## _wp13241

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---

### Introduction — motivation and main contribution
- Spikes in measures of uncertainty in late 2008 coincided with declines in bank credit growth; the paper studies the channel from uncertainty shocks to the real economy via the supply of bank credit.
- Main contribution: provides theoretical foundations and empirical evidence that aggregate uncertainty reduces bank lending through a bank self-insurance channel (banks cut lending to rebuild capital when uncertainty increases).
- Key conceptual points:
  - Bank capital mitigates agency costs between creditors and the bank; lower bank capital → higher premium creditors demand.
  - Non-linear financial frictions create a self-insurance mechanism: higher uncertainty raises the value of bank capital and induces lending cuts to strengthen balance sheets.
  - Identification strategy exploits cross-bank heterogeneity: less-capitalized banks should reduce lending more in response to uncertainty if the effect is supply-driven.

### Theoretical framework — model structure and mechanism
- Key model ingredients:
  - Limited liability and asymmetric information modeled via costly state verification.
  - Continuum of risk-neutral borrowers with one-period lives; borrowers have endowment of 1 unit of capital and borrow l_t so capital k_t = l_t + 1.
  - Production: y_{t+1} = α_{t+1} R k_t where α_{t+1} is i.i.d., mean-one, non-negative support; α is common and unknown at period start (aggregate uncertainty).
  - Bank is monopoly lender; depositors supply funds elastically; monitoring costs μ and ω satisfy 1 ≥ μ > 0 and 1 > ω > μ.
- Bank optimization:
  - Bank maximizes expected discounted dividends subject to resource constraint l_t ≤ d_t + e_t − c_t and limited liability (e_{t+1} ≥ 0).
  - First-order conditions emphasize marginal value of bank capital V′(e) and optimal lending condition; uncertainty (σ, the standard deviation of α) affects probabilities of default and loan/deposit interest rates.
- Mechanism and comparative statics:
  - Marginal value of bank capital is decreasing in capital-to-assets; uncertainty increases the marginal value of capital (shifts it up).
  - In response to higher uncertainty, banks target higher capital and reduce lending; part of lending contraction can be permanent (higher funding costs) and part temporary (overshooting while capital is rebuilt).
  - Impact depends on initial capital: if bank capital is below target when shock hits, contraction is larger and longer.

### Data — scope and variable definitions
- Sample: universe of U.S. commercial banks filing Call Reports (FFIEC031), period 1984q1–2010q2; final baseline regression sample is N = 988,123 bank-quarter observations.
- Capitalization measures:
  - Capital-to-assets ratio: total equity capital / total assets.
  - Tangible equity ratio: (total equity − intangible assets) / (total tangible assets).
- Liquidity measure: ratio of securities holdings to total assets (cash excluded).
- Baseline uncertainty measure: dispersion of real GDP growth forecasts from the Survey of Professional Forecasters (Federal Reserve Bank of Philadelphia), computed as 75th − 25th percentile of Q/Q growth projections, expressed in annualized percentage points (4-quarter-ahead; 2-quarter-ahead used in robustness checks).
- Alternative uncertainty proxies: stock market volatility (log of annualized std dev of S&P500 daily returns), Senior Loan Officer Opinion Survey (SLOOS) “outlook” reason for tightening, common factor of forecast errors (Jurado et al. (2013)).
- Macro controls: seasonally adjusted real GDP growth and CPI inflation; monetary policy proxy: change in effective federal funds rate.
- Data filtering and sample construction:
  - Only federally insured commercial banks in 50 contiguous U.S. states + DC; removed non-deposit trust companies, savings banks, credit unions, cooperative banks, industrial banks, brokers, etc.
  - Quarters with mergers set to missing for loan series; loan-growth outliers (>5 standard deviations from cross-sectional mean) set to missing; loan series included only if at least four consecutive quarterly growth rates available.
  - Final number of bank-quarter observations after filters: 1,060,335 (original 1,178,658); baseline regression uses 988,123 observations.
- Selected summary statistics (Table 1 — values preserved exactly):
  - Total loans growth: Mean 2.3 Percent; Std. Dev. 8.0; p25 -1.2; p50 1.7; p75 4.9; Min -98.0; Max 111.5
  - C&I loans growth: Mean 1.7 Percent; Std. Dev. 18.7; p25 -5.7; p50 1.1; p75 8.5; Min -166.9; Max 187.9
  - Individual loans growth: Mean 1.0 Percent; Std. Dev. 14.3; p25 -4.2; p50 0.4; p75 5.1; Min -191.0; Max 188.1
  - Real Estate loans growth: Mean 3.1 Percent; Std. Dev. 10.2; p25 -1.2; p50 1.9; p75 5.7; Min -106.7; Max 117.6
  - Total assets: Mean 11.1 Log of th. USD; Std. Dev. 1.3; p25 10.3; p50 11.0; p75 11.8; Min 0.0; Max 21.3
  - Liquidity: Mean 32.6 Percent; Std. Dev. 16.2; p25 20.9; p50 30.7; p75 42.6; Min 0.0; Max 100.0
  - Total equity / Total assets: Mean 9.7 Percent; Std. Dev. 3.4; p25 7.6; p50 8.9; p75 11.0; Min 4.0; Max 31.1
  - Tangible Equity / Tangible Assets: Mean 1.3 Percent; Std. Dev. 2.0; p25 0.3; p50 0.8; p75 1.6; Min -4.3; Max 15.3
  - Real GDP growth: Mean 3.1 Percent; Std. Dev. 1.9; p25 2.4; p50 3.1; p75 4.2; Min -4.1; Max 8.5
  - Expected GDP growth: Mean 5.7 Percent; Std. Dev. 1.3; p25 5.0; p50 5.6; p75 6.3; Min 2.0; Max 9.3
  - Inflation: Mean 0.8 Percent; Std. Dev. 0.5; p25 0.6; p50 0.8; p75 1.0; Min -2.4; Max 1.7
  - Monetary policy indicator: Mean -0.1 Percent; Std. Dev. 0.6; p25 -0.4; p50 0.0; p75 0.2; Min -2.1; Max 1.0
  - Dispersion of professional forecasts (4 quarters ahead): Logs Mean 0; Std. Dev. 0.4; p25 -0.3; p50 0.2; p75 -0.0; Min -0.9; Max 0.9
  - Stock market volatility: Log of annualized std dev Mean 2.7; Std. Dev. 0.4; p25 2.4; p50 2.6; p75 2.9; Min 1.9; Max 4.2
  - SLOOS outlook: percent Mean 30.0; Std. Dev. 45.2; p25 -4.5; p50 18.6; p75 72.2; Min -51.3; Max 100
  - Common factor of forecast errors (12-months): Mean 0; Std. Dev. 0.7; p25 -0.5; p50 0.3; p75 -0.9; Min -0.9; Max 3.3

### Empirical strategy — identification and regression specification
- Identification premise: because of convex marginal value of bank capital, less-capitalized banks should reduce lending more when aggregate uncertainty rises; differential response identifies supply effects.
- Baseline panel regression (fixed effects, regressors lagged one period unless noted):
  - Dependent variable: ∆ log(L_{i,t}) (growth of loans)
  - Key regressors: ∆ ln CPI_{t−1}, ∆ ln GDP_{t−1}, UNC_{t−1} (aggregate uncertainty), LIQ_{i,t}, lnAssets_{i,t−1}, CAP_{i,t−1} (capital-to-assets)
  - Interactions: CAP_{i,t−1} interacted with UNC_{t−1}, ∆GDP_{t−1}, ∆CPI_{t−1}, TIME, etc.
  - Controls: time trend (TIME), seasonal dummies (QUARTER), FRB regional dummies, bank fixed effects υ_i.
  - Equation shown exactly as: ∆ log(L_{i,t}) = α∆CPI_{t−1} + β∆GDP_{t−1} + γUNC_{t−1} + κLIQ_{i,t} + χlogAssets_{i,t−1} + CAP_{i,t−1}(ζ+η∆CPI_{t−1}+τ∆GDP_{t−1}+λUNC_{t−1}+μTIME) + νTIME + Σ_{k=1}^{3} ξ_k QUARTER_k + Σ_{k=1}^{11} ρ_k FRB_k + υ_i + ε_{i,t}

### Empirical results — main findings and robustness
- Main empirical regularities:
  - Increase in uncertainty is associated with a reduction in loan growth (negative coefficient on UNC).
  - Interaction CAP*Uncertainty is positive and significant: banks with higher capital-to-assets ratios experience smaller reductions in lending when uncertainty rises; equivalently, less-capitalized banks cut lending more.
  - Results robust to alternative uncertainty measures (stock volatility, SLOOS outlook, volatility of forecast errors), inclusion of macro controls (real GDP growth, inflation), inclusion of year fixed effects, alternative capitalization measure (tangible CAR), and alternative sample splits (excluding pre-1993, pre-1994, post-Lehman periods).
  - Larger banks are less affected by uncertainty: bank size dampens the effect (three proposed explanations: “too-big-to-fail” implicit guarantee, flight-to-quality from small to large banks, broader hedging options for large banks).
  - Results by loan type: coefficients are significant for individual and real estate loans; coefficients are of expected sign but not significant for commercial and industrial (C&I) loans.
- Quantitative comparison:
  - "A 1 standard deviation increase in uncertainty generates an effect in lending that is about 82 percent of what a 1 standard deviation monetary policy shock generates."
- Regression coefficients preserved from Table 2 (selected exact figures):
  - Baseline specification (Column (1), N = 988,123; Fixed effects; standard errors clustered at bank level):
    - ∆Real GDP: 0.368 (0.023) ***
    - Uncertainty: -0.647 (0.123) ***
    - Inflation: 0.078 (0.063)
    - Liquidity: 0.074 (0.002) ***
    - Bank capitalization (CAR): 0.881 (0.046) ***
    - Time trend: 0.058 (0.003) ***
    - CAR*Uncertainty: 0.073 (0.013) ***
    - Constant: 4.818 (1.766) ***
    - R^2 = 0.09
  - Tangible equity specification (Column (2), N = 989,818):
    - Uncertainty: -0.478 (0.038) ***
    - CAR*Uncertainty (using TCAR): 0.226 (0.026) ***
    - Tangible CAR (TCAR): 1.098 (0.092) ***
  - Robustness notes:
    - Column (3) excluding time trend: Uncertainty: -1.832 (0.128) ***; CAR*Uncertainty: 0.065 (0.012) ***.
    - Column (4) includes ∆ ln assets: ∆ ln assets coefficient 13.423 (0.346) ***; Uncertainty: -0.712 (0.113) ***; CAR*Uncertainty: 0.077 (0.013) ***
    - Column (5) includes expected GDP growth (E[GDP growth]): Uncertainty: -0.673 (0.123) ***; CAR*E[∆Real GDP] coefficient reported as -0.009 (0.006).

### Alternative measures of uncertainty and robustness
- Literature overview: multiple measures exist (crossectional volatility of firm profitability, stock market volatility or the VIX, surveys of professional forecasts, volatility of forecast errors); each has limitations.
- Table 4 (summary of robustness across definitions):
  - ∆Real GDP coefficients across six specifications: 0.369, 0.372, 0.350, 0.352, 0.161, 0.375.
  - Uncertainty coefficients across six specifications: -0.673, -0.642, -0.204, -0.233, -0.005, -0.037.
  - CAR*Uncertainty across six specifications: 0.077, 0.065, 0.032, 0.037, 0.001, 0.038.
- Definitions used in Table 4:
  - (1) dispersion of 4-quarters-ahead professional forecasts of real GDP growth;
  - (2) same as (1) at a horizon of 2 quarters;
  - (3) Common factor in volatility of forecast errors of 279 macroeconomic and financial variables at a horizon of 2 quarters taken from Jurado et al. (2013);
  - (4) same as (3) at a horizon of 4 quarters;
  - (5) net fraction of banks responding that they have tightened lending standards due to an uncertain outlook, from the Survey of Senior Loan Officers;
  - (6) average (quarterly) volatility of S&P daily returns.
- Note: Expected GDP growth is measured as the mean among 4-quarters-ahead professional forecasts of real GDP growth.

### Different loan types and bank size heterogeneity
- Loan-type heterogeneity:
  - Main results hold for individual and real estate loans.
  - For C&I loans coefficients are of expected sign but CAR*Uncertainty is not significant—consistent with many C&I loans being drawings on pre-existing commitments.
- Empirical coefficients from Table 5 (selected exact figures):
  - ∆Real GDP: C&I loans 0.664; Individual 0.550; R.E. loans 0.166; Size specification 0.402.
  - Uncertainty: C&I loans -0.701; Individual -1.539; R.E. loans -0.435; Size specification -4.924.
  - CAR*Uncertainty: C&I 0.027; Individual 0.112; R.E. loans 0.057; Size 0.386.
  - Interaction terms involving ln assets and uncertainty (Size column):
    - CAR*ln assets = -0.154
    - Uncertainty*ln assets = 0.386
    - CAR*ln assets*Uncertainty = -0.029
- Interpretation:
  - Interaction between assets and uncertainty is significant and positive, suggesting large banks better shield business against increases in uncertainty.
  - Negative and statistically significant triple interaction (CAR*ln assets*Uncertainty) suggests the self-insurance effect of bank capital weakens for large banks.

### Monetary policy, omitted variable concerns, and quantitative comparison
- Regressions augmented to control for monetary policy shocks (change in effective federal funds rate) and include 4 lags of ∆FF and 4 lags of uncertainty.
- Table 6 (selected sums of lag coefficients — exact figures):
  - ∑4_{j=1} ∆FF rate_{t−j}: column (1) -.193; column (2) -.588.
  - ∑4_{j=1} Uncertainty_{t−j}: column (1) -0.708; column (2) -0.743.
  - CAR*∑4_{j=1} ∆FF rate_{t−j}: column (1) 0.041; column (2) 0.023.
  - CAR*∑4_{j=1} Uncertainty_{t−j}: column (1) 0.051; column (2) 0.052.
- Main findings:
  - The bank capital channel and the bank lending channel of monetary policy are present and statistically significant.
  - Controlling for monetary policy shocks does not remove the effect of uncertainty on lending.
  - Quantitative comparison reiterated: the differential response of lending between a bank at the 75th percentile of CAR and one at the 25th percentile, to a 1 standard deviation increase in uncertainty, is 82 percent of the effect on lending generated by a 1 standard deviation monetary policy shock.

### Conclusions — policy-relevant takeaways
- Aggregate uncertainty shocks can materially compress bank credit supply through a self-insurance mechanism: banks cut lending to rebuild capital buffers when uncertainty rises.
- Less-capitalized and smaller banks are more vulnerable to uncertainty shocks, implying that capital regulation and macroprudential policy can affect the transmission of uncertainty to credit supply.
- Because C&I lending often operates under commitments (less sensitive to supply shocks), the composition of lending matters for how uncertainty translates to aggregate credit quantities.
- The self-insurance mechanism can operate even under limited liability and contributes to the amplification of credit cycles.

### Appendix: numerical solution method and parameter values
- Parameters used (Table 7):
  - β 0.995
  - R 1.01
  - μ 0.13
  - ω 0.20
  - ρ 1.00
- Solution approach:
  - The model is solved by backwards induction using Carroll (2006)’s endogenous gridpoints method.
  - Steps: solve for q and l satisfying equations (12) and (13); choose grid for q; determine whether no-equity finance constraint binds; compute dividends and lending via closed-form or root-finding; recover beginning-of-period bank capital e_t from q = e − c; construct c(e) and l(e) via piecewise linear interpolation; construct marginal value function; iterate backward through equations (14) and (15) until maximum absolute difference between dividend and lending functions and previous period counterparts is below 0.001.

*Source: _wp13241 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .*

### References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### _wp13241 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### Introduction — motivation and main contribution
- Spikes in measures of uncertainty in late 2008 coincided with declines in bank credit growth; the paper studies the channel from uncertainty shocks to the real economy via the supply of bank credit.
- Main contribution: provides theoretical foundations and empirical evidence that aggregate uncertainty reduces bank lending through a bank self-insurance channel (banks cut lending to rebuild capital when uncertainty increases).
- Key conceptual points:
  - Bank capital mitigates agency costs between creditors and the bank; lower bank capital → higher premium creditors demand.
  - Non-linear financial frictions create a self-insurance mechanism: higher uncertainty raises the value of bank capital and induces lending cuts to strengthen balance sheets.
  - Identification strategy exploits cross-bank heterogeneity: less-capitalized banks should reduce lending more in response to uncertainty if the effect is supply-driven.

### Theoretical framework — model structure and mechanism
- Key model ingredients:
  - Limited liability and asymmetric information modeled via costly state verification.
  - Continuum of risk-neutral borrowers with one-period lives; borrowers have endowment of 1 unit of capital and borrow l_t so capital k_t = l_t + 1.
  - Production: y_{t+1} = α_{t+1} R k_t where α_{t+1} is i.i.d., mean-one, non-negative support; α is common and unknown at period start (aggregate uncertainty).
  - Bank is monopoly lender; depositors supply funds elastically; monitoring costs μ and ω satisfy 1 ≥ μ > 0 and 1 > ω > μ.
- Bank optimization:
  - Bank maximizes expected discounted dividends subject to resource constraint l_t ≤ d_t + e_t − c_t and limited liability (e_{t+1} ≥ 0).
  - First-order conditions emphasize marginal value of bank capital V′(e) and optimal lending condition; uncertainty (σ, the standard deviation of α) affects probabilities of default and loan/deposit interest rates.
- Mechanism and comparative statics:
  - Marginal value of bank capital is decreasing in capital-to-assets; uncertainty increases the marginal value of capital (shifts it up).
  - In response to higher uncertainty, banks target higher capital and reduce lending; part of lending contraction can be permanent (higher funding costs) and part temporary (overshooting while capital is rebuilt).
  - Impact depends on initial capital: if bank capital is below target when shock hits, contraction is larger and longer.

### Data — scope and variable definitions
- Sample: universe of U.S. commercial banks filing Call Reports (FFIEC031), period 1984q1–2010q2; final baseline regression sample is N = 988,123 bank-quarter observations.
- Capitalization measures:
  - Capital-to-assets ratio: total equity capital / total assets.
  - Tangible equity ratio: (total equity − intangible assets) / (total tangible assets).
- Liquidity measure: ratio of securities holdings to total assets (cash excluded).
- Baseline uncertainty measure: dispersion of real GDP growth forecasts from the Survey of Professional Forecasters (Federal Reserve Bank of Philadelphia), computed as 75th − 25th percentile of Q/Q growth projections, expressed in annualized percentage points (4-quarter-ahead; 2-quarter-ahead used in robustness checks).
- Alternative uncertainty proxies: stock market volatility (log of annualized std dev of S&P500 daily returns), Senior Loan Officer Opinion Survey (SLOOS) “outlook” reason for tightening, common factor of forecast errors (Jurado et al. (2013)).
- Macro controls: seasonally adjusted real GDP growth and CPI inflation; monetary policy proxy: change in effective federal funds rate.
- Data filtering and sample construction:
  - Only federally insured commercial banks in 50 contiguous U.S. states + DC; removed non-deposit trust companies, savings banks, credit unions, cooperative banks, industrial banks, brokers, etc.
  - Quarters with mergers set to missing for loan series; loan-growth outliers (>5 standard deviations from cross-sectional mean) set to missing; loan series included only if at least four consecutive quarterly growth rates available.
  - Final number of bank-quarter observations after filters: 1,060,335 (original 1,178,658); baseline regression uses 988,123 observations.

- Selected summary statistics (Table 1 — values preserved exactly):
  - Total loans growth: Mean 2.3 Percent; Std. Dev. 8.0; p25 -1.2; p50 1.7; p75 4.9; Min -98.0; Max 111.5
  - C&I loans growth: Mean 1.7 Percent; Std. Dev. 18.7; p25 -5.7; p50 1.1; p75 8.5; Min -166.9; Max 187.9
  - Individual loans growth: Mean 1.0 Percent; Std. Dev. 14.3; p25 -4.2; p50 0.4; p75 5.1; Min -191.0; Max 188.1
  - Real Estate loans growth: Mean 3.1 Percent; Std. Dev. 10.2; p25 -1.2; p50 1.9; p75 5.7; Min -106.7; Max 117.6
  - Total assets: Mean 11.1 Log of th. USD; Std. Dev. 1.3; p25 10.3; p50 11.0; p75 11.8; Min 0.0; Max 21.3
  - Liquidity: Mean 32.6 Percent; Std. Dev. 16.2; p25 20.9; p50 30.7; p75 42.6; Min 0.0; Max 100.0
  - Total equity / Total assets: Mean 9.7 Percent; Std. Dev. 3.4; p25 7.6; p50 8.9; p75 11.0; Min 4.0; Max 31.1
  - Tangible Equity / Tangible Assets: Mean 1.3 Percent; Std. Dev. 2.0; p25 0.3; p50 0.8; p75 1.6; Min -4.3; Max 15.3
  - Real GDP growth: Mean 3.1 Percent; Std. Dev. 1.9; p25 2.4; p50 3.1; p75 4.2; Min -4.1; Max 8.5
  - Expected GDP growth: Mean 5.7 Percent; Std. Dev. 1.3; p25 5.0; p50 5.6; p75 6.3; Min 2.0; Max 9.3
  - Inflation: Mean 0.8 Percent; Std. Dev. 0.5; p25 0.6; p50 0.8; p75 1.0; Min -2.4; Max 1.7
  - Monetary policy indicator: Mean -0.1 Percent; Std. Dev. 0.6; p25 -0.4; p50 0.0; p75 0.2; Min -2.1; Max 1.0
  - Dispersion of professional forecasts (4 quarters ahead): Logs Mean 0; Std. Dev. 0.4; p25 -0.3; p50 0.2; p75 -0.0; Min -0.9; Max 0.9
  - Stock market volatility: Log of annualized std dev Mean 2.7; Std. Dev. 0.4; p25 2.4; p50 2.6; p75 2.9; Min 1.9; Max 4.2
  - SLOOS outlook: percent Mean 30.0; Std. Dev. 45.2; p25 -4.5; p50 18.6; p75 72.2; Min -51.3; Max 100
  - Common factor of forecast errors (12-months): Mean 0; Std. Dev. 0.7; p25 -0.5; p50 0.3; p75 -0.9; Min -0.9; Max 3.3

### Empirical strategy — identification and regression specification
- Identification premise: because of convex marginal value of bank capital, less-capitalized banks should reduce lending more when aggregate uncertainty rises; differential response identifies supply effects.
- Baseline panel regression (fixed effects, regressors lagged one period unless noted):
  - Dependent variable: ∆ log(L_{i,t}) (growth of loans)
  - Key regressors: ∆ ln CPI_{t−1}, ∆ ln GDP_{t−1}, UNC_{t−1} (aggregate uncertainty), LIQ_{i,t}, lnAssets_{i,t−1}, CAP_{i,t−1} (capital-to-assets)
  - Interactions: CAP_{i,t−1} interacted with UNC_{t−1}, ∆GDP_{t−1}, ∆CPI_{t−1}, TIME, etc.
  - Controls: time trend (TIME), seasonal dummies (QUARTER), FRB regional dummies, bank fixed effects υ_i.
  - Equation shown exactly as: ∆ log(L_{i,t}) = α∆CPI_{t−1} + β∆GDP_{t−1} + γUNC_{t−1} + κLIQ_{i,t} + χlogAssets_{i,t−1} + CAP_{i,t−1}(ζ+η∆CPI_{t−1}+τ∆GDP_{t−1}+λUNC_{t−1}+μTIME) + νTIME + Σ_{k=1}^{3} ξ_k QUARTER_k + Σ_{k=1}^{11} ρ_k FRB_k + υ_i + ε_{i,t}

### Empirical results — main findings and robustness
- Main empirical regularities:
  - Increase in uncertainty is associated with a reduction in loan growth (negative coefficient on UNC).
  - Interaction CAP*Uncertainty is positive and significant: banks with higher capital-to-assets ratios experience smaller reductions in lending when uncertainty rises; equivalently, less-capitalized banks cut lending more.
  - Results robust to alternative uncertainty measures (stock volatility, SLOOS outlook, volatility of forecast errors), inclusion of macro controls (real GDP growth, inflation), inclusion of year fixed effects, alternative capitalization measure (tangible CAR), and alternative sample splits (excluding pre-1993, pre-1994, post-Lehman periods).
  - Larger banks are less affected by uncertainty: bank size dampens the effect (three proposed explanations: “too-big-to-fail” implicit guarantee, flight-to-quality from small to large banks, broader hedging options for large banks).
  - Results by loan type: coefficients are significant for individual and real estate loans; coefficients are of expected sign but not significant for commercial and industrial (C&I) loans — consistent with many C&I loans being under pre-existing commitments and thus less sensitive to supply shocks.
- Quantitative comparison:
  - A 1 standard deviation increase in uncertainty generates an effect in lending that is about 82 percent of what a 1 standard deviation monetary policy shock generates (exact phrasing preserved: "about 82 percent of what a 1 standard deviation monetary policy shock generates").
- Regression coefficients preserved from Table 2 (selected exact figures):
  - Baseline specification (Column (1), N = 988,123; Fixed effects; standard errors clustered at bank level):
    - ∆Real GDP: 0.368 (0.023) ***
    - Uncertainty: -0.647 (0.123) ***
    - Inflation: 0.078 (0.063)
    - Liquidity: 0.074 (0.002) ***
    - Bank capitalization (CAR): 0.881 (0.046) ***
    - Time trend: 0.058 (0.003) ***
    - CAR*Uncertainty: 0.073 (0.013) ***
    - Constant: 4.818 (1.766) ***
    - R^2 = 0.09
  - Tangible equity specification (Column (2), N = 989,818):
    - Uncertainty: -0.478 (0.038) ***
    - CAR*Uncertainty (using TCAR): 0.226 (0.026) ***
    - Tangible CAR (TCAR): 1.098 (0.092) ***
  - Robustness notes:
    - Column (3) excluding time trend: Uncertainty coefficient magnitude increases (Uncertainty: -1.832 (0.128) ***; CAR*Uncertainty: 0.065 (0.012) ***).
    - Column (4) includes ∆ ln assets: ∆ ln assets coefficient 13.423 (0.346) ***; Uncertainty: -0.712 (0.113) ***; CAR*Uncertainty: 0.077 (0.013) ***
    - Column (5) includes expected GDP growth (E[GDP growth]): Uncertainty: -0.673 (0.123) ***; CAR*E[∆Real GDP] coefficient reported as -0.009 (0.006) (not significant).
- Interpretation and robustness to alternative explanations:
  - Results are unlikely driven by demand effects because (i) cross-sectional breadth of banks and borrowers makes systematic correlation between borrower capitalization and lender capitalization implausible, and (ii) heterogeneity by bank capitalization and bank size matches supply-side predictions.
  - Tests excluding sample periods around major regulatory changes (Basel I, Interstate Banking and Branching Act, post-Lehman/Basel III discussion) and including year fixed effects do not overturn main findings.

### Practical implications and policy-relevant takeaways
- Aggregate uncertainty shocks can materially compress bank credit supply through a self-insurance mechanism: banks cut lending to rebuild capital buffers when uncertainty rises.
- Less-capitalized and smaller banks are more vulnerable to uncertainty shocks, implying that capital regulation and macroprudential policy can affect the transmission of uncertainty to credit supply.
- Because C&I lending often operates under commitments (less sensitive to supply shocks), the composition of lending matters for how uncertainty translates to aggregate credit quantities.

*Source: _wp13241 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .*

### 1.  Alternative Measures of Uncertainty

### _wp13241 - 1.  Alternative Measures of Uncertainty

### Alternative measures of uncertainty and robustness
- The literature contains multiple measures of uncertainty: crossectional volatility of firm profitability, stock market volatility or the VIX, surveys of professional forecasts, and volatility of forecast errors. Each measure has limitations (e.g., forward-lookingness, liquidity effects, disagreement vs. uncertainty, ex-post measurement).
- The paper shows results are robust across several alternative uncertainty measures:
  - Table 4: coefficient on ∆Real GDP across six specifications: 0.369, 0.372, 0.350, 0.352, 0.161, 0.375 (standard errors shown in table).
  - Table 4: coefficient on Uncertainty across six specifications: -0.673, -0.642, -0.204, -0.233, -0.005, -0.037 (standard errors shown in table).
  - Table 4: coefficient on CAR*Uncertainty across six specifications: 0.077, 0.065, 0.032, 0.037, 0.001, 0.038 (standard errors shown in table).
- Definitions of uncertainty used in Table 4:
  - (1) dispersion of 4-quarters-ahead professional forecasts of real GDP growth;
  - (2) same as (1) at a horizon of 2 quarters;
  - (3) Common factor in volatility of forecast errors of 279 macroeconomic and financial variables at a horizon of 2 quarters taken from Jurado et al. (2013);
  - (4) same as (3) at a horizon of 4 quarters;
  - (5) net fraction of banks responding that they have tightened lending standards due to an uncertain outlook, from the Survey of Senior Loan Officers;
  - (6) average (quarterly) volatility of S&P daily returns.
- Note: Expected GDP growth is measured as the mean among 4-quarters-ahead professional forecasts of real GDP growth.

### Different types of loans and bank size
- Loan-type heterogeneity:
  - Main results hold for individual and real estate loans.
  - For commercial and industrial (C&I) loans the coefficients are of expected sign, but the interaction between capital and uncertainty is not significant—likely because a large fraction of C&I loans are drawings on pre-existing commitments.
- Empirical coefficients from Table 5 (selected):
  - ∆Real GDP: C&I loans 0.664; Individual 0.550; R.E. loans 0.166; Size specification 0.402.
  - Uncertainty: C&I loans -0.701; Individual -1.539; R.E. loans -0.435; Size specification -4.924.
  - CAR*Uncertainty: C&I 0.027; Individual 0.112; R.E. loans 0.057; Size 0.386.
  - Interaction terms involving ln assets and uncertainty (Size column):
    - CAR*ln assets = -0.154
    - Uncertainty*ln assets = 0.386
    - CAR*ln assets*Uncertainty = -0.029
- Interpretation:
  - The interaction between assets and uncertainty is significant and positive, suggesting large banks better shield business against increases in uncertainty.
  - The negative and statistically significant triple interaction (CAR*ln assets*Uncertainty) suggests that the self-insurance effect of bank capital weakens for large banks (large banks have alternative hedges and potential too-big-to-fail effects).

### Monetary policy, omitted variable concerns, and quantitative comparison
- Regression specification augmented to control for monetary policy shocks:
  - Monetary policy measured as the change in the effective federal funds rate; regressions include 4 lags of the change in the effective federal funds rate and 4 lags of uncertainty.
- Table 6 (selected sums of lag coefficients):
  - ∑4_{j=1} ∆FF rate_{t−j}: column (1) -.193; column (2) -.588.
  - ∑4_{j=1} Uncertainty_{t−j}: column (1) -0.708; column (2) -0.743.
  - CAR*∑4_{j=1} ∆FF rate_{t−j}: column (1) 0.041; column (2) 0.023.
  - CAR*∑4_{j=1} Uncertainty_{t−j}: column (1) 0.051; column (2) 0.052.
- Main findings:
  - The bank capital channel and the bank lending channel of monetary policy are present and statistically significant.
  - Controlling for monetary policy shocks does not remove the effect of uncertainty on lending.
  - Quantitative comparison: the differential response of lending between a bank at the 75th percentile of the distribution of capital-to-asset ratios and one at the 25th percentile, to a 1 standard deviation increase in uncertainty, is 82 percent of the effect on lending generated by a 1 standard deviation monetary policy shock. In other words, the effect of uncertainty through the supply of bank credit is almost as strong as the effect of the bank capital and lending channels of monetary policy.

### Conclusions
- The paper provides theoretical foundations and empirical evidence for a channel through which uncertainty shocks affect the real economy via financial intermediaries.
- Key mechanisms:
  - Financial frictions imply a self-insurance mechanism: increases in macroeconomic uncertainty induce banks to adjust capitalization and lending, exacerbating credit cycles.
  - This self-insurance mechanism can operate even under limited liability.
- Empirical support:
  - There is a robust supply-side response of credit to changes in macroeconomic uncertainty, stronger for banks with lower levels of capital.
  - A 1 standard deviation uncertainty shock generates an effect on lending that is almost (82 percent) as large as a 1 standard deviation monetary policy shock.

### Appendix: numerical solution method and parameter values
- Parameters used (Table 7):
  - β 0.995
  - R 1.01
  - μ 0.13
  - ω 0.20
  - ρ 1.00
- Solution approach:
  - The model is solved by backwards induction using Carroll (2006)’s endogenous gridpoints method.
  - Steps: solve for q and l satisfying equations (12) and (13); choose grid for q; determine whether no-equity finance constraint binds; compute dividends and lending via closed-form or root-finding; recover beginning-of-period bank capital e_t from q = e − c; construct c(e) and l(e) via piecewise linear interpolation; construct marginal value function; iterate backward through equations (14) and (15) until maximum absolute difference between dividend and lending functions and previous period counterparts is below 0.001.

*Source: _wp13241 - 1.  Alternative Measures of Uncertainty*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13241.pdf_