## 8. Estimation Results for Linear Panel Estimation ## Source details **Canonical URL:** [8. Estimation Results for Linear Panel Estimation](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1664.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1664.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1664.pdf.json) --- ### Estimation context and objectives - Purpose: assess how balance-sheet based measures of bank fundamentals (solvency, asset quality, liquidity, profitability) affect measures of bank funding cost (average funding cost and wholesale funding cost) using panel estimation. - Datasets: - FDIC Call Report: quarterly data for approximately 10,000 U.S. banks over 21 years (1993–2013); averaged to yearly frequency for analysis. - SNL Financial global dataset: yearly data for approximately 2,700 banks from 80 countries (used for supplementary/global evidence). - Motivation: identify cross-sectional sensitivity of creditors (fundamentalist investors) to changes in bank fundamentals to inform stress-testing and understanding of amplification between asset risk and funding risk. ### Variable construction and data treatment - Funding cost proxies: - Average funding cost: ratio of total interest expense to total liabilities (quarterly for FDIC, annualized and averaged to yearly). - Wholesale funding cost (proxy): interbank funding cost = ratio of interest expense on federal funds and repos purchased to the total stock of federal funds and repos purchased (quarterly then averaged to yearly). This measure represents on average about 9 percent of U.S. banks’ total liabilities. - Note: unsecured federal funds represent ~70 percent of the liabilities summarized in the wholesale funding measure; term segment estimates range from 10 percent to 50 percent. - Bank fundamental measures: constructed via Principal Component Analysis (PCA) aggregating constituent balance-sheet variables into CAMELS-type dimensions: - Solvency (SO): Tier 1 capital ratio, total regulatory capital ratio, leverage ratio. - Asset quality (AQ): Net-charge offs to loans, credit loss provisions to net charge offs, earnings coverage of net-charge offs, noncurrent loans to total loans. (Constituent variables negatively associated with asset quality were multiplied by “-1” so that lower values of asset quality are preferred.) - Liquidity (LIQ): liquid assets to total assets, volatile liabilities to total liabilities, deposits to assets. - Profitability (PROF): Return on equity (ROE), return on assets (ROA), net interest margin. - Additional controls: - Relative size (RE): log of the ratio of the bank’s total assets to total assets in the sample. - Fraction of insured deposits (FI). - Temporal controls: primarily year dummies; alternative specifications replace year dummies with VIX and 10-year U.S. T-bill rate. - Data cleaning (U.S. banks): observations dropped if any of the following conditions met: - Funding cost in excess of 30 percent per quarter. - ROE larger than 50 percent or smaller than -50 percent. - ROA larger than 5 percent or smaller than -5 percent. - Net interest margin larger than 10 percent. - Any variable with an invalid value (e.g., ratio larger than one or wrong sign). ### Estimation methodology - Models estimated: - Logit: to validate that constructed fundamental measures relate to bank default probability (using FDIC bank default dataset). - Linear panel models: baseline linear panel relating funding cost to solvency, asset quality, liquidity, profitability, with year dummies; extended specifications include bank controls RE and FI, and variants replacing year dummies with VIX and U.S. T-bill rate. - Linear estimation repeated by year (OLS) to obtain yearly sensitivity coefficients. - Non-linear specifications on the entire panel: - Piecewise linear regression with variable threshold SOC to test for increased sensitivity of funding cost below a solvency critical value. - Two-parameter non-linear transformation of solvency with offset parameter and exponent to allow funding cost sensitivity to diverge below a critical solvency offset and increase at a rate determined by the exponent. - Estimation technique for panel models: Arellano-Bond Generalized Method of Moments (GMM) to handle large cross-section, bank fixed effects, weakly exogenous regressors, and lags of dependent variable. ### Key empirical findings (linear and nonlinear estimation) - Relationship between fundamentals and default: - All constructed proxies for bank fundamentals are significant at the 1 percent level in logit specifications; coefficients have expected signs. Bank-specific controls (RE and FI) are not significant in the default probit/logit. - Main linear-panel results (baseline and robustness checks): - (i) Solvency is negatively related to both average and wholesale funding cost. - This result is robust to inclusion of RE and FI and to replacing year dummies with VIX and U.S. T-bill rate. - The identified relationship is driven by cross-sectional variation rather than temporal variation; translating cross-sectional sensitivities into time-paths for individual banks may not be straightforward. - (ii) Sensitivity of wholesale funding cost to solvency shocks is somewhat larger than sensitivity of average funding cost: - Estimated solvency coefficients in Table 4 (baseline specifications with fundamentals and year dummies): - Wholesale funding cost solvency coefficient: -0.0004 - Average funding cost solvency coefficient: -0.0002 - Interpretation (as provided in source): suggests that banks with a 1 percentage point lower level of solvency tend to pay [text ends in source and does not provide the remainder of the sentence]. - Supplementary summary result statements: - A solvency shock of 5 percentage points would lead to an average increase in interbank funding cost of approximately 0.2 percentage points. - The solvency measure (comprised of the leverage ratio, Tier 1 capital ratio, and total regulatory capital ratio) is negatively and significantly related to both average and wholesale measures of funding cost. - On average, the coefficient linking funding cost and solvency is larger in magnitude for wholesale funding cost than for average funding cost. - The magnitude of the relationship from linear panel estimation is small, but shocks to bank solvency can have a non negligible impact on the bank’s profitability. - Wholesale funding cost tends to “explode” for a relatively small number of banks with low values of the solvency measure, while the vast majority of ‘weak’ banks still exhibit relatively low funding costs. ### State dependence and non-linearity - Distributional pattern: as solvency decreases the wholesale funding cost distribution becomes more dispersed with a growing number of observations with very large wholesale funding cost while most observations remain at relatively low wholesale funding cost; suggests potential for large increases in wholesale funding cost at low solvency levels. - Piecewise linear regression: estimating threshold SOC for solvency identifies whether sensitivity of lenders increases below a critical solvency value; T-tests are used to assess significance of coefficient differences below and above SOC. - Two-parameter non-linear transformation: offset parameter determines a critical solvency value where funding cost may diverge; exponent determines rate at which sensitivity increases as solvency drops; model parameters are selected by best statistical fit (pseudo R-square). - Specific state-dependent findings: - The sensitivity of interbank (wholesale) funding cost to solvency shocks varies significantly over the cycle; sensitivity is stronger during times of crisis. - Yearly re-estimation shows wholesale funding cost is more sensitive to solvency during periods of economic stress than in normal times; by contrast the sensitivity for average funding cost is relatively stable over time. - Piecewise linear panel estimations show the sensitivity of wholesale funding cost to solvency is larger in magnitude for banks with low levels of solvency than for strong banks; the sensitivity for poorly capitalized banks is more than twice as large as for well-capitalized banks. - A t-test indicates that the difference in sensitivities between poorly capitalized and well-capitalized banks is statistically significant at a 5 percent significance level. - A two-parameter non-linear transformation of solvency does not improve model fit relative to the piecewise linear estimation, likely because a majority of observations remain concentrated at low levels of wholesale funding cost even for low solvency. - Conclusion from the data: low solvency is a necessary but not sufficient condition for very high wholesale funding costs. ### Implications for stress testing and risk management - The state-contingent strength of the interaction between solvency and funding cost introduces additional uncertainty into stress testing: historically estimated average sensitivities may be invalid during severe crises and underestimate true sensitivity. - Estimates of the sensitivity of bank funding cost to solvency shocks are critical for stress-testing, but sensitivity of wholesale lenders varies considerably over time and is particularly elevated during times of crisis. - Linear-model-based funding cost elasticity can be used as a conservative tool in stress-testing to compute changes in overall interest expense from an exogenous solvency shock and to recompute net interest margin and secondary solvency shocks via the funding cost channel. - Linear models are likely to underestimate the actual impact of solvency shocks on ‘weak’ banks and cannot identify threshold values at which banks may be shut out from wholesale funding markets altogether. - For funding cost stress testing, an alternative approach is recommended: consider the probability of a particular funding cost spike conditional on a level of solvency rather than focusing on the mean of a heavily skewed conditional distribution. ### Policy and operational recommendations - Increasing capital ratios may help banks reduce their cost of funding; depending on the cost of raising additional capital it may be beneficial for banks to increase their capital to drive down average funding cost—though it is unclear whether the magnitude of reduction identified would be sufficient to induce banks to raise capital. - Better data on wholesale funding cost (e.g., actual bond yields) and events of extreme funding liquidity tightening (including cases where liquidity completely dried out) are crucial to understand non-linear responses and to identify potentially state-contingent solvency thresholds for access to wholesale funding markets. - Stress-testing frameworks should account for non-linearities and skewness in the conditional distribution of funding cost; calibration should explicitly allow for elevated sensitivities in crisis states and for thresholds that could lead to wholesale market exclusion. ### Extension to a global dataset - Re-estimation of the baseline model using SNL data for banks from advanced and emerging economies finds that funding cost is significantly negatively associated with changes in bank solvency, consistent with results from the FDIC dataset. - Models estimated include fixed effects and Arellano-Bond GMM estimators; year dummies were included to capture temporal effects. *Source: _wp1664 - 8. Estimation Results for Linear Panel Estimation _________________________________24_* ### REFERENCES ____________________________________________________________29 ### REFERENCES ____________________________________________________________29 ### FIGURES - 1. Distribution of Wholesale Funding Cost (proxied by Interbank Funding Cost) ________25 - 2. Estimated Coefficients of the Solvency Measure ________________________________26 - 3. Estimated Coefficients and p-values of Coefficients of the Solvency Measure _________27 - 4. Goodness of Fit (as measured by the Pseudo R-squared) __________________________28 ### TABLES - 1. Pairwise Correlations of Funding Cost Variables ________________________________17 - 2. Summary Statistics for FDIC Dataset _________________________________________18 - 3. Estimates of Logit Model Linking Measures of Bank Fundamentals ________________19 - 4. Estimation Results for Linear Panel Estimation _________________________________20 - 5. Estimation Results for Piecewise Panel Estimation ______________________________21 - 6. Estimation Results for Panel Estimation_______________________________________22 - 7. Summary Statistics for SNL Dataset _________________________________________23 *_wp1664 - REFERENCES ____________________________________________________________29_* ### 8. Estimation Results for Linear Panel Estimation _________________________________24 ### 8. Estimation Results for Linear Panel Estimation _________________________________24 ### Estimation context and objectives - Purpose: assess how balance-sheet based measures of bank fundamentals (solvency, asset quality, liquidity, profitability) affect measures of bank funding cost (average funding cost and wholesale funding cost) using panel estimation. - Datasets: - FDIC Call Report: quarterly data for approximately 10,000 U.S. banks over 21 years (1993–2013); averaged to yearly frequency for analysis. - SNL Financial global dataset: yearly data for approximately 2,700 banks from 80 countries (used for supplementary/global evidence). - Motivation: identify cross-sectional sensitivity of creditors (fundamentalist investors) to changes in bank fundamentals to inform stress-testing and understanding of amplification between asset risk and funding risk. ### Variable construction and data treatment - Funding cost proxies: - Average funding cost: ratio of total interest expense to total liabilities (quarterly for FDIC, annualized and averaged to yearly). - Wholesale funding cost (proxy): interbank funding cost = ratio of interest expense on federal funds and repos purchased to the total stock of federal funds and repos purchased (quarterly then averaged to yearly). This measure represents on average about 9 percent of U.S. banks’ total liabilities. - Note: unsecured federal funds represent ~70 percent of the liabilities summarized in the wholesale funding measure; term segment estimates range from 10 percent to 50 percent. - Bank fundamental measures: constructed via Principal Component Analysis (PCA) aggregating constituent balance-sheet variables into CAMELS-type dimensions: - Solvency (SO): Tier 1 capital ratio, total regulatory capital ratio, leverage ratio. - Asset quality (AQ): Net-charge offs to loans, credit loss provisions to net charge offs, earnings coverage of net-charge offs, noncurrent loans to total loans. (Constituent variables negatively associated with asset quality were multiplied by “-1” so that lower values of asset quality are preferred.) - Liquidity (LIQ): liquid assets to total assets, volatile liabilities to total liabilities, deposits to assets. - Profitability (PROF): Return on equity (ROE), return on assets (ROA), net interest margin. - Additional controls: - Relative size (RE): log of the ratio of the bank’s total assets to total assets in the sample. - Fraction of insured deposits (FI). - Temporal controls: primarily year dummies; alternative specifications replace year dummies with VIX and 10-year U.S. T-bill rate. - Data cleaning (U.S. banks): observations dropped if any of the following conditions met: - Funding cost in excess of 30 percent per quarter. - ROE larger than 50 percent or smaller than -50 percent. - ROA larger than 5 percent or smaller than -5 percent. - Net interest margin larger than 10 percent. - Any variable with an invalid value (e.g., ratio larger than one or wrong sign). ### Estimation methodology - Models estimated: - Logit: to validate that constructed fundamental measures relate to bank default probability (using FDIC bank default dataset). - Linear panel models: baseline linear panel relating funding cost to solvency, asset quality, liquidity, profitability, with year dummies; extended specifications include bank controls RE and FI, and variants replacing year dummies with VIX and U.S. T-bill rate. - Linear estimation repeated by year (OLS) to obtain yearly sensitivity coefficients. - Non-linear specifications on the entire panel: - Piecewise linear regression with variable threshold SOC to test for increased sensitivity of funding cost below a solvency critical value. - Two-parameter non-linear transformation of solvency with offset parameter and exponent to allow funding cost sensitivity to diverge below a critical solvency offset and increase at a rate determined by the exponent. - Estimation technique for panel models: Arellano-Bond Generalized Method of Moments (GMM) to handle large cross-section, bank fixed effects, weakly exogenous regressors, and lags of dependent variable. ### Key empirical findings (linear and nonlinear estimation) - Relationship between fundamentals and default: - All constructed proxies for bank fundamentals are significant at the 1 percent level in logit specifications; coefficients have expected signs. Bank-specific controls (RE and FI) are not significant in the default probit/logit. - Main linear-panel results (baseline and robustness checks): - (i) Solvency is negatively related to both average and wholesale funding cost. - This result is robust to inclusion of RE and FI and to replacing year dummies with VIX and U.S. T-bill rate. - The identified relationship is driven by cross-sectional variation rather than temporal variation; translating cross-sectional sensitivities into time-paths for individual banks may not be straightforward. - (ii) Sensitivity of wholesale funding cost to solvency shocks is somewhat larger than sensitivity of average funding cost: - Estimated solvency coefficients in Table 4 (baseline specifications with fundamentals and year dummies): - Wholesale funding cost solvency coefficient: -0.0004 - Average funding cost solvency coefficient: -0.0002 - Interpretation (as provided in source): suggests that banks with a 1 percentage point lower level of solvency tend to pay [text ends in source and does not provide the remainder of the sentence]. - Non-linear evidence and interpretation: - Distributional pattern: as solvency decreases the wholesale funding cost distribution becomes more dispersed with a growing number of observations with very large wholesale funding cost while most observations remain at relatively low wholesale funding cost; suggests potential for large increases in wholesale funding cost at low solvency levels. - Piecewise linear regression: estimating threshold SOC for solvency identifies whether sensitivity of lenders increases below a critical solvency value; T-tests are used to assess significance of coefficient differences below and above SOC. - Two-parameter non-linear transformation: offset parameter determines a critical solvency value where funding cost may diverge; exponent determines rate at which sensitivity increases as solvency drops; model parameters are selected by best statistical fit (pseudo R-square). ### Practical implications and interpretation - Cross-sectional link: creditors appear to price solvency differences across banks — lower solvency is associated with higher funding costs, with stronger effects for wholesale (interbank) funding than for average funding (dominated by retail deposits). - Stress-testing relevance: the balance-sheet based sensitivity estimates are directly applicable to stress tests that rely on balance sheet data; non-linearities imply that solvency deterioration can trigger disproportionate increases in wholesale funding cost, potentially amplifying losses and triggering feedbacks. - Caution: main empirical identification uses cross-sectional variation; temporal extrapolation to individual banks requires care. *Source: _wp1664 - 8. Estimation Results for Linear Panel Estimation _________________________________24* ### 0.04 percentage points more on their wholesale liabilities. Similarly, they tend to pay ### _wp1664 - 0.04 percentage points more on their wholesale liabilities. Similarly, they tend to pay ### Key empirical findings on solvency and funding cost - A solvency shock of 5 percentage points would lead to an average increase in interbank funding cost of approximately 0.2 percentage points. - The solvency measure (comprised of the leverage ratio, Tier 1 capital ratio, and total regulatory capital ratio) is negatively and significantly related to both average and wholesale measures of funding cost. - On average, the coefficient linking funding cost and solvency is larger in magnitude for wholesale funding cost than for average funding cost. - The magnitude of the relationship from linear panel estimation is small, but shocks to bank solvency can have a non negligible impact on the bank’s profitability. - Wholesale funding cost tends to “explode” for a relatively small number of banks with low values of the solvency measure, while the vast majority of ‘weak’ banks still exhibit relatively low funding costs. ### State dependence and non-linearity - The sensitivity of interbank (wholesale) funding cost to solvency shocks varies significantly over the cycle; sensitivity is stronger during times of crisis. - Yearly re-estimation shows wholesale funding cost is more sensitive to solvency during periods of economic stress than in normal times; by contrast the sensitivity for average funding cost is relatively stable over time. - Piecewise linear panel estimations show the sensitivity of wholesale funding cost to solvency is larger in magnitude for banks with low levels of solvency than for strong banks; the sensitivity for poorly capitalized banks is more than twice as large as for well-capitalized banks. - A t-test indicates that the difference in sensitivities between poorly capitalized and well-capitalized banks is statistically significant at a 5 percent significance level. - A two-parameter non-linear transformation of solvency does not improve model fit relative to the piecewise linear estimation, likely because a majority of observations remain concentrated at low levels of wholesale funding cost even for low solvency. - Conclusion from the data: low solvency is a necessary but not sufficient condition for very high wholesale funding costs. ### Implications for stress testing and risk management - The state-contingent strength of the interaction between solvency and funding cost introduces additional uncertainty into stress testing: historically estimated average sensitivities may be invalid during severe crises and underestimate true sensitivity. - Estimates of the sensitivity of bank funding cost to solvency shocks are critical for stress-testing, but sensitivity of wholesale lenders varies considerably over time and is particularly elevated during times of crisis. - Linear-model-based funding cost elasticity can be used as a conservative tool in stress-testing to compute changes in overall interest expense from an exogenous solvency shock and to recompute net interest margin and secondary solvency shocks via the funding cost channel. - Linear models are likely to underestimate the actual impact of solvency shocks on ‘weak’ banks and cannot identify threshold values at which banks may be shut out from wholesale funding markets altogether. - For funding cost stress testing, an alternative approach is recommended: consider the probability of a particular funding cost spike conditional on a level of solvency rather than focusing on the mean of a heavily skewed conditional distribution. ### Policy and operational recommendations - Increasing capital ratios may help banks reduce their cost of funding; depending on the cost of raising additional capital it may be beneficial for banks to increase their capital to drive down average funding cost—though it is unclear whether the magnitude of reduction identified would be sufficient to induce banks to raise capital. - Better data on wholesale funding cost (e.g., actual bond yields) and events of extreme funding liquidity tightening (including cases where liquidity completely dried out) are crucial to understand non-linear responses and to identify potentially state-contingent solvency thresholds for access to wholesale funding markets. - Stress-testing frameworks should account for non-linearities and skewness in the conditional distribution of funding cost; calibration should explicitly allow for elevated sensitivities in crisis states and for thresholds that could lead to wholesale market exclusion. ### Extension to a global dataset - Re-estimation of the baseline model using SNL data for banks from advanced and emerging economies finds that funding cost is significantly negatively associated with changes in bank solvency, consistent with results from the FDIC dataset. - Models estimated include fixed effects and Arellano-Bond GMM estimators; year dummies were included to capture temporal effects. *Source: IMF Working Paper content provided in the supplied PDF excerpt.* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1664.pdf_