous investment responses:
  - High-MRPK firms expand capital while keeping a more stable leverage ratio.
  - Firms with average MRPK issue relatively more debt: approximately 1.25% and 2% for capital and debt two years after the shock, respectively.
  - Example: 2 years after a one standard deviation monetary policy easing shock, high-MRPK high-leverage firms increase capital by around 2.25%, while low-MRPK high-leverage firms increase capital by about 0.5%.
- Conclusion: Traditional measures of financial frictions do not outperform MRPK in explaining firms’ sensitivity to monetary policy; MRPK captures additional heterogeneity.

### Extensive margin — entry and exit (aggregate and composition)
- Construction:
  - Entry and exit rates at industry j in period t+h: y_{j,t+h} = n_{j,t+h} / N_{j,t+h−1} (equation (5)).
  - High- and low-MRPK entrants defined by MRPK_i,t+h > MRPK_j and its complement; when MRPK_i,t missing, impute indicator by firm-average T_i^{-1} ∑_t 1[MRPK_i,t > MRPK_j].
- Entry (Figure 8, left chart):
  - Average entry rate in the sector-year sample: 5.6%.
  - Approximately half of entrants are high-MRPK firms.
  - A standard monetary policy easing shock: entry rate increases by approximately 0.12 p.p. at impact.
  - The increase is less persistent than in some U.S. studies; it turns slightly negative (not statistically significant) after three years.
  - Composition: high- and low-MRPK firms contribute similarly to the positive on-impact entry effect; the contribution of high-MRPK firms is marginally more persistent but differences are not statistically significant.
  - Economic interpretation: the on-impact entry effect is relatively small when measured against the average entry rate or its standard deviation.
- Exit (Figure 8, right chart):
  - Average exit rate in the sample: 3.5%; standard deviation: 1.9%.
  - A standard monetary policy easing shock: decreases firm exit rate at impact by about 0.6 p.p.
  - The decrease persists one year after the shock but shrinks by a factor of three, and turns zero from horizon two onward.
  - Composition: at impact, the decrease in exit is homogeneous among high- and low-MRPK firms; one year after the shock, about 2/3 of the decrease in exit comes from high-MRPK firms not exiting.
  - Net assessment: effects on exit and entry are of similar magnitudes but economically small; evidence points to a mild improvement in allocation via the extensive margin one year after the shock (relatively fewer high-MRPK firms exiting and relatively more high-MRPK firms entering), but effects are small.
- Overall conclusion on extensive margin: Monetary policy easing increases entry and decreases exit in the short run (quantitatively small). There is no strong adverse change in composition that would support widespread “zombification”; main effects operate through the intensive margin.

### Robustness checks (Appendix C.I references)
- Missing MRPK indicator imputation:
  - Baseline: replace missing 1[MRPK_i,t > MRPK_j] with firm-average T_i^{-1} ∑_t 1[MRPK_i,t > MRPK_j].
  - Alternatives tested: round the firm-average value to 0 or 1; use 0.5 instead of the average value.
  - Results: robustness checks yield results very similar to baseline.
- Alternative dependent-variable constructions:
  - Instead of contributions to industry-level entry/exit rates, use entry and exit rates for high- and low-MRPK groups as dependent variables in equation (5); results are qualitatively very similar to baseline findings.
- Additional robustness material and further details referenced in Appendix C.K and Appendix C.J.

### Robustness and sample adjustments (Appendix C.L)
- Years 2018–2019 are excluded from parts of the analysis to mitigate the concern that the likelihood of mis-classifying non-reporting (but active) firms as exiters is increasing by the end of the main sample.
- The paper reports the results of alternative definitions in Appendix C.M and states that none of these alternative definitions affect significantly the main findings.

### Main findings on monetary policy and misallocation (Appendix C.L)
- Central banks’ accommodative monetary policy stance has been a fundamental feature of the post-Global Financial Crisis policy scenario.
- Expansionary monetary policy, while essential as a crisis response to smooth the downturn and boost the recovery, raised concerns about potential unintended adverse effects on aggregate productivity.
- Empirical findings using detailed micro-level data for Spain:
  - Expansionary monetary policy decreases misallocation at the within-sector level.
  - At the firm level, more productive firms with higher MRPK increase their investment relatively more after an expansionary monetary policy shock.
  - Capital expansion by productive firms is accompanied by an increase in debt issuance.
  - Firms’ MRPK outperforms the standard measures of financial constraints in explaining investment sensitivity to monetary policy shocks.
  - Expansionary monetary policy affects firms’ entry and exit decisions only marginally and does not change the composition of high- and low-MRPK entrants and exiters.
- Overall conclusion: evidence points to a decrease in misallocation following an expansionary monetary policy shock, mainly driven by a relatively larger increase in investment of high-MRPK firms thanks to the relaxation of financial frictions.

### Data preparation and sample construction
- Data sources:
  - Central de Balances Integrada (CBI) — administrative database of detailed financial reports of firms in Spain (Bank of Spain, CB Department).
  - Directorio Central de Empresas (DIRCE) — maintained by Instituto Nacional de Estadística (INE); records covering 1995–2020.
- Key processing steps (applied in listed order):
  1. Keep firms that have a status of a joint-stock company or a limited liability company, and index firms by their tax identifiers. Use all characters of the Spanish tax identifier (CIF) following the first letter of the id; treat observations of a joint-stock company switching to a limited liability company (or vice versa) as observations of the same firm.
  2. When a firm has multiple records of its primary industry, use the most frequent one.
  3. When firms have multiple records for the same year extracted from the current and future year’s filings:
     - If financial reports are missing for some firm-years, use the following year’s filings from which past year’s accounting data are extracted.
     - If both the current and the following period’s filings are available, prioritize the current year’s values unless they are flagged as a “low-quality record” by the CB Department of the Bank of Spain, in which case use the following year’s values.
  4. When firms report multiple years of entry, take the earliest year as the actual year of the start of the firm’s business.
  5. Drop records with negative capital, and deflate capital using industry-specific capital deflators. Deflate debt using industry-level value-added deflators.
  6. Treat negative values of cash and debt as zero values. Winsorize investment at 1% level, and trim cash holdings and leverage at the level 1 and 2, respectively.
- Exclude from the analysis all records flagged as a “low-quality record” by the CBI.

### Variable definitions (excerpt)
- Firm-level:
  - Investment: Change in the log of tangible capital — Source: CBI
  - MRPK: Log of value added over capital — Source: CBI
  - Employment: The average number of employees of a firm during the year — Source: CBI
  - Age: The number of years since the firm’s incorporation — Source: CBI
  - Leverage: The ratio of total interest-bearing debt to total assets — Source: CBI
  - Cash holdings: The ratio of cash holdings to total assets — Source: CBI
  - Entry year: The minimum of (i) first entry year recorded in DIRCE, (ii) incorporation year self-reported by firms to the CBI — Sources: CBI, DIRCE
  - Exit year: The last year of exit recorded in DIRCE, unless bankruptcy, in which case exit year is first year of bankruptcy as reported in CBI, CIR, or RPC — Sources: CBI, DIRCE, CIR, RPC
- Industry-level:
  - Industry sales growth: The log growth rate of industry total sales — Source: CBI
  - Entry rate: Number of firms entering the industry divided by lagged number of active firms in that industry
  - Exit rate: Number of firms exiting the industry divided by lagged number of active firms in that industry
- Aggregate:
  - Unemployment rate: The rate of unemployment, all ages, all regions — Source: INE
  - Inflation rate: HICP inflation rate — Source: INE
  - Monthly MP shocks: MP surprises from the updated series of Jarociński and Karadi (2020) — Source: Marek Jarociński’s Github page
  - Annual MP shocks: Monthly MP shocks aggregated to the annual frequency with the weighting scheme (1)

### Descriptive statistics (excerpt)
- Firm-year panel: number of observations is 9,047,649; number of firms is 1,323,799.
- Firm panel (Mean, SD, 25%, 50%, 75%):
  - Investment: −0.0035, 0.582, −0.20, −0.06, 0.04
  - MRPK: 0.640, 2.024, −0.60, 0.64, 1.91
  - Employment, # employees: 12.722, 29.91, 1.03, 3.0, 7.41
  - log(1 + Employment): 1.495, 1.108, 0.69, 1.392, 2.13
  - Age, # years: 12.33, 9.304, 4.0, 10.0, 17.0
  - log(1 + Age): 2.358, 0.710, 1.95, 2.40, 2.89
  - Leverage: 0.225, 0.288, 0.000, 0.100, 0.37
  - Cash holdings: 0.144, 0.189, 0.010, 0.070, 0.20
  - Industry sales growth: −0.032, 1.954, −0.73, −0.04, 0.76
  - Unemployment rate: 16.28, 5.656, 11.25, 15.26, 21.39
  - Inflation rate: 1.958, 1.459, 0.70, 2.45, 3.20
  - Annual MP shocks: −2.814, 9.425, −6.96, 0.68, 3.88
- Industry panel (Mean, SD, 25%, 50%, 75%):
  - log Var MRPK: 1.157, 0.357, 0.93, 1.18, 1.40
  - Entry rate: 0.056, 0.048, 0.029, 0.045, 0.072
  - Exit rate: 0.039, 0.019, 0.018, 0.033, 0.049
  - Industry sales growth: −0.036, 11.654, −0.43, −0.01, 0.34
  - Unemployment rate: 15.89, 5.526, 11.25, 15.26, 19.86
  - Inflation rate: 2.110, 1.409, 1.41, 2.45, 3.20
  - Annual MP shocks: −2.643, 9.580, −6.96, 0.68, 3.88
- Monthly MP shock†: −0.329, 3.293, −1.54, 0.00, 0.78
- † Summary statistics of the time series of monetary policy surprises from the updated file of Jarociński and Karadi (2020).

### Aggregation of monetary policy (MP) shocks — method and intuition
- High-frequency (HF) process specification (monthly HF index τ):
  - y_τ = ∑_{k=0}^K ̃β^k x_{τ−k} + ̃u_τ
  - In context: y_τ = monthly log growth of capital; x_τ = monetary policy shock in month τ.
- LF period spans L HF-periods; K ≡ L P, where P is number of LF lags (in main analysis L = 12 months, P = 4 years).
- Restriction across HF marginal effects:
  - ̃β_{L p + l} = ˇβ_p for all p = 0,...,P and l = 0,...,L−1 (assumes marginal effects constant within LF-period).
- LF aggregate defined:
  - Y_t = ∑_{p=0}^P ˇβ_p ˇX_{t−p} + u_t
  - ˇX_{t−p} = ∑_{j=0}^{L−1} (L − j) x_{L(t−p) ± j}
- Normalization used in main analysis:
  - X_{t−p} ≡ L^{−1} ˇX_{t−p}
  - LF specification: Y_t = ∑_{p=0}^P β_p X_{t−p} + u_t
  - Interpretation: β_p are dynamic effects on LF aggregate of a unit change in HF shock happening at the very beginning of the LF period; this motivates using one SD of HF shocks as the normal “quantity” when quantifying effects.

### Theoretical framework: Hsieh and Klenow (2009)
- Baseline model: firms under monopolistic competition; industry output aggregated by CES with elasticity σ; firm production with constant returns and productivity z_{is}; capital share at industry level α_s.
- Frictions enter as wedges τ^k_{ist} and τ^l_{ist} affecting returns:
  - MRPK_{ist} = α_s ((σ−1)/σ) (P_{sit} Y_{sit} / K_{sit}) = r_t / (1 + τ^k_{ist})
  - MRPL_{ist} = (1−α_s) ((σ−1)/σ) (P_{sit} Y_{sit} / L_{sit}) = w_t / (1 + τ^l_{ist})
- Dispersion of MRPK across firms is used as proxy for capital misallocation; dispersion of MRPK and MRPL maps into dispersion of revenue TFPR:
  - TFPR_{ist} = P_{ist} z_{ist} ∝ (MRPK_{ist})^{α_s} (MRPL_{ist})^{1−α_s}
- Aggregate log(TFP_st) relation:
  - log(TFP_st) = 1/(1−σ) log(∑_{i=1}^{M_s} z_{ist}^{σ−1}) − σ/2 var(log(TFPR_{ist}))
- Implication: an increase in var(log(MRPK_{ist})) increases var(log(TFPR_{ist})), reducing aggregate TFP.
- Caveats from literature:
  - MRPK-dispersion may reflect technology or adjustment costs (Asker, Collard-Wexler, Loecker, 2014).
  - MRPK-based misallocation metric sensitive to model mis-specification and could reflect demand shifts (Haltiwanger, Kulick, Syverson, 2018).
- Authors’ stance:
  - Physical/technological frictions unlikely to be influenced by MP shocks in short run.
  - Correlation between MRPK dispersion and MP shocks provides insights into capital misallocation.
  - Empirical checks: reduced-form estimates for dispersion of TFPR show reduction at least one year post-shock; interaction estimates using TFPR yield qualitatively similar results to MRPK findings.

### Robustness checks — overview and additional checks
- Alternative aggregation of MP shocks (current-year shocks with linear decreasing weights, excluding previous-year shocks): qualitatively similar, slightly quantitatively larger effects on MRPK dispersion and firm investment.
- Intensive margin: alternative demeaning of MRPK (no demeaning and demeaning by firm) produce similar qualitative investment effects.
- Additional interactions (firm MRPK with sector sales growth, unemployment, inflation; MP shocks with lagged firm characteristics): results unchanged.
- Fixed effects, lag specifications, Global Financial Crisis control (indicator for years 2008–2009), instrumenting interest rate changes (1 month OIS instrumented with MP shocks; a 1 SD decrease in 1 month OIS is 4.26bp), quarterly panel checks: results robust.
- Financing margin checks:
  - Change in net debt (log(debt) change minus log(cash) change) used as dependent variable: pattern nearly identical to debt results — cash holdings do not change significantly after expansionary MP shock; increase in debt is main financing source for investment by high-MRPK firms.
- Extensive margin of debt:
  - Probability of obtaining credit (conditional on previously having none) and maintaining credit (conditional on previously having debt) respond to MP easing; dynamic patterns plotted and reported.
- Alternative imputation for missing MRPK observations: main fractional assignment, rounded assignment, equal weight (0.5) — only minor impact on entry/exit composition effects.
- Entry and exit rate specifications: effects estimated on log entry and exit rates for all firms, high MRPK firms, and low MRPK firms — dynamic effects plotted and consistent.
- Shorter sample excluding years 2018–2019 (sample covers 2000–2016): consistent composition results.

### Key empirical quantities and sample sizes
- Firm-year panel: 9,047,649 observations; 1,323,799 firms.
- Industry-year panel: 1,512 observations; 72 industries.
- Annual MP shocks descriptive: Mean −2.814, SD 9.425, 25% −6.96, 50% 0.683, 75% 3.88.
- Monthly MP shock descriptive: Mean −0.329, SD 3.293, 25% −1.54, 50% 0.00, 75% 0.78.
- 1 month OIS 1 SD used in IV exercise: 4.26bp.

*Source: wpiea2024121-print-pdf (Appendix C.I and Appendix C.L) — A Tale of Two Margins: Monetary Policy and Capital Misallocation — Working Paper No. WP/2024/121.*

### Appendix C.I we run the same specification as the one behind Figure 4 with the only difference that we use

### wpiea2024121-print-pdf - Appendix C.I we run the same specification as the one behind Figure 4 with the only difference that we use

### MRPK and financial frictions (summary)
- MRPK is evaluated as a proxy for heterogeneous sensitivity of firm investment to monetary policy shocks, compared to traditional proxies of financial constraints (age, leverage, liquidity).
- Empirical specification (equation (4)) interacts a high-MRPK dummy (MRPK above industry average) with indicators of financial constraints: firm age (below/above industry median, which range between 5 to 16 years old), leverage (below/above industry median, which range between 0 to 0.5), liquidity/cash holdings (below/above industry median, which range between 0.02 to 0.2), and the interaction of highly-leveraged and cash-poor firms.
- Main empirical finding: independently of the measure of financial constraints considered, only high-MRPK firms show higher sensitivity to monetary policy shocks.

### Intensive margin — effects on firm investment and debt
- Cash holdings: Re-estimating the specification behind Figure 4 using net debt on the left-hand side yields average and differential effects similar to those in Figure 4, pointing at cash holdings not changing significantly after an expansionary monetary policy shock.
- Heterogeneous investment responses:
  - High-MRPK firms expand capital while keeping a more stable leverage ratio.
  - Firms with average MRPK issue relatively more debt: approximately 1.25% and 2% for capital and debt two years after the shock, respectively.
  - Example (from Figure 6 / Figure 7 discussion): 2 years after a one standard deviation monetary policy easing shock, high-MRPK high-leverage firms increase capital by around 2.25%, while low-MRPK high-leverage firms increase capital by about 0.5%.
- Conclusion: Traditional measures of financial frictions do not outperform MRPK in explaining firms’ sensitivity to monetary policy; MRPK captures additional heterogeneity.

### Extensive margin — entry and exit (aggregate and composition)
- Construction:
  - Entry and exit rates at industry j in period t+h: y_{j,t+h} = n_{j,t+h} / N_{j,t+h−1} (equation (5)).
  - High- and low-MRPK entrants defined by MRPK_i,t+h > MRPK_j and its complement; when MRPK_i,t missing, impute indicator by firm-average T_i^{-1} ∑_t 1[MRPK_i,t > MRPK_j].
- Entry (Figure 8, left chart):
  - Average entry rate in the sector-year sample: 5.6%.
  - Approximately half of entrants are high-MRPK firms.
  - A standard monetary policy easing shock: entry rate increases by approximately 0.12 p.p. at impact.
  - The increase is less persistent than in some U.S. studies; it turns slightly negative (not statistically significant) after three years.
  - Composition: high- and low-MRPK firms contribute similarly to the positive on-impact entry effect; the contribution of high-MRPK firms is marginally more persistent but differences are not statistically significant.
  - Economic interpretation: the on-impact entry effect is relatively small when measured against the average entry rate or its standard deviation.
- Exit (Figure 8, right chart):
  - Average exit rate in the sample: 3.5%; standard deviation: 1.9%.
  - A standard monetary policy easing shock: decreases firm exit rate at impact by about 0.6 p.p.
  - The decrease persists one year after the shock but shrinks by a factor of three, and turns zero from horizon two onward.
  - Composition: at impact, the decrease in exit is homogeneous among high- and low-MRPK firms; one year after the shock, about 2/3 of the decrease in exit comes from high-MRPK firms not exiting.
  - Net assessment: effects on exit and entry are of similar magnitudes but economically small; evidence points to a mild improvement in allocation via the extensive margin one year after the shock (relatively fewer high-MRPK firms exiting and relatively more high-MRPK firms entering), but effects are small.
- Overall conclusion on extensive margin: Monetary policy easing increases entry and decreases exit in the short run (quantitatively small). There is no strong adverse change in composition that would support widespread “zombification”; main effects operate through the intensive margin.

### Robustness checks
- Missing MRPK indicator imputation:
  - Baseline: replace missing 1[MRPK_i,t > MRPK_j] with firm-average T_i^{-1} ∑_t 1[MRPK_i,t > MRPK_j].
  - Alternatives tested: round the firm-average value to 0 or 1; use 0.5 instead of the average value.
  - Results: robustness checks yield results very similar to baseline.
- Alternative dependent-variable constructions:
  - Instead of contributions to industry-level entry/exit rates, use entry and exit rates for high- and low-MRPK groups as dependent variables in equation (5); results are qualitatively very similar to baseline findings.
- Additional robustness material and further details referenced in Appendix C.K and Appendix C.J.

*Source: wpiea2024121-print-pdf (Appendix C.I) — https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024121-print-pdf.pdf*

### Appendix C.L.

### Appendix C.L.

### Robustness and sample adjustments
- Years 2018–2019 are excluded from parts of the analysis to mitigate the concern that the likelihood of mis-classifying non-reporting (but active) firms as exiters is increasing by the end of the main sample.
- The paper reports the results of alternative definitions in Appendix C.M and states that none of these alternative definitions affect significantly the main findings.

### Main findings on monetary policy and misallocation
- Central banks’ accommodative monetary policy stance has been a fundamental feature of the post-Global Financial Crisis policy scenario.
- Expansionary monetary policy, while essential as a crisis response to smooth the downturn and boost the recovery, raised concerns about potential unintended adverse effects on aggregate productivity.
- Using detailed micro-level data for Spain, the paper estimates the impact of monetary policy on capital misallocation and the transmission mechanisms:
  - Expansionary monetary policy decreases misallocation at the within-sector level.
  - At the firm level, more productive firms with higher MRPK increase their investment relatively more after an expansionary monetary policy shock.
  - Capital expansion by productive firms is accompanied by an increase in debt issuance.
  - Firms’ MRPK outperforms the standard measures of financial constraints in explaining investment sensitivity to monetary policy shocks.
  - Expansionary monetary policy affects firms’ entry and exit decisions only marginally and does not change the composition of high- and low-MRPK entrants and exiters.
- Overall conclusion: evidence points to a decrease in misallocation following an expansionary monetary policy shock, mainly driven by a relatively larger increase in investment of high-MRPK firms thanks to the relaxation of financial frictions.

### Data preparation (CBI data) — key processing steps
- Data source: Central de Balances Integrada (CBI), an administrative database of detailed financial reports of firms in Spain maintained by the CB Department of the Bank of Spain.
- Preparation steps (applied in the order listed):
  1. Keep firms that have a status of a joint-stock company or a limited liability company, and index firms by their tax identifiers. Use all characters of the Spanish tax identifier (CIF) following the first letter of the id; treat observations of a joint-stock company switching to a limited liability company (or vice versa) as observations of the same firm.
  2. When a firm has multiple records of its primary industry, use the most frequent one.
  3. When firms have multiple records for the same year extracted from the current and future year’s filings:
     - If financial reports are missing for some firm-years, use the following year’s filings from which past year’s accounting data are extracted.
     - If both the current and the following period’s filings are available, prioritize the current year’s values unless they are flagged as a “low-quality record” by the CB Department of the Bank of Spain, in which case use the following year’s values.
  4. When firms report multiple years of entry, take the earliest year as the actual year of the start of the firm’s business.
  5. Drop records with negative capital, and deflate capital using industry-specific capital deflators. Deflate debt using industry-level value-added deflators.
  6. Treat negative values of cash and debt as zero values. Winsorize investment at 1% level, and trim cash holdings and leverage at the level 1 and 2, respectively.

*Source: wpiea2024121-print-pdf - Appendix C.L.*

### 7.  We exclude from the analysis all records flagged as a “low-quality record” by the CB

### 7.  We exclude from the analysis all records flagged as a “low-quality record” by the CB

### Data sources and sample construction
- Primary firm population source: Directorio Central de Empresas (DIRCE), maintained by Instituto Nacional de Estadística (INE). We extract records covering 1995–2020.
- DIRCE microdata contain firm fiscal identifier and years of firm “activation” (altas) and “deactivation” (bajas).
- DIRCE is compiled using data from the tax office, the central mercantile registry, and the social security office.
- For each tax identifier, when multiple activations/deactivations are observed, we keep only the first year of activation and the last year of deactivation and discard other status changes.
- We exclude from analysis all records flagged as a “low-quality record” by the Central de Balances Integrada (CBI) — Department of the Bank of Spain.

### Variable definitions (excerpt from Table 1)
- Firm-level variables:
  - Investment: Change in the log of tangible capital — Source: CBI
  - MRPK: Log of value added over capital — Source: CBI
  - Employment: The average number of employees of a firm during the year — Source: CBI
  - Age: The number of years since the firm’s incorporation — Source: CBI
  - Leverage: The ratio of total interest-bearing debt to total assets — Source: CBI
  - Cash holdings: The ratio of cash holdings to total assets — Source: CBI
  - Entry year: The minimum of (i) first entry year recorded in DIRCE, (ii) incorporation year self-reported by firms to the CBI — Sources: CBI, DIRCE
  - Exit year: The last year of exit recorded in DIRCE, unless bankruptcy, in which case exit year is first year of bankruptcy as reported in CBI, CIR, or RPC — Sources: CBI, DIRCE, CIR, RPC
- Industry-level variables:
  - Industry sales growth: The log growth rate of industry total sales — Source: CBI
  - Entry rate: Number of firms entering the industry divided by lagged number of active firms in that industry
  - Exit rate: Number of firms exiting the industry divided by lagged number of active firms in that industry
- Aggregate variables:
  - Unemployment rate: The rate of unemployment, all ages, all regions — Source: INE
  - Inflation rate: HICP inflation rate — Source: INE
  - Monthly MP shocks: MP surprises from the updated series of Jarociński and Karadi (2020) — Source: Marek Jarociński’s Github page
  - Annual MP shocks: Monthly MP shocks aggregated to the annual frequency with the weighting scheme (1)

### Descriptive statistics (excerpt from Table 2)
- Upper panel: firm-year panel; number of observations is 9,047,649; number of firms is 1,323,799.
- Firm panel (Mean, SD, 25%, 50%, 75%):
  - Investment: −0.0035, 0.582, −0.20, −0.06, 0.04
  - MRPK: 0.640, 2.024, −0.60, 0.64, 1.91
  - Employment, # employees: 12.722, 29.91, 1.03, 3.0, 7.41
  - log(1 + Employment): 1.495, 1.108, 0.69, 1.392, 2.13
  - Age, # years: 12.33, 9.304, 4.0, 10.0, 17.0
  - log(1 + Age): 2.358, 0.710, 1.95, 2.40, 2.89
  - Leverage: 0.225, 0.288, 0.000, 0.100, 0.37
  - Cash holdings: 0.144, 0.189, 0.010, 0.070, 0.20
  - Industry sales growth: −0.032, 1.954, −0.73, −0.04, 0.76
  - Unemployment rate: 16.28, 5.656, 11.25, 15.26, 21.39
  - Inflation rate: 1.958, 1.459, 0.70, 2.45, 3.20
  - Annual MP shocks: −2.814, 9.425, −6.96, 0.68, 3.88
- Industry panel (Mean, SD, 25%, 50%, 75%):
  - log Var MRPK: 1.157, 0.357, 0.93, 1.18, 1.40
  - Entry rate: 0.056, 0.048, 0.029, 0.045, 0.072
  - Exit rate: 0.039, 0.019, 0.018, 0.033, 0.049
  - Industry sales growth: −0.036, 11.654, −0.43, −0.01, 0.34
  - Unemployment rate: 15.89, 5.526, 11.25, 15.26, 19.86
  - Inflation rate: 2.110, 1.409, 1.41, 2.45, 3.20
  - Annual MP shocks: −2.643, 9.580, −6.96, 0.68, 3.88
- Monthly MP shock†: −0.329, 3.293, −1.54, 0.00, 0.78
- † Summary statistics of the time series of monetary policy surprises from the updated file of Jarociński and Karadi (2020).

### Aggregation of monetary policy (MP) shocks — method and intuition
- High-frequency (HF) process specification (monthly HF index τ):
  - y_τ = ∑_{k=0}^K ̃β^k x_{τ−k} + ̃u_τ
  - In context: y_τ = monthly log growth of capital; x_τ = monetary policy shock in month τ.
- LF period spans L HF-periods; K ≡ L P, where P is number of LF lags (in main analysis L = 12 months, P = 4 years).
- Restriction across HF marginal effects:
  - ̃β_{L p + l} = ˇβ_p for all p = 0,...,P and l = 0,...,L−1 (assumes marginal effects constant within LF-period).
- LF aggregate defined:
  - Y_t = ∑_{p=0}^P ˇβ_p ˇX_{t−p} + u_t
  - ˇX_{t−p} = ∑_{j=0}^{L−1} (L − j) x_{L(t−p) ± j}
- Normalization used in main analysis:
  - X_{t−p} ≡ L^{−1} ˇX_{t−p}
  - LF specification: Y_t = ∑_{p=0}^P β_p X_{t−p} + u_t
  - Interpretation: β_p are dynamic effects on LF aggregate of a unit change in HF shock happening at the very beginning of the LF period; this motivates using one SD of HF shocks as the normal “quantity” when quantifying effects.
- Illustrative example for L = 3, P = 0 provided to show how HF shocks mount around early HF periods within LF aggregate.

### Theoretical framework: Hsieh and Klenow (2009)
- Baseline model: firms under monopolistic competition; industry output aggregated by CES with elasticity σ; firm production with constant returns and productivity z_{is}; capital share at industry level α_s.
- Frictions enter as wedges τ^k_{ist} and τ^l_{ist} affecting returns:
  - MRPK_{ist} = α_s ((σ−1)/σ) (P_{sit} Y_{sit} / K_{sit}) = r_t / (1 + τ^k_{ist})
  - MRPL_{ist} = (1−α_s) ((σ−1)/σ) (P_{sit} Y_{sit} / L_{sit}) = w_t / (1 + τ^l_{ist})
- Dispersion of MRPK across firms is used as proxy for capital misallocation; dispersion of MRPK and MRPL maps into dispersion of revenue TFPR:
  - TFPR_{ist} = P_{ist} z_{ist} ∝ (MRPK_{ist})^{α_s} (MRPL_{ist})^{1−α_s}
- Aggregate log(TFP_st) relation:
  - log(TFP_st) = 1/(1−σ) log(∑_{i=1}^{M_s} z_{ist}^{σ−1}) − σ/2 var(log(TFPR_{ist}))
- Implication: an increase in var(log(MRPK_{ist})) increases var(log(TFPR_{ist})), reducing aggregate TFP.
- Caveats from literature:
  - MRPK-dispersion may reflect technology or adjustment costs (Asker, Collard-Wexler, Loecker, 2014).
  - MRPK-based misallocation metric sensitive to model mis-specification and could reflect demand shifts (Haltiwanger, Kulick, Syverson, 2018).
- Authors’ stance:
  - Physical/technological frictions unlikely to be influenced by MP shocks in short run.
  - Correlation between MRPK dispersion and MP shocks provides insights into capital misallocation.
  - Empirical checks: reduced-form estimates for dispersion of TFPR show reduction at least one year post-shock; interaction estimates using TFPR yield qualitatively similar results to MRPK findings.

### Robustness checks — overview of approaches and qualitative outcomes
- Alternative aggregation of MP shocks:
  - Aggregate current-year shocks with linear decreasing weights, excluding previous-year shocks.
  - Result: qualitatively similar, slightly quantitatively larger effects on MRPK dispersion and firm investment.
- Intensive margin: alternative demeaning of MRPK
  - No demeaning (MRPK in levels scaled to unit SD) and demeaning by firm produce similar qualitative investment effects.
- Additional interactions:
  - Interact firm MRPK with aggregate controls (sector sales growth, unemployment, inflation) — results unchanged.
  - Interact MP shocks with lagged firm characteristics (employment, leverage, age, liquidity) — results unchanged.
- Fixed effects and lag specifications:
  - Industry-year fixed effects included — results robust.
  - Lagged macro- and sector-level controls; lagged dependent variable included — results robust.
- Global Financial Crisis control:
  - Indicator for years 2008–2009 included — results robust.
- Instrumenting interest rate changes:
  - Changes in 1 month OIS instrumented with MP shocks (annual shocks aggregated similarly) — results very similar; a 1 SD decrease in 1 month OIS is 4.26bp.
- Quarterly panel checks:
  - Subsample with quarterly data and alternative quarterly aggregation — comparable effects to annual analysis.
- Financing margin checks:
  - Change in net debt (log(debt) change minus log(cash) change) used as dependent variable in investment specification.
  - Observation: pattern nearly identical to debt results — cash holdings do not change significantly after expansionary MP shock; increase in debt is main financing source for investment by high-MRPK firms.
- Extensive margin of debt:
  - Probability of obtaining credit (conditional on previously having none) and maintaining credit (conditional on previously having debt) respond to MP easing; dynamic patterns plotted and reported.
- Alternative imputation for missing MRPK observations:
  - Main imputation: missing MRPK status set to average of observed periods (fractional assignment).
  - Alternatives: rounded average assignment; equal weight (0.5) for missing.
  - Result: only minor impact on entry/exit composition effects.
- Entry and exit rate specifications:
  - Effects estimated on log entry and exit rates for all firms, high MRPK firms, and low MRPK firms — dynamic effects plotted and discussed.
  - Shorter sample excluding years 2018–2019 (sample covers 2000–2016) yields consistent composition results.

### Key empirical quantities and sample sizes
- Firm-year panel: 9,047,649 observations; 1,323,799 firms.
- Industry-year panel: 1,512 observations; 72 industries.
- Annual MP shocks descriptive: Mean −2.814, SD 9.425, 25% −6.96, 50% 0.683, 75% 3.88.
- Monthly MP shock descriptive: Mean −0.329, SD 3.293, 25% −1.54, 50% 0.00, 75% 0.78.
- 1 month OIS 1 SD used in IV exercise: 4.26bp.

*Source: A Tale of Two Margins: Monetary Policy and Capital Misallocation — Working Paper No. WP/2024/121 (content excerpt from section 7 and appendices of the PDF).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024121-print-pdf.pdf_