## APPENDIX I: MODEL DETAILS

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### Behavioral framework and purpose
- Embeds a standard stress-testing framework based on individual banks’ data in a semi-structural macroeconomic model.
- Captures endogenous linkages between the real economy and the banking sector to:
  - Analyze impact of macroeconomic shocks on individual banks’ balance sheets.
  - Allow feedback from banking-sector solvency shocks to the real economy.
- Combines granularity of balance-sheet stress tests with general-equilibrium macro modeling; illustrated using Brazil (1999q1–2016q2) for a baseline scenario at aggregate banking-sector level.

### Stochastic processes, definitions, and key equations
- Produces scenarios that account for endogenous feedback effects; can extract business and credit cycles via historical shock decompositions and impulse responses.
- Credit-gap and credit-shock structure (notation preserved):
  - Equation (1): ܿ௜௧௣௕ = νܿ௜௧ିଵ௣௕ߥ + ଶݕ௧ିଵ߳ + ௜௧௖೛್
  - Equation (2): ܿ௜௧௣௥߬ = ଵܿ௜௧ିଵ௣௥߬ + ଶݕ௧ିଵ߳ + ௜௧௖೛ೝ
  - Credit shocks:
    - (3) ߳௜௧௖೛్ߥ = ଷܭ௜௧௣௕ܧ + ௜௧௖೛్
    - (4) ߳௜௧௖೛್߬ = ଷܭ௜௧௣௥ܧ + ௜௧௖೛ೝ
  - Capital buffer:
    - (5) ܭ௜௧ = ܮܣܶܫܲܣܥ௜௧ = ௜௧ܭെሺ௧∗ܭ + ഥ௜)
    - Regulatory requirement (random-walk process):
      - (6) ܭ௧∗ܭ = ௧ିଵ∗ߝ + ௧௄∗
  - Financial conditions index (FCI):
    - (7) ݂௧߯ = ଵ݂௧ିଵെ߯ + ଶݕሺ௧ାଶݕെ௧ିଵ߳ሻ + ௧௙
    - Shock: (8) ߳௧௙߯ = ଷ߳௧ିଵ௙߱ + ௧௙ (AR(1))
  - Output (demand) equation:
    - (9) ݕ௧ߩ = ଵݕ௧ିଵߩ + ଶݕ௧ାଵߩെଷݎ௧ߩ + ସݖ௧ߩ + ହݕ௧∗ߩ + ଺(∑ݓ௜௣௕߳௜௧௖೛್) + (∑ݓ௝௣௥߳௝௧௖೛ೝ)െߩ + ଻߳௧௙ + ߝ௧௬
    - Weights ݓ௜௣௕ and ݓ௝௣௥ reflect shares of public and private banks in total real credit.
- Macro module includes IS curve, Phillips curve (non-regulated-price and regulated-price inflation), policy rule, Fisher equation, real credit gaps (private and public), FCI and other standard blocks; shocks ε_{x,t} are i.i.d. white noise.

### Stress-testing module (banks) — structure and mechanics
- Capital accumulation and net income over a 5-year testing horizon:
  - Capital ratio = CET1 (equation (10)); net income (equation (11)) = sum of three revenue streams and five expense types plus AOCI and Basel III deductions.
  - Net income components (ratios to RWAs): interest revenue, non-interest revenue, net other income; expenses: interest expenses, non-interest expenses, taxes, dividend payments, provisions.
  - Brazil-specific: 100 percent of AOCI incorporated into CET1 in 2016 and onwards.
- Panel estimation of income-statement items:
  - General model (equation (12)) regressors: Δy_t (quarterly change in output gap), Δi_t (nominal interest rate gap), ΔFCI_t, capital buffer_t, steady-state X*, and idiosyncratic shock ε_{i,t}.
  - Endogenous reaction parameter γ_i captures banks’ adjustment of income/expenses to past capital (two specifications: γ_i = 0 and γ_i ≠ 0).
  - Restrictions on coefficients: e.g., coefficients on lagged X in (0,1); provisions coefficient on Δy in (−∞,0); Basel III deductions (0,1) on lagged X and 0 on macro regressors.

- Feedback loop sequencing (stylized):
  - t-1: macro variables (GDP growth, interest rate, FCI)
  - t: stress-testing projects income statement(t), capital(t); capital(t) affects credit contemporaneously
  - t: credit shock determined by capital ratio, output, credit lag → affects output via demand function
  - t+1: updated macro variables feed back into stress-testing → repeat

- Scaling: capital and income items scaled by RWAs for stationarity; resulting approximation errors treated as deductions to preserve historical accounting identities.

### Model estimation, data, and sample
- Estimation method: Bayesian estimation using Kalman Filter; prior distributions and posterior means/SDs reported in Tables 3–5.
- Brazil data: 1999q1–2016q2.
- Stress-testing sample: consolidated, publicly-available financial reports of Brazil’s six largest banks; adjusted for mergers and acquisitions.
- Appendices provide additional macro specifications and parameter estimates; stress-testing details referenced in Brazil 2016 Staff Report.

### Selected estimated and calibrated parameter values (preserved exactly)
- Calibrated parameters (Table 3):
  - g = 2.00
  - g^k = 5.00
  - ρ_g = 0.05
  - ρ_q = 0.05
  - σ (unspecified) = 0.05
  - α = 0.75
  - Note: shock SDs for trends calibrated using HP filter smoothing parameter 1600.

- Selected posteriors (Table 4):
  - σ1 — Posterior: 0.38, Std Dev.: 0.04
  - σ2 — Posterior: 0.24, Std Dev.: 0.03
  - σ3 — Posterior: 0.08, Std Dev.: 0.03
  - ρ1 — Posterior: 0.66, Std Dev.: 0.05
  - ρ2 — Posterior: 0.07, Std Dev.: 0.02
  - ρ3 — Posterior: 0.31, Std Dev.: 0.04
  - ρ4 — Posterior: 0.02, Std Dev.: 0.01
  - ρ5 — Posterior: 0.28, Std Dev.: 0.07
  - ρ6 — Posterior: 0.07, Std Dev.: 0.03
  - ρ7 — Posterior: 1.07, Std Dev.: 0.06
  - θ1 — Posterior: 0.76, Std Dev.: 0.03
  - θ2 — Posterior: 1.51, Std Dev.: 0.10
  - θ3 — Posterior: 0.20, Std Dev.: 0.05
  - υ1 — Posterior: 0.58, Std Dev.: 0.10
  - υ2 — Posterior: 0.78, Std Dev.: 0.07
  - υ3 — Posterior: 1.14, Std Dev.: 0.04
  - λ1 — Posterior: 0.52, Std Dev.: 0.14
  - λ2 — Posterior: 0.65, Std Dev.: 0.29
  - λ3 — Posterior: 1.01, Std Dev.: 0.09
  - ω1 — Posterior: 0.26, Std Dev.: 0.10
  - ω2 — Posterior: 0.43, Std Dev.: 0.07
  - δ1 — Posterior: 0.54, Std Dev.: 0.09
  - δ2 — Posterior: 0.25, Std Dev.: 0.03
  - ι1 — Posterior: 0.41, Std Dev.: 0.05
  - φ2 — Posterior: 0.48, Std Dev.: 0.07
  - ι3 — Posterior: 0.74, Std Dev.: 0.06
  - ρ_π — Posterior: 0.42, Std Dev.: 0.05
  - ζ — Posterior: 0.60, Std Dev.: 0.08
  - η — Posterior: 0.75, Std Dev.: 0.04

- Estimated shock SDs (Table 4):
  - σ_{ε_y} = 0.36, Std Dev.: 0.04
  - σ_{ε_π^e} = 1.60, Std Dev.: 0.14
  - σ_{ε_π^r} = 0.90, Std Dev.: 0.04
  - σ_{ε_i} = 0.44, Std Dev.: 0.04
  - σ_{ε_q} = 1.40, Std Dev.: 0.25
  - σ_{ε_u} = 0.14, Std Dev.: 0.02
  - σ_{ε_cu} = 0.12, Std Dev.: 0.02
  - σ_{ε_y^*} = 0.19, Std Dev.: 0.02
  - σ_{ε_x} = 0.17, Std Dev.: 0.02

- Selected banking-parameter posteriors (Table 5):
  - κ_{bank,1} — Posterior: 0.61, Std Dev.: 0.01
  - κ_{bank,2} — Posterior: 0.28, Std Dev.: 0.04
  - ψ_{bank,1} — Posterior: 0.10, Std Dev.: 0.04
  - ι_{bank,1} — Posterior: 0.00, Std Dev.: 0.00
  - κ_{bank,trend} — Posterior: 0.25, Std Dev.: 0.00
  - κ_{asset} — Posterior: 0.41
  - κ_{liquid} — Posterior: 0.94
  - κ_{reserves} — Posterior: 0.91
  - κ_{capital} — Posterior: 0.77, Std Dev.: 0.03
  - d_{capital} — Posterior: 0.20, Std Dev.: 0.04
  - ψ_{capital} — Posterior: 0.08
  - κ_{stress} — Posterior: 0.35
  - κ_{risk} — Posterior: 0.94, Std Dev.: 0.03
  - κ_{profitability} — Posterior: 0.91, Std Dev.: 0.04
  - κ_{leverage} — Posterior: 0.76
  - ψ_{other} — Posterior: 0.86, Std Dev.: 0.03
  - Note: error SD priors for banking model drawn from σ(1,∞); full posterior estimates available on request.

### Key comparative projection findings (baseline and specification differences)
- Four specifications compared: macro-feedback vs. no macro-feedback; income-statement adjustment vs. no adjustment.
- No income-statement adjustment:
  - "No macro-feedback effects and no income statement adjustments" projects capital ratio falls by 50 bps over next two years mostly due to lower output gap.
  - Including macro-feedback effects leads to an additional fall of 90 bps (total additional), as lower capital buffers reduce credit supply and output (by about 100 bps).
  - Interest rates projected to drop by almost 200 bps more in the model with macro-feedback effects due to larger falls in output and inflation.
- With income-statement adjustment:
  - Aggregate capital ratio projected to reach 16 percent by the end of the stress-testing horizon as banks increase net income in response to higher Basel III deductions and higher supervisory thresholds.
  - Feedback effects remain important but are smaller because banks’ adjustments mitigate initial capital fall; credit and output decrease less than in no-adjustment case.
  - Despite higher capital ratios, credit and output still fall because credit depends on difference between capital ratio and regulatory threshold; Basel III transition raises regulatory requirements.

### Impulse responses, historical decompositions, and numerical experiments
- Impulse-response highlights:
  - Credit responds more to output than output responds to credit; peak impacts occur around one year and persist for about 2 years.
  - Macro-feedback effects double private and public credit responses to a demand shock.
  - Private credit increases by 2 percent following a positive output shock; public credit increases by around 1.5 percent.
  - A 1 percent shock to capital across all banks causes credit to drop by 6 percent and output to fall by 0.6 percent (capital → credit → output).
- Selected shock effects on income statements:
  - Positive demand shock: raises net interest income, non-interest revenue, non-interest expense, unrealized gains; reduces provisions → higher net income and capital.
  - Positive interest-rate shock: lowers profits and aggregate capital (higher funding costs, credit losses, unrealized losses); interest income does not necessarily increase.
  - FCI shock similar to interest-rate shock (via funding spread).
  - Provisions shock: 1 percent provisions shock causes capital ratio trough fall of 1.2 percent in model without macro-feedback; with macro-feedback capital ratio falls an additional 20 bps and output falls by 0.4 percent.
- Solvency–liquidity interactions:
  - Interlinking solvency and liquidity raises the impact on aggregate capital ratio by 60 bps; output drops by 1.3 percent when solvency–liquidity interactions included — "three times as large" as without the link.
  - Extension: include aggregate capital buffer in FCI equation (equation (13)) to capture solvency effects on liquidity.

- Historical decomposition (Brazil, 1999 onward):
  - Private credit shocks boosted output pre-2008; public credit boosted output in 2009–10.
  - Both public and private capital shocks weakened output and credit during 2008/2009; private capital shocks supported post-crisis recovery via equity issuance and lending cuts.
  - Financial conditions shocks crucial during 2008/2009 and 2009 recovery; tightening in 2013 impeded recovery.
  - Since early 2015, public and private credit and FCI shocks have been a drag on output; private and public capital shocks positive contributors since mid-2015 (new equity issuance, dividend cuts, slowdown in RWAs).

### Forecast accuracy and model performance
- Root-mean-squared forecast errors (Table 2):
  - Models with income-statement adjustment have better forecast accuracy than models with no adjustment.
  - Including macro-feedback effects does not significantly improve forecast accuracy in Brazil, possibly because real-world links between credit, capital and output are more complex than modeled.

### Policy simulations, implications, and applications
- Applications:
  - Strengthening stress testing: generate endogenous scenarios that account for feedbacks between banking sector and real economy.
  - Strengthening macro-financial analysis: medium-term projections, business and credit cycles, systemic-risk analysis, policy simulation (monetary and macroprudential).
- Policy simulations illustrated:
  - Higher capital requirement for all banks → reduces output by lowering gap between bank capital and regulatory threshold → drop in credit supply and aggregate demand. Estimated impacts in Brazil align with literature on higher capital effects on credit and output.
  - Interaction with monetary policy: higher capital → lower output and inflation → induces monetary policy loosening.
  - Endogenous income-statement adjustments allow banks to meet higher requirements via business-model changes; without adjustments, banks may end up with lower capital ratios after higher requirements.
- Macroprudential modeling:
  - Framework can incorporate countercyclical capital buffer linked to credit gap or credit growth to increase requirements in good times and decrease in recessions; coordinated macroprudential and monetary policy response reduces impacts of solvency shocks.

### Contagion, liquidity, non-linearities, and recommended extensions
- Contagion:
  - Can embed counterparty losses using interbank bilateral exposure data via loss-cascading iterative mechanism (Sole and Espinosa, 2013) or entropy-maximization simulation (Anand and others, 2014) if data missing.
- Non-linearities:
  - Propose adding a credit-buffer term (possibly quadratic) to capture stronger lending constraints as capital approaches regulatory thresholds.
  - Calibration challenges for non-linear effects if banks have not experienced solvency episodes historically; cross-country elasticity calibration is imperfect.
- Microprudential improvements (data-rich FSAP-style extensions):
  - (i) net interest income via maturity-gap analysis;
  - (ii) provisions using PDs and LGDs by asset class;
  - (iii) realized/unrealized gains/losses on AFS securities via duration approach;
  - (iv) credit-risk RWAs for each asset class using IRB formula with projected PDs, LGDs;
  - (v) impose dividend distribution rule.
- Future work: incorporate liquidity risk, network contagion/spillovers, explicit modeling of RWAs and income-statement items to reduce approximation error; move toward structural, dynamic balance-sheet models acknowledging modeling and data challenges.

### Identification caveats and robustness
- Uses credit stock (not flow) due to short flow series; GDP growth more closely related to flow of new credit than to credit stock growth.
- Identification of supply vs. demand effects remains challenging:
  - Bank capital, income, expenses, and loan growth are endogenously determined.
  - Lagging capital insufficient because banks may preemptively change capital.
  - Recommendation: employ identification strategies from literature (e.g., Carlson and others, 2011; Jimenez and others, 2012).

*Source: APPENDIX I: MODEL DETAILS, wp17149*

### APPENDIX I: MODEL DETAILS _____________________________________________________ 37

### APPENDIX I: MODEL DETAILS

### Behavioral Equations
- The framework embeds a standard stress-testing framework based on individual banks’ data in a semi-structural macroeconomic model.
- Allows for endogenous linkages between the real economy and the banking sector to:
  - Analyze the impact of macroeconomic shocks on balance sheets of individual banks.
  - Allow for feedback effects from banking-sector solvency shocks to the real economy.
- Ensures consistency in the relationships between macroeconomic and financial variables and individual banks’ balance sheets.
- The model is illustrated using macroeconomic and banking data for Brazil in section IV (illustrative; outputs are not an assessment of the current state of the Brazilian banking sector).
- For illustrative purposes in the paper, the model’s output is examined for a baseline scenario only and banking sector variables are presented at the aggregate level.

### Stochastic Processes and Definitions
- The framework produces scenarios that ensure linkages between all variables are taken into account in a consistent way to accommodate endogenous feedback effects.
- The model can be used to extract business and credit cycles via historical shock decompositions and impulse responses.
- It captures macro-feedback effects that stem from banks’ solvency problems impacting the real economy, which can amplify and propagate the effects of shocks.
- The framework integrates macroprudential stress-testing improvements by better incorporating macro-feedback effects into stress tests that traditionally rely on exogenous macroeconomic scenarios, behavioral ad-hoc assumptions, and reduced-form mapping of macro shocks to bank risks.

### Estimated Parameters
- The model provides consistent macroeconomic frameworks with feedback loops between real and financial sectors and can be used as inputs into standard stress-testing frameworks.
- It addresses a key missing element identified in prior work: explicit incorporation of linkages between individual banks’ balance sheets and the real economy.
- The framework estimates and uses behavioral and banking parameters that are subject to restrictions (see "Restrictions on the Banking Parameters" referenced in the source table of contents).
- The paper contains estimated parameters and estimated banking parameters in separate tables (see contents: "Estimated Parameters", "Estimated Banking Parameters").

### Applications and Policy-Relevant Uses
- Strengthening stress testing:
  - Capture and analyze importance of macro-feedback effects for individual banks, the overall banking sector, and the real economy.
  - Generate scenarios that incorporate endogenous feedback effects consistently.
- Strengthening macro-financial analysis:
  - Generate baseline and stress scenarios in a consistent way.
  - Strengthen systemic risk analysis by estimating effects of banking-sector solvency on the real economy.
  - Extract business and credit cycles to inform risk analysis.
  - Measure impact of financial variables on the real economy and vice versa through historical shock decompositions and impulse responses.
  - Analyze impact of micro- and macro-prudential capital measures on the banking sector and the real economy, including impacts on monetary policy.
- Overall role:
  - Support IMF efforts to mainstream macro-financial analysis into bilateral surveillance.

*Source: APPENDIX I: MODEL DETAILS, wp17149*

### Section V provides some

### Section V provides some

### Related literature
- The literature related to stress-testing models that incorporate feedback effects from the financial sector to the real economy remains is very limited. Notable contributions discussed:
  - Bank of Korea (2012): systemic risk assessment model links decline in banks’ capital ratios to lower credit supply and higher borrowers’ probabilities of default that bring about second-round “credit crunch” losses. The model does not explicitly take account of the transmission of banking sector shocks to the real economy.
  - Bank of Japan (2014): macroeconomic stress tests incorporate macro-feedback effects from adverse shocks in the financial sector through higher lending rates driven by worsening non-banking sector balance sheets and lower credit supply from a credit crunch. Limitations: not fully based on theoretical considerations; focuses only on dynamics of real output without endogenous feedback loops for inflation and interest rates; equation-by-equation OLS estimation with simultaneity problems; some bank income statement items not explicitly modeled (e.g., non-interest income, realized and unrealized gains/losses on securities holdings).
  - Kida (2008): model incorporates transmission of banking sector solvency shocks to the real economy, but the macro module consists only of an equation linking output growth to credit growth; framework is calibrated so macro-feedback effects are imposed rather than estimated.
  - Gray and others (2013): contingent claims analysis embedded in a global VAR to study endogenous interactions between banking sector, corporate sector, sovereign, and output and credit growth. Limitations: risk indicators derived from equity prices rather than accounting principles, making source of vulnerabilities hard to disentangle; market-price indicators sensitive to short-term swings and not applicable where market price data are limited.
  - Many DSGE models have been complemented with a stress-testing exercise to analyze how a solvency shock would affect the real economy. Footnote: the solvency shock in these models typically derive from credit quality problems only, and the models are not formally linked to the stress-testing frameworks.

### Model — overview
- Purpose: embed a simple stress-testing model in a semi-structural macro model to fully capture endogenous macro-feedback effects; combine granularity of individual-bank balance-sheet stress tests with general-equilibrium macroeconomic modeling.
- Overall framework components (linked by the concept of credit crunch; see Figure 2):
  - Macro module: variant of models developed in Carabenciov and others (2008) and Krznar and Matheson (2017). Characterizes an open economy and jointly determines output, inflation, unemployment, interest rates, credit, financial conditions, foreign demand, and the real exchange rate.
  - Stress testing module: simple balance sheet-based approach to assess solvency through changes in net income and risk-weighted assets. Panel regression models describe individual banks’ income and expenses as functions of key macro variables. Some specifications allow endogenous reaction of banks’ incomes and expenses to changes in their capital buffers. Dividend distribution and Basel III phase-ins and phase-outs are taken into account; new share issuance and share buy backs are not considered.
  - Credit equation (link): panel credit equations link individual banks’ capital (from the stress-testing module) to bank credit and output (from the macro module) based on a credit crunch mechanism deriving from the failure of the Modigliani-Miller propositions.

### Credit channel and theoretical foundation
- Credit channel view:
  - The bank credit channel derives from the failure of the Modigliani-Miller propositions: issuing new equity is costly, so undercapitalized banks may restore targeted leverage by cutting lending rather than raising equity, hurting economic growth.
  - Empirical/theoretical literature supports that capital affects lending when regulatory-threshold breaches are costly and equity issuance is difficult. Examples cited include Jimenez and others (2009), Aiyar and others (2014), Brun and others (2013), Gambacorta and Marques-Ibanez (2011), Adrian and Shin (2010), Carlson and others (2011), Bridges and others (2014), De Nicolo (2015), Berrospide and Edge (2010), Calza and Sousa (2005), Cingano and others (2013), Barone and others (2016), Dimelis and others (2013), Meeks (2014).

### Macro module — key structure and equations
- Total credit is modeled separately for public and private banks to account for behavioral differences in Brazil.
- Credit gap equations (public and private banks) — notation and structure preserved as in source:
  - Equation (1): ܿ௜௧௣௕ = νܿ௜௧ିଵ௣௕ߥ + ଶݕ௧ିଵ߳ + ௜௧௖೛್
  - Equation (2): ܿ௜௧௣௥߬ = ଵܿ௜௧ିଵ௣௥߬ + ଶݕ௧ିଵ߳ + ௜௧௖೛ೝ
  - Shocks to credit supply unrelated to past aggregate demand and past credit adjustments are denoted ௜௧௖೛್ and ௜௧௖೛ೝ.
- Credit shocks specification:
  - Equations (3) and (4): ߳௜௧௖೛್ߥ = ଷܭ௜௧௣௕ܧ + ௜௧௖೛್ and ߳௜௧௖೛್߬ = ଷܭ௜௧௣௥ܧ + ௜௧௖೛ೝ with ܭ௜௧௣௕ and ܭ௜௧௣௥ being capital buffers for public and private banks, respectively; ܧ terms are white-noise shocks to real credit.
- Capital buffer definition:
  - Equation (5): ܭ௜௧ = ܮܣܶܫܲܣܥ௜௧ = ௜௧ܭെሺ௧∗ܭ + ഥ௜) — where ܮܣܶܫܲܣܥ௜௧ is common equity tier 1 capital ratio, ܭ௧∗ is the regulatory capital requirement, and ܭഥ௜ is each bank’s historical capital buffer.
  - Regulatory requirement process:
    - Equation (6): ܭ௧∗ܭ = ௧ିଵ∗ߝ + ௧௄∗ (modeled as a random walk to examine shocks to the capital requirement; treated as observable over history and the projection horizon).
- Financial conditions index (FCI):
  - Equation (7): ݂௧߯ = ଵ݂௧ିଵെ߯ + ଶݕሺ௧ାଶݕെ௧ିଵ߳ሻ + ௧௙
  - Shock process:
    - Equation (8): ߳௧௙߯ = ଷ߳௧ିଵ௙߱ + ௧௙ (AR(1) shock to financial conditions).
- Output (demand) equation with credit and FCI shocks feeding in:
  - Equation (9): ݕ௧ߩ = ଵݕ௧ିଵߩ + ଶݕ௧ାଵߩെଷݎ௧ߩ + ସݖ௧ߩ + ହݕ௧∗ߩ + ଺(∑ݓ௜௣௕߳௜௧௖೛್) + (∑ݓ௝௣௥߳௝௧௖೛ೝ)െߩ + ଻߳௧௙ + ߝ௧௬
  - Where ߝ௧௬ is an idiosyncratic demand shock; weights ݓ௜௣௕ and ݓ௝௣௥ reflect shares of public and private banks in total real credit.
- Interpretation: an autonomous expansion in public or private credit unrelated to past demand raises demand, while an autonomous tightening of the FCI reduces demand. Credit shocks in the demand function model feedback effects between the output gap and other macro variables via the credit crunch assumption.

### Estimation and data
- The combined framework (macro + stress testing module) is estimated using Bayesian methods.
- Data for Brazil: ranging from the beginning of 1999q1 to 2016q2.
- Stress-testing module data: publicly-available, consolidated data from the financial reports of Brazil’s six largest banks; data adjusted for mergers and acquisitions by the banks included in the stress test.
- Appendices A and B (referenced) provide more details on macro module specifications and parameter estimates; further stress testing details referenced in the Brazil 2016 Staff Report and Selected Issues paper on stress testing.

*Italic: Source — wp17149, Section V (excerpt).*

### Appendix I.

### Appendix I. Banks: Stress-Testing Module

### Capital accumulation and net income
- Capital ratios of individual banks are projected over a 5-year testing horizon using projections of net income and accumulated other comprehensive income (AOCI) together with Basel III capital deductions (Chart).
- For each bank i, the capital ratio accumulates according to equation (10) (capital ratio = CET1), where:
  - CET1 is common equity tier 1 capital ratio.
  - Net income (equation (11)) is a main driver and is expressed as the sum of three revenue streams and five expense types.
  - AOCI is accumulated other comprehensive income.
  - Basel III deductions are denoted and treated as fixed (see mechanics).
- Net income components (expressed as ratios to risk-weighted assets):
  - Revenues: interest revenue, non-interest revenue, net other income.
  - Expenses: interest expenses, non-interest expenses, taxes, dividend payments, provisions.
- Brazil-specific note: Consistent with Brazilian regulation, 100 percent of accumulated other comprehensive income (AOCI) was incorporated into CET1 capital in 2016 and onwards.

### Panel models, estimation and parameter structure
- A set of simple panel models estimates main components of each bank’s income statement to capture effects of lagged macroeconomic variables and financial conditions determined in the macro module in period t-1.
- General model for each income-statement item X is equation (12), where regressors include:
  - Quarterly change in the output gap (Δy_t).
  - Change in the nominal interest rate gap (Δi_t — nominal rate less its trend).
  - Change in the FCI (ΔFCI_t).
  - Bank capital buffer (capital buffer_t).
  - Steady-state of each income-statement variable X* (historical average).
  - Idiosyncratic shock (ε_{i,t}).
- Endogenous reaction of banks to shocks to past capital levels is captured by parameter γ_i in equation (12); two specifications are analyzed: with and without this endogenous reaction (i.e., γ_i = 0 or γ_i ≠ 0).
- Restrictions imposed for estimation (Table 1) include sign and range constraints on coefficients for revenues and expenses (examples from table, preserved as in source):
  - Coefficients on lagged X: (0,1) for many items.
  - Coefficients on Δy: ranges include (0,∞), (−∞,0) depending on item (e.g., provisions (−∞,0)).
  - Coefficients on Δi and ΔFCI: ranges vary by item; Basel III deductions treated as (0,1) on lagged X and 0 on macro regressors.

### Mechanics of the model and feedback loops
- Sequence (stylized, Figure 3):
  - t-1: Macro variables (GDP growth, interest rate, FCI).
  - t: Stress-testing module projects income statement(t), capital(t); capital(t) affects credit contemporaneously.
  - t: Credit shock determined by capital ratio, output, credit lag → affects output via demand function.
  - t+1: Macro variables in t enter stress-testing to generate capital adequacy ratios in t+1; cycle repeats.
- Scaling: capital and income-statement items are scaled by risk-weighted assets to ensure stationarity; approximation errors from scaling are assumed part of deductions to ensure accounting identities hold historically.

### Model specifications and comparative projections
- Four specifications examined:
  - Macro feedback effects vs. no macro feedback effects.
  - Income-statement adjustment (endogenous reaction to capital) vs. no adjustment.
- Key projection outcomes under baseline (comparative findings):
  - Models with no income-statement adjustment:
    - "No macro-feedback effects and no income statement adjustments" projects capital ratio falls by 50 bps over next two years mostly due to lower output gap.
    - Including macro-feedback effects leads to an additional fall of 90 bps (total additional), as lower capital buffers reduce credit supply and output (by about 100 bps).
    - Interest rates projected to drop by almost 200 bps more in the model with macro-feedback effects due to larger falls in output and inflation.
  - Models with income-statement adjustment:
    - Aggregate capital ratio projected to reach 16 percent by the end of the stress-testing horizon as banks increase net income in response to higher Basel III deductions and higher supervisory thresholds.
    - Feedback effects remain important but are smaller because initial capital fall is mitigated by banks’ adjustments; credit and output decrease less than in no-adjustment case.
    - Despite higher capital ratios, credit and output still fall because credit depends on difference between capital ratio and regulatory threshold; Basel III transition raises regulatory requirements.

### Forecast accuracy and applications
- Root-mean-squared forecast errors (Table 2) indicate models with income-statement adjustment have better forecast accuracy than models with no adjustment.
- Including macro-feedback effects does not significantly improve forecast accuracy in Brazil, possibly because links between credit, capital and output are more complicated than modeled.
- Uses:
  - Strengthening stress testing: generate endogenous scenarios accounting for feedbacks between banking sector and real economy.
  - Strengthening macro-financial analysis: medium-term projections, business and credit cycles, linkages between banks and real economy, policy simulation (monetary and macroprudential).
  - Building consistent baseline and risk scenarios for surveillance.

### Impulse responses and historical decompositions (selected quantitative findings)
- Impulse responses to 1 percent shocks (aggregate/bank-level):
  - Credit responds more to output than output responds to credit; peak impacts occur around one year and persist for about 2 years.
  - Macro-feedback effects double private and public credit responses to a demand shock due to impact on capital buffers.
  - Private credit increases by 2 percent following a positive output shock; public credit increases by around 1.5 percent.
  - A 1 percent shock to capital across all banks causes credit to drop by 6 percent and output to fall by 0.6 percent (capital → credit → output).
- Effects of specific shocks on banking income statement (Figure 7):
  - Positive demand shock: increases net interest income, non-interest revenue, non-interest expense, unrealized gains; reduces provisions → higher net income and capital.
  - Positive interest rate shock: lowers profits and aggregate capital (higher funding costs, credit losses, unrealized losses); interest income does not necessarily increase.
  - FCI shock effects are similar to interest rate shock (funding spread impact).
- Provisions (asset-quality) shock:
  - Increases provisions → lowers net income, aggregate capital, credit supply, output and inflation; monetary easing partially offsets via demand and bank balance-sheet channels.
  - Macro-feedback effects prolong negative impacts through larger effects on output and FCI.
- Historical decomposition for Brazil (1999 onward):
  - Private credit shocks boosted output pre-2008; public credit boosted output following the crisis (2009–10).
  - Both public and private capital shocks weakened output and credit during 2008/2009; private capital shocks supported post-crisis recovery via equity issuance and lending cuts raising capital ratios.
  - Financial conditions shocks important during 2008/2009 and recovery; looser conditions aided 2009 recovery until tightening in 2013.
  - Since early 2015, public and private credit and FCI shocks have been a drag on output; private and public capital shocks have been positive contributors since mid-2015 (new equity issuance, dividend cuts, slowdown in RWAs).

### Policy simulations and implications
- The framework can simulate:
  - Higher capital buffer imposed on an individual bank.
  - Countercyclical capital buffer for the banking sector as a macroprudential policy reaction.
- Illustration: A higher capital requirement for all banks reduces output by lowering the gap between bank capital and regulatory threshold, inducing a drop in credit supply and aggregate demand.
  - Estimated impacts in Brazil are in line with literature measuring higher capital effects on credit and output.
- Interaction with monetary policy: Higher capital → lower output and inflation → induces monetary policy loosening.
- Endogenous income-statement adjustments matter: with such adjustments banks can quickly change business models to increase net income and meet capital requirements; without adjustments, banks may have lower capital ratios after higher requirements.

### Future work: possible extensions
- Model can incorporate additional channels at the cost of complexity; key extensions include:
  - Non-linearities between capital and credit.
  - Liquidity risk (funding liquidity, market liquidity) and their interaction with solvency.
  - Network contagion and spillover effects from counterparty losses.
  - Explicit modeling of risk-weighted assets, capital, and income-statement items to remove approximation error (acknowledged approximation error when scaling by RWAs is treated as deductions in current model).
- Enhancing microprudential and macroprudential aspects would improve stress-testing, notably capturing solvency–liquidity–contagion interactions that can generate crisis-era non-linearities.

*Appendix I. Banks: Stress-Testing Module*

### references therein.

### wp17149 - references therein

### Macro-feedback and capital–credit interactions
- The framework combines a solvency stress test with a macroeconomic model to capture first-round effects on individual banks and amplified feedback effects on the economy via bank deleveraging.
- Key quantified impacts from model experiments:
  - Capital ratio falls by 1.2 percent (trough) following a provisions shock of 1 percent in the model with no macro-feedback effects and no link between solvency and liquidity.
  - Considering macro-feedback effects, the capital ratio goes down by an additional 20 bps and output falls by 0.4 percent.
  - The impact of the interlinkages between solvency and liquidity on the aggregate capital ratio amounts to 60 bps.
  - Output drops by 1.3 percent following a capital ratio shock when solvency–liquidity interactions are included — described as "the impact of interaction between solvency and liquidity on output is three times as large as in the model without the link between solvency and liquidity."

### Incorporating liquidity risks and solvency–liquidity link
- Funding costs are modeled using the financial conditions index (FCI), which depends on the expectation of future output growth (relative to trend).
- An extension to capture solvency–liquidity interactions:
  - The FCI equation is extended to include the aggregate capital buffer (ܭ
௧
), a weighted average of capital buffer of the six banks examined, to capture solvency effects on liquidity and subsequent consequences for future solvency (equation (13) in the source).
- Impulse responses indicate that the solvency–liquidity link enlarges macro-feedback effects (see quantified impacts above).

### Contagion and second-round effects
- Contagion can be embedded by adding counterparty losses due to contagion in banks’ income statements.
  - If interbank bilateral exposure data are available, use a loss-cascading, iterative mechanism (Sole and Espinosa, 2013) to calculate counterparty credit losses and account for liquidity losses and fire-sale effects.
  - If interbank data are missing, use simulation techniques based on entropy maximization (Anand and others, 2014).
- Non-linearities between capital and credit:
  - Banks’ loan-supply responses vary with proximity of capital ratios to regulatory thresholds; lending decisions are more constrained as capital approaches or falls below regulatory thresholds.
  - Proposal: add an additional credit buffer term (possibly quadratic) that affects credit supply when the capital buffer is close to zero or negative to proxy non-linearities from solvency, liquidity, and contagion interactions.
  - Calibration challenges: non-linear effects are difficult to estimate if banks have not experienced solvency problems historically; cross-country calibration of elasticities is an imperfect alternative.

### Macroprudential policy modeling and monetary interaction
- The macro module can be extended to incorporate a countercyclical capital buffer that depends on credit gap or credit growth:
  - Countercyclical capital buffer increases capital requirements in good times and decreases them in recessions.
  - Reaction of macroprudential and monetary policy following adverse events would reduce the impact of initial solvency shocks on the real economy and mitigate macro-feedback effects.

### Modeling banks’ optimizing behavior and structural approaches
- Structural, general-equilibrium frameworks can replace ad-hoc portfolio allocation and dividend policies with optimizing behavior:
  - Corbae and others (2016) build a structural framework for stress testing and show structural predictions for capital shortfalls deviate from static ad-hoc counterparts.
  - ECB stress-testing framework embeds a dynamic balance sheet module to model optimizing responses but does not model macro-feedback loops.
- Challenges: building fully endogenous models of liquidity and solvency interactions is difficult given definitional and modeling complexities.

### Improving micro-prudential aspects of the stress-testing module
- The paper’s stress-testing module is intentionally simple to be operational with publicly-available data, but can be extended toward an FSAP-style supervisory test if supervisory data are available:
  - Suggested extensions in an ideal data-rich case:
    - (i) calculating net interest income using maturity gap analysis;
    - (ii) modeling provisions for credit losses using PDs, LGDs for each asset class;
    - (iii) calculating realized and unrealized gains/losses on available for sale securities using the duration approach;
    - (iv) modeling credit risk RWAs for each asset class using the IRB formula and projected PDs, LGDs;
    - (v) imposing a dividend distribution rule.

### Additional caveats and identification issues
- Use of credit stock versus flow:
  - The framework uses the stock of credit because data on the flow of new credit are too short; GDP growth tends to be more closely related to growth in the flow of new credit rather than to growth in the credit stock.
- Identification of credit demand vs credit supply:
  - Uncovering the effect of capital on credit growth requires disentangling supply effects from demand effects; simultaneity and endogeneity issues remain:
    - Banks’ capital, income, expenses, and loan growth are endogenously determined.
    - Lagging capital does not solve identification because banks may increase capital preemptively in anticipation of loan volume increases.
  - Recommended: use identification strategies from the literature (e.g., Carlson and others, 2011; Jimenez and others, 2012).

### Policy implications and recommended model extensions
- To strengthen systemic risk analysis and stress testing:
  - Incorporate interactions between solvency, liquidity, and contagion risks to quantify likely impacts of adverse conditions on banking sector and real economy.
  - Model countercyclical capital buffers and study interactions with monetary policy.
  - Move toward structural, dynamic balance-sheet models that endogenously determine bank reactions, while acknowledging the current infancy of structural stress-testing.
  - Improve microprudential detail when supervisory data permit, following the FSAP-style checklist above.

*Source: wp17149 - references therein.*

### APPENDIX I: MODEL DETAILS

### APPENDIX I: MODEL DETAILS

### Behavioral Equations
- IS Curve:
  - y_t^p = (equation form with lags, real interest rate r_t, credit gap, foreign output and other terms as specified in the source).
- Phillips Curve (Non-Regulated-Price inflation):
  - π_t^e = γ_1 π_{t-1}^e + (1−γ_1) π_{t+1}^e + γ_2 y_t + γ_3 Δk_t + shocks (as specified).
- Regulated-Price Inflation:
  - π_t^r = π_t^{r*} + (θ_1−1) π_t^r + shock.
- Policy Rule:
  - i_t = ρ_1 i_{t-1} + (ρ_1−1) [φ(i_t, π̄_t, π_t, π_t^*)] + shock (policy rule structure as specified).
- Real Interest Rate (Fisher Equation):
  - r_t = i_t − E_t π_{t+1}.
- Real Credit Gaps:
  - For private credit: ψ_{p,t}^{gap} = ν_1 ψ_{p,t-1}^{gap} + ν_2 y_{t-1} + other lags (structure as specified).
  - For public credit: ψ_{g,t}^{gap} = ν_1 ψ_{g,t-1}^{gap} + ν_2 y_{t-1} + other lags (structure as specified).
- Financial Conditions:
  - FCI and related financial blocks included (structure and shocks as specified).

Note: All shocks (denoted ε_{x,t} for variable x_t) are assumed to be independently and identically distributed white noise processes.

### Stochastic Processes and Definitions
- Output gap:
  - y_t = y_t^o − y_t^p, where y_t is log real GDP and y_t^p is potential output.
- Potential output:
  - y_t^p = y_{t-1}^p + 1/4 g_t + η_{p,t}.
- Potential output growth:
  - g_t = (ρ_g) g_{t-1} + η_g,t, where g is steady state annual real GDP growth.
- Real credit gap:
  - c_{t}^{gap} = c_{t} − c_{t}^{trend}, where c_t is log real credit and c_t^{trend} is trend real credit = f(x̄).
- Real credit trend:
  - c_{t}^{trend} = c_{t-1}^{trend} + 1/4 q_t^{c} + η_{c,t}.
- Real credit trend growth:
  - q_t^{c} = ρ_{q} q_{t-1}^{c} + η_{q,t}, where g^c is steady state annual real credit growth.
- Inflation target:
  - π_t^* = π_{t-1}^* + η_{π^*,t}.
- Headline inflation:
  - π_t = π_t^e + (α_1) π_t^r, where π_t^e is non-regulated-price inflation and π_t^r is regulated-price inflation.
- Annual headline inflation:
  - π_t^{annual} = 1/4 [π_t + π_{t-1} + π_{t-2} + π_{t-3}].
- Real interest rate gap:
  - r_t^{gap} = r_t − r_t^{trend}, where r_t is real interest rate and r_t^{trend} is trend real interest rate.
- Trend real interest rate:
  - r_t^{trend} = r_{t-1}^{trend} + η_{r,t}.
- Unemployment gap:
  - u_t^{gap} = u_t − u_t^{NAIRU}, where u_t is unemployment rate and u_t^{NAIRU} is NAIRU.
- NAIRU:
  - u_t^{NAIRU} = u_{t-1}^{NAIRU} + η_{u,t}.
- Capacity utilization gap:
  - cu_t^{gap} = cu_t − cu_t^{trend}, where cu_t is (log) capacity utilization and cu_t^{trend} is its trend.
- Trend capacity utilization:
  - cu_t^{trend} = cu_{t-1}^{trend} + η_{cu,t}.
- Real exchange rate gap:
  - q_t^{gap} = q_t − q_t^{trend}, where q_t is (log) real effective exchange rate and q_t^{trend} is trend.
- Trend real exchange rate:
  - q_t^{trend} = q_{t-1}^{trend} + η_{q,t}.
- Foreign output gap:
  - y_t^{*} = y_t^{*,o} − y_t^{*,p}, where y_t^{*} is log U.S. real GDP and y_t^{*,p} is foreign potential output.
- Foreign potential output:
  - y_t^{*,p} specified with lags and shock terms (structure as specified).

### Estimated and Calibrated Parameters — Key Values Preserved Exactly

- Table 3. Calibrated Parameters:
  - g = 2.00
  - g^k = 5.00
  - ρ_g = 0.05
  - ρ_q = 0.05
  - σ (unspecified) = 0.05
  - α = 0.75
  - Note: The shock standard deviations for the trends of all variables are calibrated based on trends extracted using a standard HP filter (i.e., with smoothing parameter of 1600).

- Table 4. Estimated Parameters (selected transitory parameters and posteriors):
  - σ1 (prior β(0.2,0.05)) — Posterior: 0.38, Std Dev.: 0.04
  - σ2 (prior σ(0.35,0.05)) — Posterior: 0.24, Std Dev.: 0.03
  - σ3 (prior σ(0.1,0.025)) — Posterior: 0.08, Std Dev.: 0.03
  - ρ1 (prior β(0.8,0.05)) — Posterior: 0.66, Std Dev.: 0.05
  - ρ2 (prior β(0.1,0.025)) — Posterior: 0.07, Std Dev.: 0.02
  - ρ3 (prior σ(0.35,0.05)) — Posterior: 0.31, Std Dev.: 0.04
  - ρ4 (prior σ(0.05,0.025)) — Posterior: 0.02, Std Dev.: 0.01
  - ρ5 (prior σ(0.5,0.2)) — Posterior: 0.28, Std Dev.: 0.07
  - ρ6 (prior σ(0.5,0.2)) — Posterior: 0.07, Std Dev.: 0.03
  - ρ7 (prior σ(1,0.2)) — Posterior: 1.07, Std Dev.: 0.06
  - θ1 (prior β(0.8,0.025)) — Posterior: 0.76, Std Dev.: 0.03
  - θ2 (prior σ(1.5,0.05)) — Posterior: 1.51, Std Dev.: 0.10
  - θ3 (prior σ(0.2,0.025)) — Posterior: 0.20, Std Dev.: 0.05
  - υ1 (prior β(0.5,0.1)) — Posterior: 0.58, Std Dev.: 0.10
  - υ2 (prior σ(0.8,0.2)) — Posterior: 0.78, Std Dev.: 0.07
  - υ3 (prior σ(1.5,0.1)) — Posterior: 1.14, Std Dev.: 0.04
  - λ1 (prior β(0.5,0.1)) — Posterior: 0.52, Std Dev.: 0.14
  - λ2 (prior σ(0.8,0.2)) — Posterior: 0.65, Std Dev.: 0.29
  - λ3 (prior σ(1.5,0.1)) — Posterior: 1.01, Std Dev.: 0.09
  - ω1 (prior β(0.5,0.1)) — Posterior: 0.26, Std Dev.: 0.10
  - ω2 (prior σ(0.5,0.2)) — Posterior: 0.43, Std Dev.: 0.07
  - δ1 (prior β(0.5,0.1)) — Posterior: 0.54, Std Dev.: 0.09
  - δ2 (prior σ(0.5,0.2)) — Posterior: 0.25, Std Dev.: 0.03
  - ι1 (prior β(0.5,0.1)) — Posterior: 0.41, Std Dev.: 0.05
  - φ2 (prior σ(0.8,0.2)) — Posterior: 0.48, Std Dev.: 0.07
  - ι3 (prior σ(0.8,0.05)) — Posterior: 0.74, Std Dev.: 0.06
  - ρ_π (prior β(0.5,0.1)) — Posterior: 0.42, Std Dev.: 0.05
  - ζ (prior β(0.5,0.1)) — Posterior: 0.60, Std Dev.: 0.08
  - η (prior β(0.5,0.1)) — Posterior: 0.75, Std Dev.: 0.04

- Table 4. Estimated Shocks (posterior standard deviations, priors drawn from σ_{ε} ~ σ(1,∞)):
  - σ_{ε_y} = 0.36, Std Dev.: 0.04
  - σ_{ε_π^e} = 1.60, Std Dev.: 0.14
  - σ_{ε_π^r} = 0.90, Std Dev.: 0.04
  - σ_{ε_i} = 0.44, Std Dev.: 0.04
  - σ_{ε_q} = 1.40, Std Dev.: 0.25
  - σ_{ε_u} = 0.14, Std Dev.: 0.02
  - σ_{ε_cu} = 0.12, Std Dev.: 0.02
  - σ_{ε_y^*} = 0.19, Std Dev.: 0.02
  - σ_{ε_x} = 0.17, Std Dev.: 0.02

- Table 5. Estimated Banking Parameters (selected posterior values):
  - κ_{bank,1} (prior σ(0.9,0.1)) — Posterior: 0.61, Std Dev.: 0.01
  - κ_{bank,2} (prior σ(0.4,0.05)) — Posterior: 0.28, Std Dev.: 0.04
  - ψ_{bank,1} (prior σ(0.1,0.05)) — Posterior: 0.10, Std Dev.: 0.04
  - ι_{bank,1} (prior σ(~0.1,0.05)) — Posterior: 0.00, Std Dev.: 0.00
  - κ_{bank,trend} (prior σ(0.9,0.1)) — Posterior: 0.25, Std Dev.: 0.00
  - Various other banking parameters reported with posteriors at or near 0.00 or small positive/negative values; examples include:
    - κ_{asset} Posterior: 0.41
    - κ_{liquid} Posterior: 0.94
    - κ_{reserves} Posterior: 0.91
    - κ_{capital} Posterior: 0.77, Std Dev.: 0.03
    - d_{capital} Posterior: 0.20, Std Dev.: 0.04
    - ψ_{capital} Posterior: 0.08
    - κ_{stress} Posterior: 0.35
    - κ_{risk} Posterior: 0.94, Std Dev.: 0.03
    - κ_{profitability} Posterior: 0.91, Std Dev.: 0.04
    - κ_{leverage} Posterior: 0.76
    - ψ_{other} Posterior: 0.86, Std Dev.: 0.03
  - Note: All prior distributions of the error standard deviations for the banking model are drawn from σ(1,∞). These posterior estimates are available on request.

### Model Estimation Notes
- The model is estimated using the Kalman Filter and Bayesian estimation.
- Prior distributions and posterior means/standard deviations are reported in Tables 3–5 (selected values above).
- For Bayesian estimation methodology reference, see Herbst and Schorfheide (2015) (manuscript cited in source).

*Source: APPENDIX I: MODEL DETAILS, wp17149 - APPENDIX I: MODEL DETAILS.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17149.pdf_