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---

### Overview and motivation
- Research question: How are disruptions in corporate bond markets transmitted to the rest of the financial system and real economic activity?
- Context: Systemic importance of corporate bond markets highlighted during the COVID-19 crisis, when the US Federal Reserve purchased corporate bonds for the first time in its history.
- Theoretical motivation: Builds on theories linking borrower balance-sheet quality to access to external finance (Bernanke and Gertler 1989; Kiyotaki and Moore, 1997; Bernanke et al., 1999) and on empirical evidence that credit spreads predict economic activity (Gilchrist and Zakrajšek 2012).

### Research questions addressed
- How do excess bond premium shocks affect commercial bank lending?
- Do commercial banks act as conduits for the transmission of these shocks?
- Does including commercial banking variables in a Gilchrist–Zakrajšek (GZ) VAR model affect estimated responses of output, consumption, and investment to excess bond premium shocks?

### Key contributions
- Provides empirical evidence that corporate bond market shocks generate significant contagion to the commercial banking sector.
- Extends the GZ VAR framework to explicitly include commercial banking activity (bank lending quantity and bank lending rate).
- Uses Call Report micro data to show heterogeneity: banks with high exposure to the corporate bond market reduce lending more aggressively than banks with low exposure.

### Main empirical findings and quantitative outcomes
- A 20 basis point (one standard deviation) shock in the excess bond premium results in:
  - a 4 percentage point contraction in the level of real commercial bank lending after six to eight quarters.
  - a 40 basis point increase in the real lending rate after six to eight quarters.
- Implication: corporate bond shocks predominantly affect the supply of bank loans; the supply effect more than offsets loan-demand effects.
- Forecast performance improvements when banking variables are included:
  - Reduces root mean square errors (RMSEs) of forecasts for consumption, investment, and output by 10-15 percent at four to eight quarter horizons.
  - Reductions in RMSEs for other variables of up to 50 percent at four to eight quarter horizons.
  - For longer-term forecasts, reductions remain large for the GDP deflator, the 10-year treasury yield, and the federal funds rate.
- Correlations of bank lending with reduced-form residuals from the GZ model range from +0.29 to -0.11.
  - Ljung-Box tests indicate statistically significant correlations (at the 10 percent significance level) of bank lending with consumption, output, and the federal funds rate.
- Correlations of the lending rate with GZ residuals range from +0.07 to -0.02; Ljung-Box tests do not reject no correlation for the lending rate.

### Methodology and VAR model specification
- Model: An “extended” VAR that augments the GZ VAR with banking sector variables (change in log of real commercial bank lending and average commercial bank nominal lending rate).
- GZ VAR variables included: log-difference of real personal consumption expenditure, log-difference of real business fixed investment, log-difference of real GDP, log-difference of the GDP price deflator (inflation), quarterly average excess bond premium (ebp), quarterly (value-weighted) excess stock market return (esr), 10-year nominal treasury yield, effective nominal federal funds rate.
- Identification: Recursive ordering assumes an excess bond premium shock (ebp) can affect economic activity and inflation only with a lag, while esr, federal funds rate, 10-year yield, and banking variables can react contemporaneously.
- Estimation details: VAR estimated with two lags using quarterly data for the period 1972:Q4 to 2019:Q4. Data sources include FRED, Global Financial Data, and the Federal Reserve’s webpage (ebp).

### Tests for inclusion of banking variables in the GZ model
- Three complementary approaches:
  - Correlation analysis between GZ residuals and banking variables (Canova, 2009).
  - Forecast comparison using out-of-sample RMSEs for the last 48 quarters.
  - Formal tests: Granger-causality, Sims (1972) exogeneity tests, and “fundamentalness” tests (Forni and Gambetti, 2014; Canova and Hamidi Sahneh, 2018).
- Results summary:
  - Adding bank lending alone: system-wide Granger causality test rejects no causality from bank lending to GZ variables at the 5 percent level; system-wide Sims exogeneity test rejects unidirectional causality of GZ variables on bank lending.
  - Adding lending rate alone: system-wide Granger causality test rejects absence of causality; individual-equation tests indicate the lending rate Granger-causes most GZ variables (exceptions: consumption, ebp, esr); system-wide Sims exogeneity test rejects exogeneity of GZ variables with respect to the lending rate.
  - Forecast and statistical tests jointly indicate the GZ VAR should be augmented with commercial banking variables.

### Interpretation and policy implications
- Commercial banks act as conduits for corporate bond market shocks; banks do not reliably act as a “spare tire” when corporate bond markets seize up.
- Larger corporate bond shocks could pose system-wide stability threats unless mitigated by policy interventions or backstops.
- Policy implications:
  - Evidence can justify macroprudential regulations on non-banks.
  - Supports deployment of liquidity backstops (such as central bank interventions during the March 2020 “dash-for-cash” episode) to prevent contagion to commercial banks.
  - Recommendation focus: mitigate vulnerabilities of corporate bond dealers and investment funds that reduce risk-bearing capacity during stress to limit need for central banks to act as market makers of last resort.

### Appendix II — Impulse Response Analysis (IRFs) to an ebp shock
- IRFs in the extended model are qualitatively similar to the GZ study: a positive ebp shock leads to significant declines in consumption, investment, output, and equity valuations despite reductions in policy and treasury interest rates.
- Banking-sector quantified responses to a 20 basis point shock:
  - Bank credit: a 3½–4 percentage point decline in the level of bank credit after 6-8 quarters.
  - Nominal lending rate: declines by about 25 basis points.
  - Federal funds rate: declines by 40 basis points, so the “intermediation” spread widens.
  - Real lending rate: increases by about 40 basis points for ten quarters because the decline in inflation more than offsets the decline in the nominal lending rate.
- Variance decompositions: ebp shocks explain 16 percent of variation in commercial bank lending and 14 percent of variation in the lending rate.
- Interpretation: The bank credit channel is an important mechanism transmitting ebp shocks to the real economy; transmission lags peak after 8-10 quarters.

### Appendix II — Heterogeneity in bank exposure to the corporate bond market
- Hypothesis: Banks with higher exposure to the corporate bond market on the funding side reduce lending more aggressively in response to ebp shocks.
- Empirical strategy:
  - Use Call Report micro data to compute non-deposit debt-to-total debt ratio as proxy for exposure.
  - Define groups: “Low” (below 25th percentile), “Medium” (25th–75th percentile), “High” (above 75th percentile).
  - Call Report data begin in 2002:Q1; sample constraints limit full augmented GZ estimation by group.
- Alternative method: near-VAR (Basu et al., 2006) over 2002:Q1-2019:Q4.
- Findings:
  - For all banks: bank lending declines and real lending rate increases over time.
  - By group: real lending rates increase for all groups; only “high” exposure banks reduce bank lending in response to the shock.
  - Confidence bands wider due to shorter sample period.
- Caveats: near-VAR has limited dynamics and omits broader macro factors; results are suggestive that high-exposure banks drive aggregate lending declines.

### Conclusion — methodology, core findings, and robustness
- Extended GZ framework to include commercial banks; estimated near-VARs for bank groups and extended-system VAR over 1972:Q4-2019:Q4.
- Bootstrap-after-bootstrap approach used for confidence intervals:
  - 2,000 bootstrap replications for most extended-model IRFs and variance decompositions.
  - For near-VARs by bank exposure: 1,000 replications to compute bias followed by 2,000 replications to compute confidence bands.
  - Reported confidence bands correspond to the 68 and 95 percent bias-corrected confidence intervals.
- Core empirical conclusions:
  - Excess bond premium (ebp) shocks can trigger reductions in commercial bank lending consistent with loan supply tightening.
  - Measurable contagion from corporate bond markets to commercial banks demonstrates a systemic transmission channel from non-banks to banks.
  - Extended model improves out-of-sample forecast RMSEs relative to the GZ model at multiple horizons.
- Robustness and diagnostics: omission, exogeneity, and fundamentalness tests; Granger-Causality tests; Sims exogeneity tests; Canova-Hamidi and Forni-Gambetti tests; Ljung-Box cross-correlation tests; forecast-error variance decompositions.

### Chi-squared test of the leads in (testing procedures)
- Individual banking variables (equation-by-equation):
  - Model: x_t = b(L) x_{t-i} + d(L) e_{GZ,t} v_t + ...
  - b(L) is a lag polynomial of order 2; d(L) is a (row) vector of two-sided lag polynomials of order 2.
  - Tests: Chi-squared test of the leads in (equation-by-equation); i = 1, 2,...,8; L = 0 appears in some specifications.
- Both banking variables (system-wide):
  - Model: y_t = B(L) y_{t-1} + D(L) e_{GZ,t} v_t + ...
  - B(L) is a matrix of lag polynomials of order 2; D(L) is a matrix of two-sided lag polynomials of order 2.
  - Tests: Chi-squared test of the leads in (system-wide).
- Forni-Gambetti variants:
  - Include structural innovations ε_{ebp,t} in specifications and perform Chi-squared tests using the Forni-Gambetti approach.
  - Notation from source (preserved verbatim): instances such as L = 0; 5,5; x bL = 5,5; GZ_{t} ε_{ebp}() GL () x bL (  )    0. appear in the original algebraic expressions.
- Implementation notes:
  - All Chi-squared tests performed as Wald tests.
  - Equation-by-equation test statistic constructed by multiplying a standard (excluded variables) F-statistic by the degrees of freedom in the numerator.
  - System-wide tests use an analogous calculation.
  - The Canova-Hamidi tests include lagged dependent variables and are implemented as in the Geweke, Messe and Dent (1983) version of the Sims test.

*IMF WORKING PAPERS — Do Corporate Bond Shocks Affect Commercial Bank Lending?*

### Introduction ...........................................................................................................

### Introduction

### Overview and motivation
- Research question: How are disruptions in corporate bond markets transmitted to the rest of the financial system and real economic activity?
- Context: The systemic importance of corporate bond markets was highlighted during the COVID-19 crisis, when the US Federal Reserve purchased corporate bonds for the first time in its history.
- Theoretical motivation: Builds on theories linking borrower balance-sheet quality to access to external finance (Bernanke and Gertler 1989; Kiyotaki and Moore, 1997; Bernanke et al., 1999) and on empirical evidence that credit spreads predict economic activity (Gilchrist and Zakrajšek 2012, among others).

### Research questions addressed
- How do excess bond premium shocks affect commercial bank lending?
- Do commercial banks act as conduits for the transmission of these shocks?
- Does including commercial banking variables in a Gilchrist–Zakrajšek (GZ) VAR model affect estimated responses of output, consumption, and investment to excess bond premium shocks?

### Key contributions
- Provides empirical evidence that corporate bond market shocks generate significant contagion to the commercial banking sector.
- Extends the GZ VAR framework to explicitly include commercial banking activity (bank lending quantity and bank lending rate).
- Uses Call Report micro data to show heterogeneity: banks with high exposure to the corporate bond market reduce lending more aggressively than banks with low exposure.

### Main empirical findings and quantitative outcomes
- A 20 basis point (one standard deviation) shock in the excess bond premium results in:
  - a 4 percentage point contraction in the level of real commercial bank lending after six to eight quarters.
  - a 40 basis point increase in the real lending rate after six to eight quarters.
- Implication: corporate bond shocks predominantly affect the supply of bank loans; the supply effect more than offsets loan-demand effects.
- Forecast performance improvements when banking variables are included:
  - Reduces root mean square errors (RMSEs) of forecasts for consumption, investment, and output by 10-15 percent at four to eight quarter horizons.
  - Reductions in RMSEs for other variables of up to 50 percent at four to eight quarter horizons.
  - For longer-term forecasts, reductions remain large for the GDP deflator, the 10-year treasury yield, and the federal funds rate.
- Correlations of bank lending with reduced-form residuals from the GZ model range from +0.29 to -0.11.
  - Ljung-Box tests indicate statistically significant correlations (at the 10 percent significance level) of bank lending with consumption, output, and the federal funds rate.
- Correlations of the lending rate with GZ residuals range from +0.07 to -0.02; Ljung-Box tests do not reject no correlation for the lending rate.

### Methodology and VAR model specification
- Model: An “extended” VAR that augments the GZ VAR with banking sector variables (change in log of real commercial bank lending and average commercial bank nominal lending rate).
- GZ VAR variables included: log-difference of real personal consumption expenditure, log-difference of real business fixed investment, log-difference of real GDP, log-difference of the GDP price deflator (inflation), quarterly average excess bond premium (ebp), quarterly (value-weighted) excess stock market return (esr), 10-year nominal treasury yield, effective nominal federal funds rate.
- Identification: Recursive ordering assumes an excess bond premium shock (ebp) can affect economic activity and inflation only with a lag, while esr, federal funds rate, 10-year yield, and banking variables can react contemporaneously.
- Estimation details: VAR estimated with two lags (as in the GZ study) using quarterly data for the period 1972:Q4 to 2019:Q4. Data sources: FRED database (most series), Global Financial Data (stock market return), and the Federal Reserve’s webpage (ebp).

### Tests for inclusion of banking variables in the GZ model
- Three complementary approaches:
  - Correlation analysis between GZ residuals and banking variables (Canova, 2009).
  - Forecast comparison using out-of-sample RMSEs for the last 48 quarters.
  - Formal tests: Granger-causality, Sims (1972) exogeneity tests, and “fundamentalness” tests (Forni and Gambetti, 2014; Canova and Hamidi Sahneh, 2018).
- Results summary:
  - Adding bank lending alone: system-wide Granger causality test rejects no causality from bank lending to GZ variables at the 5 percent level; system-wide Sims exogeneity test rejects unidirectional causality of GZ variables on bank lending.
  - Adding lending rate alone: system-wide Granger causality test rejects absence of causality; individual-equation tests indicate the lending rate Granger-causes most GZ variables (exceptions: consumption, ebp, esr); system-wide Sims exogeneity test rejects exogeneity of GZ variables with respect to the lending rate.
  - Forecast and statistical tests jointly indicate the GZ VAR should be augmented with commercial banking variables.

### Interpretation and policy implications
- Commercial banks act as conduits for corporate bond market shocks; banks do not reliably act as a “spare tire” when corporate bond markets seize up.
- Policy-relevant magnitudes indicate that larger corporate bond shocks could pose system-wide stability threats unless mitigated by policy interventions or backstops.
- Implication for policy: evidence can justify macroprudential regulations on non-banks and the deployment of liquidity backstops (such as central bank interventions during the March 2020 “dash-for-cash” episode) to prevent contagion to commercial banks.
- Recommendation focus: mitigate vulnerabilities of corporate bond dealers and investment funds that reduce risk-bearing capacity during stress to limit need for central banks to act as market makers of last resort.

*IMF Working Paper — Introduction (wpiea2023156-print-pdf).*

### Appendix II describes the implementation of these tests in detail.

### wpiea2023156-print-pdf - Appendix II describes the implementation of these tests in detail.

### Inclusion of banking variables and Granger-causality
- System-wide and equation-by-equation tests show banking sector variables Granger-cause the variables in the GZ model and reject the exogeneity of the GZ variables vis-à-vis the banking sector variables.
- Conclusion: the test results point to the need to include banking variables in the GZ model.

### Fundamentalness tests: approach and findings
- Concept:
  - A VAR model’s structural moving average representation is “non-fundamental” if the variables in the model do not contain enough information to recover structural shocks (the “true” innovations).
  - Beaudry et al. (2019) note rejection of fundamentalness may be explained by a “small” informational deficiency of the VAR; in that case a sub-set of estimated impulse responses “could” still provide useful information.
- Implementation differences:
  - Unlike Forni and Gambetti (2014) and Canova and Hamidi Sahneh (2018), which base tests on information comprising dozens or hundreds of time series with principal components, this study’s fundamentalness tests focus on two banking sector variables.
  - Note: a model can be found non-fundamental whenever the fundamentalness hypothesis is rejected, even if the test is based on a single variable.
- Tests and results:
  - Canova-Hamidi (system-wide) tests for the GZ model consistently reject the null hypothesis of fundamentalness for inclusion of bank lending, the lending rate, and both variables together — providing evidence of the need to include banking sector information in the GZ model to “recover” structural innovations.
  - Forni-Gambetti (orthogonality) tests fail to reject the null of fundamentalness in this study; this is attributed to the limited information set (two banking variables), which does not preclude rejection with a broader information set.
- Conclusion: tests suggest banking sector variables should be included in the VAR model and doing so leads to substantial gains in forecast performance.

### Impulse Response Analysis (IRFs) to an ebp shock
- General:
  - IRFs for the extended model (including both banking sector variables) are shown for variables in the original GZ model (Figure 1) and for banking sector variables (Figure 2).
  - Responses for variables defined as log-differences (dC, dI, dY, dP, dl) have been accumulated and so has the response of the excess stock market return (esr).
- Comparison with GZ:
  - IRFs are qualitatively the same as those in the GZ study: a positive shock to the excess bond premium leads to significant declines in consumption, investment, output, and equity valuations despite reductions in the policy and treasury interest rates.
  - Unreported results: these responses are qualitatively unchanged for the shorter sample period used in the original GZ study (1974 Q4 – 2010 Q4).
  - The 68 percent confidence intervals for the differences between the IRFs of the extended model and the GZ model consistently included zero (extended model used as data generating process).
- Banking-sector responses (Figure 2):
  - Key variables examined: (real) bank lending, the nominal and real bank lending rates, and the “intermediation” spread between the nominal bank lending rate and the federal funds rate.
  - Quantified responses to a 20 basis point shock:
    - Bank credit: a 3½–4 percentage point decline in the level of bank credit after 6-8 quarters (panel a).
    - Nominal lending rate: declines by about 25 basis points (panel b).
    - Federal funds rate: declines substantially more, 40 basis points (Figure 1 panel g), so the “intermediation” spread widens (panel c).
    - Real lending rate: increases by about 40 basis points for ten quarters (panel d) because the decline in inflation resulting from the shock more than offsets the decline in the nominal lending rate.
  - Interpretation:
    - An ebp shock is partially transmitted to the real economy through a tightening of bank credit supply: for bank lending (quantity) to decline and the real lending rate (price) to increase, the effect on commercial banks’ loan supply must more than offset the effect on loan demand.
    - Transmission lags: responses of bank lending and the lending rate increase over time, peaking after 8-10 quarters; lags likely reflect maturity transformation and staggered loan maturities versus faster-moving short-term funding costs.
- Variance decompositions (Figure 3):
  - ebp shocks account for significant portions of variation in other variables.
  - ebp shocks explain a substantial share of the variation in:
    - Commercial bank lending: 16 percent.
    - Lending rate: 14 percent.
- Summary: results suggest the bank credit channel is an important mechanism through which ebp shocks are transmitted to the real economy.

### Heterogeneity in bank exposure to the corporate bond market
- Hypothesis:
  - If ebp shocks transmit via a bank loan supply channel, banks more highly exposed to the corporate bond market on the funding side should reduce lending more aggressively than less exposed banks; less exposed banks might increase lending if higher loan demand from firms substituting bank loans for bonds outweighs adverse funding-cost effects.
- Empirical strategy:
  - Use Call Report micro data to compute the non-deposit debt-to-total debt ratio for individual banks as a proxy for relative exposure to the corporate bond market.
  - Define three groups:
    - “Low” exposure: below the 25th percentile of the cross-bank distribution of non-deposit debt funding shares.
    - “High” exposure: above the 75th percentile.
    - “Medium” exposure: between the 25th and 75th percentiles.
  - For each group and for all banks combined, compute series of bank lending and lending rate (see Appendix I for data sources and definitions).
  - Practical constraint: Call Report data begin in 2002:Q1 and the curse of dimensionality prevents estimating augmented GZ models with group variables.
- Alternative method and findings:
  - Use near-VAR methodology (Basu et al., 2006) to assess heterogeneity of bank responses to an exogenous structural ebp shock (Figure 4).
  - Findings:
    - For all banks, responses of bank lending and real lending rate exhibit the pattern from Figure 2: bank lending declines and the real lending rate increases over time.
    - For bank groups:
      - Real lending rates increase for all groups over time.
      - Only banks with “high” exposure to the corporate bond market reduce bank lending in response to the shock (rows 2 to 4).
    - Confidence bands are wider due to the shorter sample period.
- Caveats:
  - near-VAR has a limited dynamic structure; macroeconomic factors likely integral to transmission are missing due to data length limitations.
  - Results are suggestive: Figure 4 provides some evidence that the aggregate decline in bank lending caused by an ebp shock is driven by the loan supply response of banks with high exposure to the corporate bond market.

*IMF Working Paper — Appendix II: implementation details of fundamentalness and heterogeneity tests (extracted content).*

### Conclusion

### Conclusion (wpiea2023156-print-pdf)

### Methodology and model extensions
- Extended the empirical framework of Gilchrist and Zakrajšek (2012) to account for the role of commercial banks in the transmission of corporate bond credit spread shocks.
- Estimated a three-equation near-VAR model for each bank group:
  - The first equation regresses the (exogenous) excess bond premium (ebp) shock on a constant.
  - The other two equations regress real bank lending growth and the nominal lending rate on their own lags, the lags of the other banking variable, and the ebp shock.
  - The near-VAR for bank-group heterogeneity uses quarterly data for the period 2002:Q1-2019:Q4, with SUR techniques following Basu et al. (2006).
- Extended-system VAR estimated using the “full” sample period 1972:Q4-2019:Q4.
- Confidence intervals for impulse responses and variance decompositions constructed using the “bootstrap-after-bootstrap” approach of Kilian (1998) with:
  - 2,000 bootstrap replications for most extended-model IRFs and variance decompositions.
  - For near-VARs by bank exposure, 1,000 replications to compute bias followed by 2,000 replications to compute confidence bands.
- Reported confidence bands correspond to the 68 and 95 percent bias-corrected confidence intervals.

### Core empirical findings
- Excess bond premium (ebp) shocks in the corporate bond market:
  - Can trigger a reduction in commercial bank lending activity consistent with a tightening of loan supply.
  - Produce measurable contagion from the corporate bond market to commercial banks, demonstrating a channel by which systemic risk propagates within the financial sector from non-banks to commercial banks.
- Impulse response evidence:
  - IRFs reported for consumption, investment, output, prices, excess bond premium, cumulative excess market return, ten-year Treasury yield, and the federal funds rate to a one standard deviation structural ebp shock.
  - IRFs for bank variables include bank lending, nominal lending rate, the “intermediation” spread (Nominal Lending Rate - Federal Funds Rate), and the real lending rate.
- Forecast performance and model comparison:
  - Out-of-sample forecasts start in 2008:Q1 using recursive regressions with an initial estimation period spanning 1972:Q4 to 2007:Q4.
  - The extended model improves forecast RMSEs relative to the GZ model at multiple horizons (reported as percent differences in RMSE of the GZ model).
- Heterogeneity by bank exposure:
  - Dynamic responses of bank lending and real lending rate are reported separately for Total (all banks) and for banks with High, Medium, and Low exposure to the corporate bond market, using near-VARs estimated over 2002:Q1-2019:Q4.

### Robustness, tests, and diagnostics
- Omission, exogeneity, and fundamentalness tests applied to assess whether banking variables are omitted from the GZ model and whether banking variables are exogenous with respect to the GZ reduced-form residuals.
- Conducted equation-by-equation and system-wide tests, including:
  - Granger-Causality tests, Sims exogeneity tests (Geweke, Messe and Dent (1983) version), Canova-Hamidi exogeneity tests, and Forni-Gambetti orthogonality tests.
- Ljung-Box tests reported for cross-correlations of bank lending growth and bank lending rate (lagged) with the reduced-form residuals of the GZ model equations; presented alongside correlation coefficients for lags 0, 1, and 2.
- Forecast-error variance decompositions (based on the extended model, full sample 1972:Q4-2019:Q4) show contributions of ebp shocks to the variability of consumption, investment, output, prices, ebp itself, cumulative excess market return, ten-year Treasury yield, federal funds rate, bank lending, and the nominal lending rate.

### Policy implications
- Evidence of contagion from the corporate bond market to commercial banks highlights systemic risk transmission channels within the financial sector.
- Study underscores the importance and value of crisis containment policies in mitigating such contagion, citing the high importance and value of policies such as those deployed during the March 2020 “dash-for-cash” episode.

*IMF Working Paper — Conclusion (wpiea2023156-print-pdf)*

### 2. Chi-squared test of the leads in

### 2. Chi-squared test of the leads in

### Individual banking variables
- Model specification (GZ VAR, equation-by-equation):
  - x_t = b(L) x_{t-i} + d(L) e_{GZ,t} v_t + ...
  - Reduced-form residuals: e_{GZ,t} is the vector of reduced form residuals of the GZ VAR.
  - b(L) is a lag polynomial of order 2.
  - d(L) is a (row) vector of two-sided lag polynomials of order 2.
  - Tests performed: Chi-squared test of the leads in (equation-by-equation).
- Indexing and notation present in source:
  - i = 1, 2,...,8
  - L = 0 appears in some specifications.

### Both banking variables
- System specification (GZ VAR, system-wide):
  - y_t = B(L) y_{t-1} + D(L) e_{GZ,t} v_t + ...
  - e_{GZ,t} is the vector of reduced form residuals of the GZ VAR.
  - B(L) is a matrix of lag polynomials of order 2.
  - D(L) is a matrix of two-sided lag polynomials of order 2.
  - Tests performed: Chi-squared test of the leads in (system-wide).

### Forni-Gambetti variants
- Individual banking variables:
  - Specification includes structural innovations to the excess bond premium: ε_{ebp,t}.
  - Model: x_t = b(L) x_{t-i} + d(L) ε_{ebp,t} + ...
  - b(L) and d(L) are lag polynomials of order 2.
  - Tests performed: Chi-squared test of (leads) using the Forni-Gambetti approach.
- Both banking variables:
  - System specification uses structural innovations ε_{ebp,t} in the GZ VAR.
  - b(L) is a lag polynomial of order 2 and matrices of lag polynomials combine in the system.
  - Tests performed: Chi-squared test of (leads) using the Forni-Gambetti approach.
- Additional notation preserved from source:
  - GL, Lb, and combinations like (   )(   )(   ) i Lb Lb Lb appear in the original algebraic expressions.
  - Instances of explicit numeric indices: L = 0; 5,5; x bL = 5,5; GZ_{t} ε_{ebp}() GL () x bL (  )    0. (present verbatim in source).

### Notes on testing procedures and implementation details
- All Chi-squared tests are performed as Wald tests.
- Equation-by-equation test statistic construction:
  - Takes a standard (excluded variables) F-statistic and multiplies it by the degrees of freedom in the numerator.
- System-wide tests use an analogous calculation to form Chi-squared statistics.
- The Canova-Hamidi tests include lagged dependent variables (see Forni, Gambetti, and Sala (2018)), implemented as in the Geweke, Messe and Dent (1983) version of the Sims test.

*IMF WORKING PAPERS — Do Corporate Bond Shocks Affect Commercial Bank Lending?*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023156-print-pdf.pdf_