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### Literature and contribution
- Situates the paper in a large literature modeling failure of financial institutions (examples cited: Krasa and Villamil [1992]; Hirakata, Sudo, and Ueda [2011, 2013]; Zeng [2013]; Benes and Kumhof [2011]; Jin and Zeng [2011]).
- Contribution: analyzes financial sector default driven by aggregate risk (as opposed to idiosyncratic-only explanations).
- Relates to macroprudential policy literature (Lorenzoni [2008] onward); contrasts gaps identified in this paper (differences between actual and targeted levels of capital) with typical macroprudential externality rationale.

### Model overview and agents
- Six agents: households, final good firms, (entrepreneurial) firms, financial intermediaries (FIs), mutual funds, and the government.
- Financial system built from firms, financial intermediaries, and mutual funds; real economy from households, final good firms, and government.
- Household injections into firms and intermediaries: n_{F,t}, n_{B,t}; fixed equity injections ω_F and ω_B plus flexible fractions z_{u,t}(1−γ_F) and z_{k,t}(1−γ_B), subject to pay-out shocks z_{u,t} and z_{k,t}.
- Funding d_t provided to mutual funds; mutual funds fund FIs; intermediaries combine equity n_{B,t} and funding d_t to supply loans b_t (n_{B,t}+d_t = b_t).
- Firms invest equity n_{F,t} and loans b_t in productive assets k_t at price q_t (n_{F,t}+b_t = q_t k_t).
- Aggregation/market clearing: n_{F,t}+n_{B,t}+d_t = q_t k_t and c_t + i_t + g_t = y_t.

### Households — preferences, budget, and first-order conditions
- Objective: maximize E_0 Σ_{t=0}^∞ β^t U_t with CRRA and internal habit h>0; labor disutility shock z_{n,t}, parameter τ_n; household sends n_t members to work.
- Budget constraint components include q_t k_t outlays; income F_t R_{k,t} k_{t-1} + w_t n_t + R_{t-1} b_{g,t-1}; investment adjustment cost S(·) with shock z_{i,t} enters i_t(1+S(z_{i,t} i_t / i_{t-1})).
- Macro-financial feedback factor F_t appears in return on capital.
- Key first-order conditions (equation numbering preserved):
  - ∂k_t: −q_t + β E_t[ (λ_{t+1}/λ_t) F_{t+1} R_{k,t+1} ] = 0 (2)
  - R_{k,t} = r_{k,t} + (1−δ) q_t (3)
  - ∂b_t: −1 + β E_t[ (λ_{t+1}/λ_t) R ] = 0 (4)
  - ∂n_t: −z_{n,t} τ_n φ_t + λ_t w_t = 0 (5)
  - ∂c_t: (c_t − h c_{t−1})^{−σ} − h β E_t[(c_{t+1} − h c_t)^{−σ}] − λ_t = 0 (6)
  - ∂i_t: q_t − (1 + S(·) + S′(·) · ) + β E_t[ (λ_{t+1}/λ_t) S′(·) · ] = 0 (7)
  - Capital adjustment cost: S( z_{i,t} i_t / i_{t-1} ) = φ_i/2 ( z_{i,t} i_t / i_{t-1} − 1 )^2 (8)
  - Stochastic discount factor: M_{t+1} = β [λ_{t+1}/λ_t] (9).

### Final good firms
- Production: y_t = k_{t−1}^α (z_{y,t} n_t)^{1−α} (11).
- Profit maximization: max { y_t − r_t k_{t−1} − w_t n_t } (10).
- First-order conditions:
  - r_t = α y_t / k_{t−1} (12)
  - w_t = (1−α) y_t / n_t (13)
- Market clearing restated: c_t + i_t + g_t = y_t (14).

### Entrepreneurial firms — financing, returns, and default
- Firms finance q_t k_t with equity n_{F,t} and debt b_t; next period receive R_{k,t+1} and repay R_{b,t} = 1 + i_{b,t}.
- Effective capital production: z_{t+1} ε_{i,t+1} k_t with ε_{i,t+1} (idiosyncratic) and z_{t+1} (aggregate) log-normal, unit means; standard deviations σ_{z,t} and σ_{ε,t}. Shocks to these interpreted as systemic (aggregate) and credit (idiosyncratic) risk shocks.
- Firm objective (cash flow) in Lagrangian: −n_{F,t} + E_t[ M_{t+1} max( (z_{t+1} ε_{i,t+1} R_{k,t+1} k_t − R_{b,t} b_t) (1−τ), 0 ) ] (15).
- Default threshold:
  - ε_{i,t+1} < ε_{z,t+1} ≡ (R_{b,t} b_t) / ( z_{t+1} R_{k,t+1} k_t ) (16).
- Probability of firm default: PD_F( ε_{z,t+1} ) = Prob( ε_{i,t+1} < ε_{z,t+1} ) = F( ε_{z,t+1}, σ_{ε,t+1} ) (17).
- Earnings retention and loss functions:
  - Δ(ε_z) ≡ ∫_0^{ε_z} ε f(ε) dε (19)
  - Γ(ε_z) ≡ Δ(ε_z) + ε_z (1 − F(ε_z)) (20)
- Aggregate firm cash-flow (condensed): −n_{F,t} + E_t[ M_{t+1} ( z_{t+1} R_{k,t+1} k_t (1 − Γ(ε_{z,t+1})) ) (1−τ) ].

### Financial intermediaries (FIs) — structure, returns, default
- FIs: b_t = n_{B,t} + d_t; next period receive R_{B,t+1} on loans; pay R_{d,t} = 1 + i_{d,t} on funding.
- FI expected cashflow: −n_{B,t} + E_t[ M_{t+1} max( (R_{B,t+1} b_t − R_{d,t} d_t) (1−τ), 0 ) ] (21).
- Loan portfolio return accounts for non-defaulting interest and recoveries from defaulted firms with default cost fraction μ:
  - R_{B,t+1} b_t ≡ R_{b,t} b_t (1 − F(ε_{i,t+1})) + ∫_0^{ε_{z,t+1}} z_{t+1} ε_{i,t+1} R_{k,t+1} k_t f(ε) dε (1−μ) (22).
- FI cashflow rewritten using Γ and Δ: −n_{B,t} + E_t[ M_{t+1} max( z_{t+1} R_{k,t+1} k_t ( Γ(ε_{z,t+1}) − μ Δ(ε_{z,t+1}) ) − R_{d,t} d_t, 0 ) (1−τ) ] (23).
- FI default threshold z^*_{t+1} defined by:
  - z^*_{t+1} R_{k,t+1} k_t ( Γ(ε^*_{t+1}) − μ Δ(ε^*_{t+1}) ) − R_{d,t} d_t = 0 (25)
  - FI default probability: PD_B( z^*_{t+1} ) = Prob( z_{t+1} < z^*_{t+1} ) = G( z^*_{t+1}, σ_{z,t+1} ) (26).

### Productivity states and equilibrium implications for FIs
- Three realizations for productivity regions served by an FI depending on z_{t+1} relative to z^*_{t+1}:
  - Case 1 (no default): probability (1 − G(z^*_{t+1})), z_{t+1} = z_N; firms’ threshold ε_N ≡ R_{b,t} b_t z_N / ( R_{k,t+1} k_t ) (27).
  - Case 2 (threshold): z_{t+1} = z^*_{t+1}; ε^*_{t+1} ≡ R_{b,t} b_t z^*_{t+1} / ( R_{k,t+1} k_t ) (29).
  - Case 3 (default): probability G(z^*_{t+1}), z_{t+1} = z_D; ε_D ≡ R_{b,t} b_t z_D / ( R_{k,t+1} k_t ) (30).
- Each period fraction G(z^*_t) of FIs default and productivity in that part is z_D; survivors have productivity z_N.
- Aggregate FI objective (31): −n_{B,t} + E_t[ M_{t+1} ( z_N R_{k,t+1} k_t ( Γ(ε^N_{t+1}) − μ Δ(ε^N_{t+1}) ) − R_{d,t} d_t ) (1 − G(z^*_{t+1})) (1−τ) ].

### Mutual funds and intermediation returns
- Mutual funds lend d_t to FIs; payoff (32):
  - −d_t + E_t[ M_{t+1} ( R_{d,t} d_t (1 − G(z^*_{t+1})) + G(z^*_{t+1}) R_{k,t+1} k_t s^D_{B,t+1} (1 − μ_B) ) ].
- s^D_{B,t+1} defined (33): s^D_{B,t+1} = z_D ( Γ(ε^D_{t+1}) − μ Δ(ε^D_{t+1}) ).
- Mutual funds not taxed on gains; net benefit of intermediation equals τ R_{d,t} d_t (1 − G(z^*_{t+1})), which offsets default costs.

### Optimization, equilibrium constraints, and macro-financial feedback
- Aggregate resource identity: q_t k_t = n_{F,t} + n_{B,t} + d_t (34).
- Firms maximize expected cashflow subject to:
  - (i) zero-profit condition for FIs (38),
  - (ii) FI default threshold condition (40–41),
  - (iii) zero-profit condition for mutual funds (42–43).
- First-order condition for d_t:
  - ∂d_t : λ_{I,t} = E_t[ M_{t+1} R_{k,t+1} q_t X_{t+1} ] (44).
- X_{t+1} defined (45) and macro-financial feedback factor:
  - F_{t+1} ≡ X_{t+1} / λ_{I,t}.
- Interpretation: F_{t+1} depends on firm and intermediary leverage and on idiosyncratic and aggregate risk; firms choose variables to maximize F_{t+1}.

### Financial frictions and first-order conditions (selected)
- Two main frictions:
  - Payments of firms and FIs taxed at rate τ while mutual funds not taxed (can represent discount factor differences); yields benefit τ R_{d,t} d_t (1 − G(z^*_{t+1})).
  - Firm default triggers cost fraction μ of firms’ earnings.
- FI default threshold and funding chosen to trade off higher intermediation benefit vs. increased FI default likelihood G′(z^*_{t+1}) = g(z^*_{t+1}).
- First-order condition for z^*_{t+1} (47) simplified to:
  - −g(z^*_t) Z_t + λ_{V,t} ( Δ(ε^*_t) (1 − μ) + μ f(ε^*_t) ε^{*2}_t ) = 0 (52).
  - λ_{V,t} = g(z^*_t) Z_t / ( Δ(ε^*_t) (1 − μ) + μ f(ε^*_t) ε^{*2}_t ) (53).
- FI leverage and default threshold conditions equate tax benefit associated with higher collateral value with expected productivity loss from more intermediary defaults (see 54–55).

### Capital accumulation, targets, and capital gaps
- Laws of motion:
  - n_{F,t} = ( R_{k,t} q_{t−1} ( n_{F,t−1} + n_{B,t−1} + d_{t−1} ) ( s^N_{F,t} (1 − G(z^*_t)) + s^D_{F,t} G(z^*_t) ) (1−τ) (1 − γ_F) z_{u,t} ) + ω_F (56).
  - n_{B,t} = ( ( R_{k,t} q_{t−1} ( n_{F,t−1} + n_{B,t−1} + d_{t−1} ) s^N_{B,t} − R_{d,t-1} d_{t-1} ) (1 − G(z^*_t)) (1−τ) (1 − γ_{B,t}) z_{k,t} ) + ω_B (57).
- Aggregation: injections equal expected earnings minus equity payout plus ω_F and ω_B.
- Potential (target) levels from optimizing n_{F,t}, n_{B,t}, d_t, R_{d,t}, z^*_{t+1}, R_{b,t}; optimality implies λ_{B,t} = λ_{I,t} = 1 (58).
- Capital gaps computed as realized vs. target levels (targets imply higher welfare than realized cases with λ ≠ 1).

### Estimation strategy and data
- Estimation via standard Bayesian techniques per An and Schorfheide [2007].
- Sample period: 2000Q1 to 2019Q2.
- Observables: four macro variables and four financial variables.
  - Macros: real quarter-on-quarter percentage changes of GDP (dY), investment (dI), employment (dN), and consumption (dC); controlled for population growth, demeaned.
  - Financials: corporate credit spread (CCS), financial credit spread (FCS), bank Tier 1 capital growth (dnB), firm net worth growth (dnF).
- Construction details:
  - Investment = sum of total fixed investment and durable consumption.
  - Consumption = sum of non-durable consumption and consumption services.
  - CCS = percentage difference between iTraxx Industrial Corporate Bond Yield Index and 10-year treasury yield; demeaned (years 2008 and 2009 excluded from demeaning).
  - FCS = percentage difference between iTraxx Financials Yield Index and 10-year treasury yield; demeaned (years 2008 and 2009 excluded from demeaning).
  - dnB = percentage change in Tier 1 capital of all FDIC insured commercial banks and savings institutions; data from FDIC; demeaned and adjusted for inflation.
  - dnF = percentage change in the Down Jones 30 Stock Market Index; adjusted for inflation and demeaned.
- Measurement equations linking data to model variables (sensitivities retained):
  - (E(Rb,t+1) − Rt − (Rb,ss − Rss)) * 100 = CCSt / κC
  - (E(Rd,t+1) − Rt − (Rd,ss − Rss)) * 100 = FCSt / κF
  - (log(nF,t / nF,t−1)) * 100 = dnF / κnF
  - (log(nB,t / nB,t−1)) * 100 = dnB / κnB
- Concern: observed spreads and capital may imperfectly reflect lending and funding conditions (deposit funding, government guarantees can understate funding pressures).

### Shocks, priors, and estimation targets
- Model includes original RBC shocks plus four additional shock processes:
  - idiosyncratic risk shock,
  - aggregate risk shock,
  - shock to retained earnings/equity payout of firms,
  - shock to retained earnings/equity payout of FIs.
- Eight exogenous shocks defined and estimated (AR(1) log processes, preserved exactly):
  - ln(zy,t) = ρy ln(zy,t−1) + σy e y,t
  - ln(zi,t) = ρi ln(zi,t−1) + σi e i,t
  - ln(zk,t) = ρk ln(zk,t−1) + σk e k,t
  - ln(zn,t) = ρn ln(zn,t−1) + σn e n,t
  - ln(zg,t) = ρg ln(zg,t−1) + σg e g,t
  - ln(σε,t) = (1−ρε) ln(σε,ss) + ρε ln(σε,t−1) + σε e F,t
  - ln(σz,t) = (1−ρz) ln(σz,ss) + ρz ln(σz,t−1) + σε e B,t
  - ln(zu,t) = ρu ln(zu,t−1) + σu e u,t
- ej,t normally distributed. Estimated: autocorrelation coefficients ρj and standard deviations σj.
- Estimated steady-state parameters: σε,ss and σz,ss; sensitivity coefficients 1/κF, 1/κC, 1/κnF, 1/κnB; habit h; adjustment cost coefficients Φi, kF,p, kB,p.

### Calibration highlights and numeric priors/posterior notes
- Calibrated examples:
  - μ = 0.4 (recovery value 1−μ close to average recovery of senior secured bonds/loans).
  - τ = 0.2.
  - zD and zN set close to conditional expected values to limit dispersion.
  - Payout fractions γ_F and γ_B with higher payout for FIs.
- Prior of σε,ss = 0.23 so steady-state PD_F(εNt+1) ≈ 1.5 percent; posterior mode σε,ss = 0.245.
- Prior of σz,ss = 0.12 so steady-state PD_B(z∗t+1) ≈ 0.75 percent; posterior mode σz,ss = 0.157.
- Sensitivity coefficients posterior modes:
  - 1/κF = 0.12
  - 1/κC = 0.55
  - 1/κnF = 1.05
  - 1/κnB = 0.52
  - Interpretation: observed FCS maps weakly to model funding-cost variable (model changes amount to less than 12% of observed FCS changes); CCS and capital of firms and banks correspond better to the model.

### Selected calibrated and estimated parameter numeric values (examples)
- Calibrated parameters:
  - α: 0.300
  - β: 0.990
  - δ: 0.025
  - μ: 0.400
  - μ_B: 0.200
  - τ: 0.200
  - γ_F: 0.030
  - γ_B: 0.100
  - z_D: 0.700
  - z_N: 1.010
- Selected posterior modes and SDs (examples):
  - ρ_y: Post. mode 0.9537; Post. SD 0.0238
  - ρ_i: Post. mode 0.6983; Post. SD 0.0637
  - ρ_g: Post. mode 0.9337; Post. SD 0.0393
  - ρ_n: Post. mode 0.9643; Post. SD 0.0161
  - ρ_u: Post. mode 0.8575; Post. SD 0.0268
  - ρ_k: Post. mode 0.2965; Post. SD 0.0947
  - ρ_F: Post. mode 0.9605; Post. SD 0.0150
  - ρ_B: Post. mode 0.9293; Post. SD 0.0311
  - σ_y: Post. mode 0.0080; Post. SD 0.0006
  - σ_i: Post. mode 0.0269; Post. SD 0.0039
  - σ_g: Post. mode 0.0170; Post. SD 0.0013
  - σ_n: Post. mode 0.0096; Post. SD 0.0008
  - σ_k: Post. mode 0.0140; Post. SD 0.0019
  - σ_u: Post. mode 0.0070; Post. SD 0.0007
  - σ_F: Post. mode 0.0199; Post. SD 0.0026
  - σ_B: Post. mode 0.0227; Post. SD 0.0028
  - σ_F ss: Post. mode 0.2449; Post. SD 0.0122
  - σ_Bss: Post. mode 0.1570; Post. SD 0.0111
  - κ_C: Post. mode 1.8201; Post. SD 0.1993
  - κ_F: Post. mode 8.4630; Post. SD 0.7039
  - κ_nF: Post. mode 0.9501; Post. SD 0.1297
  - κ_nB: Post. mode 1.9324; Post. SD 0.1838
  - Φ_i: Post. mode 3.1199; Post. SD 0.5663
  - h: Post. mode 0.4811; Post. SD 0.0537
  - κ_F p: Post. mode 0.0552; Post. SD 0.0214
  - κ_Bp: Post. mode 0.0754; Post. SD 0.0200

### Forecast error variance decomposition — key percentages
- Output: TFP 58.42; Investment 1.42; FIs capital 0.00; Labor mkt. 27.53; Gov. spending 2.69; Credit risk 3.95; Agg. risk 1.42; Firm’s capital 4.56.
- Investment: TFP 16.32; Investment 15.46; FIs capital 0.04; Labor mkt. 6.38; Gov. spending 0.07; Credit risk 16.39; Agg. risk 0.29; Firm’s capital 36.13.
- Consumption: TFP 57.63; Investment 1.41; FIs capital 0.00; Labor mkt. 27.69; Gov. spending 4.06; Credit risk 3.50; Agg. risk 1.33; Firm’s capital 4.37.
- CCS: Credit risk 49.72; Firm’s capital 16.12; Agg. risk 11.26; TFP 8.28; Investment 4.40; FIs capital 7.08; Labor mkt. 3.07; Gov. spending 0.07.
- FCS: Agg. risk 34.16; Firm’s capital 26.81; Credit risk 23.83; TFP 6.99; Investment 4.61; FIs capital 1.14; Labor mkt. 2.39; Gov. spending 0.07.
- Firm’s probability of default (F): Credit risk 63.11; Firm’s capital 18.62; TFP 7.34; Agg. risk 5.19; Investment 2.15; others smaller.
- FIs’ probability of default (GZ): Agg. risk 35.63; Firm’s capital 28.27; Credit risk 19.07; TFP 10.02; Investment 2.67; others smaller.
- FIs capital (dnB): FIs capital 39.90; Firm’s capital 20.05; Credit risk 20.26; TFP 11.26; others smaller.
- Firms capital (dnF): Firm’s capital 59.49; TFP 21.31; Labor mkt. 7.15; others smaller.

### Main findings — historical decomposition, impulse responses, and diagnostics
- Impulse responses to a 1% increase in probability of default of both FIs and firms:
  - Output, consumption, labor and investment decrease.
  - Financial and corporate spreads substantially increase.
  - Capital gaps and probabilities of default for both FIs and firms rise.
- Historical decomposition:
  - Investment growth mainly driven by: equity price shock, idiosyncratic risk shock, aggregate risk shock, TFP shock, and investment efficiency shock.
  - CCS and FCS mainly driven by: aggregate risk shocks and firm capital shocks (confirmed by forecast error variance decomposition).
- Firm default probability: higher corporate risk associated with 2001 Dot-com bubble and the GFC; private sector risk recently driving up CCS.
- Corporate spreads and equity prices:
  - Corporate spreads described as too low (Figure 7 referenced).
  - Equity prices were too high prior to COVID-19, attributed to corporate income shocks and risk shocks (Figure 8 referenced).
- Banking sector resilience:
  - Bank capital buffers recently declined due to equity payout.
  - Three episodes with positive capital gaps (more capital needed): prior to Dot-com bubble, prior to the GFC, and the recent period.
  - Declining recent trend increases FIs vulnerability to adverse shocks.

### Counterfactual scenario (stress simulation)
- Stress test: apply GFC shocks from 2008–2011 to simulate a macro stress test over 2019–2022.
- Results:
  - Both banks and firms more resilient to adverse shocks today compared to pre-GFC period.
  - Financial intermediaries more vulnerable than firms at the current juncture.
  - Corporate and banking risk gaps elevated under adverse conditions.
  - Under adverse shocks similar to the GFC, deleveraging and severe recession would be expected.
  - Policy space noted: room to activate the countercyclical capital buffer.

### Policy implications and lessons for COVID-19
- When risk and capital gaps are identified, capital should be increased in the respective sector by earnings retention or issuance of equity, or by any policy measure that closes capital and risk gaps while accounting for general equilibrium effects.
- Model uses:
  - Signaling device for macroprudential intervention (ex-ante and contemporaneous).
  - Gauge of whether macroprudential action was successful (ex-post).
- Historical policy implication examples:
  - More capital or macroprudential policy needed before/during: Dot-com bubble (firm and FIs capital), Global Financial Crisis (firm and FIs capital), and the recent period (FIs capital).
  - Countercyclical capital buffer could have been activated prior to COVID-19 to guard against deleveraging.
- COVID-19 lessons and recommendations:
  - Under adverse conditions, default risk in both corporate and financial sectors increases while corporate equity capital and bank capital become scarce; reflected in stock market losses after COVID-19 outbreak.
  - Firms and banks should stop paying dividends and build/sustain capital levels to absorb expected losses.
  - Especially important for banks that increased equity payout prior to COVID-19, to prevent a real-economy crisis turning into a financial crisis.
  - Capital in firms and banks addresses solvency and liquidity concerns and strengthens system liquidity.

### Additional quantitative results and diagnostics
- From "Additional results":
  - Credit risk and systemic risk shocks explain an important share of investment dynamics: "16.3% and 9.3%".
  - Credit risk shock explains largest share in variation of CCS and firm’s probability of default (F).
  - Aggregate risk shock main driver of forecast variation in FCS and FIs’ probability of default (GZ).
  - Shock to firm’s income explains most variation in firm’s capital: "59.5%"; also large shares in investment ("36%"), CCS ("16.12%"), FCS ("26.81%"), F ("18.62%"), GZ ("28.27%").
  - Shock to FIs’ capital is main driver of FIs’ capital ("40%") with credit risk shock ("20.26%") and firm’s capital shock ("20%").
  - Risk shocks explain little in consumption; consumption driven mainly by TFP and labor market tightening shocks.
- Model-based measures (aggregate risk per equation 69 and FIs probability of default) move reasonably close to financial uncertainty index of Jurado, Ludvigson, and Ng [2015] despite that index not used in estimation; high correlation reported with macroeconomic uncertainty measure.

_Italic: Content based only on the supplied PDF excerpt._

### conclusions based on the application to the data).

### wpiea2020209-print-pdf - conclusions based on the application to the data).

### Literature and contribution
- Situates the paper in a large literature modeling failure of financial institutions, noting prior work (e.g., Krasa and Villamil [1992], Hirakata, Sudo, and Ueda [2011, 2013], Zeng [2013], Benes and Kumhof [2011], Jin and Zeng [2011]).
- Contribution: analyzes financial sector default driven by aggregate risk (as opposed to idiosyncratic-only explanations).
- Relates to growing macroprudential policy literature (Lorenzoni [2008] onward); contrasts gaps identified in this paper (differences between actual and targeted levels of capital) with typical macroprudential externality rationale.

### Model overview and agents
- Economy has six agents: households, final good firms, (entrepreneurial) firms, financial intermediaries, mutual funds, and the government.
- Financial system elements: firms, financial intermediaries, and mutual funds build the financial system; households, final good firms, and government build the real economy (standard part).
- Household injections into firms and intermediaries:
  - Funds invested in firms and intermediaries equal to n_{F,t} and n_{B,t}.
  - Fixed equity injections ω_F and ω_B plus flexible fractions z_{u,t}(1-γ_F) and z_{k,t}(1-γ_B) of earnings, subject to pay-out shocks z_{u,t} and z_{k,t}.
- Funding d_t is provided to mutual funds; mutual funds fund financial intermediaries; intermediaries combine equity n_{B,t} and funding d_t to provide commercial loans b_t (n_{B,t}+d_t = b_t).
- Firms invest equity n_{F,t} and loans b_t in productive assets k_t at price q_t (n_{F,t}+b_t = q_t k_t).
- Aggregation: invested funds in financial sector ultimately invested in productive assets: n_{F,t}+n_{B,t}+d_t = q_t k_t.
- Market clearing condition: c_t + i_t + g_t = y_t.

### Households (representative)
- Objective: maximize lifetime utility E_0 Σ_{t=0}^∞ β^t U_t with CRRA and internal habit h>0.
- Labor disutility shock z_{n,t} and parameter τ_n; household sends n_t members to work.
- Budget constraint components (excerpt from Lagrangian):
  - Outlays equal to q_t k_t and income equal to F_t R_{k,t} k_{t-1} plus wage w_t n_t and bond returns R_{t-1} b_{g,t-1}.
  - Investment adjustment cost S(·) with shock z_{i,t} enters gross cost term i_t(1+S(z_{i,t} i_t / i_{t-1})).
- Macro-financial feedback factor F_t appears in return on capital for household first-order condition.
- Key first-order conditions (listed with equation references preserved):
  - ∂k_t: −q_t + β E_t[ (λ_{t+1}/λ_t) F_{t+1} R_{k,t+1} ] = 0 (2)
  - R_{k,t} = r_{k,t} + (1−δ) q_t (3)
  - ∂b_t: −1 + β E_t[ (λ_{t+1}/λ_t) R ] = 0 (4)
  - ∂n_t: −z_{n,t} τ_n φ_t + λ_t w_t = 0 (5)
  - ∂c_t: (c_t − h c_{t−1})^{−σ} − h β E_t[(c_{t+1} − h c_t)^{−σ}] − λ_t = 0 (6)
  - ∂i_t: q_t − (1 + S(·) + S′(·) · ) + β E_t[ (λ_{t+1}/λ_t) S′(·) · ] = 0 (7)
  - Capital adjustment cost: S( z_{i,t} i_t / i_{t-1} ) = φ_i/2 ( z_{i,t} i_t / i_{t-1} − 1 )^2 (8)
- Defines stochastic discount factor: M_{t+1} = β [λ_{t+1}/λ_t] (9).

### Final good firms
- Production and maximization:
  - max { y_t − r_t k_{t−1} − w_t n_t } (10)
  - y_t = k_{t−1}^α (z_{y,t} n_t)^{1−α} (11)
- First-order conditions:
  - ∂k_{t−1}: r_t = α y_t / k_{t−1} (12)
  - ∂n_t: w_t = (1−α) y_t / n_t (13)
- Market clearing: c_t + i_t + g_t = y_t (14).

### Entrepreneurial firms: financing, returns, and default
- Firms finance q_t k_t with equity n_{F,t} and debt b_t from intermediaries; next period receive R_{k,t+1} and repay R_{b,t} = 1 + i_{b,t}.
- Production of effective capital: z_{t+1} ε_{i,t+1} k_t where ε_{i,t+1} (idiosyncratic) and z_{t+1} (aggregate) are log-normal with unit means; standard deviations σ_{z,t} and σ_{ε,t}.
- Shocks to law of motion of these standard deviations interpreted as systemic (aggregate) and credit (idiosyncratic) risk shocks.
- Firm objective (cash flow) in Lagrangian form includes tax at rate τ and discounting by household SDF M_{t+1}:
  - −n_{F,t} + E_t[ M_{t+1} max( (z_{t+1} ε_{i,t+1} R_{k,t+1} k_t − R_{b,t} b_t) (1−τ), 0 ) ] (15)
- Firm default threshold:
  - ε_{i,t+1} < ε_{z,t+1} ≡ (R_{b,t} b_t) / ( z_{t+1} R_{k,t+1} k_t ) (16)
- Probability of firm default: PD_F( ε_{z,t+1} ) = Prob( ε_{i,t+1} < ε_{z,t+1} ) = F( ε_{z,t+1}, σ_{ε,t+1} ) (17)
- Firms retain share 1 − Γ( ε_{z,t+1} ) of total earnings where:
  - Δ(ε_z) ≡ ∫_0^{ε_z} ε f(ε) dε (19)
  - Γ(ε_z) ≡ Δ(ε_z) + ε_z (1 − F(ε_z)) (20)
- Rewritten firm cash-flow expression (aggregate over idiosyncratic states) yields:
  - −n_{F,t} + E_t[ M_{t+1} ( z_{t+1} R_{k,t+1} k_t (1 − Γ(ε_{z,t+1})) ) (1−τ) ] (condensed from 18–20).

### Financial intermediaries (FIs): structure and default
- FIs finance loans b_t with capital n_{B,t} and funding d_t from mutual funds (b_t = n_{B,t} + d_t).
- Next period receive return R_{B,t+1} on loans and pay funding R_{d,t} = 1 + i_{d,t}.
- FI expected cashflow:
  - −n_{B,t} + E_t[ M_{t+1} max( (R_{B,t+1} b_t − R_{d,t} d_t) (1−τ), 0 ) ] (21)
- Loan portfolio return accounts for:
  - Interest/principal from non-defaulting firms and recoveries from defaulted firms, with default cost fraction μ on firms’ earnings.
  - R_{B,t+1} b_t ≡ R_{b,t} b_t (1 − F(ε_{i,t+1})) + ∫_0^{ε_{z,t+1}} z_{t+1} ε_{i,t+1} R_{k,t+1} k_t f(ε) dε (1−μ) (22)
- FI cashflow rewritten (23) in terms of Γ(·) and Δ(·): −n_{B,t} + E_t[ M_{t+1} max( z_{t+1} R_{k,t+1} k_t ( Γ(ε_{z,t+1}) − μ Δ(ε_{z,t+1}) ) − R_{d,t} d_t, 0 ) (1−τ) ].
- FI default threshold: z_{t+1} < z^*_{t+1} (24) s.t.
  - z^*_{t+1} R_{k,t+1} k_t ( Γ(ε^*_{t+1}) − μ Δ(ε^*_{t+1}) ) − R_{d,t} d_t = 0 (25)
- Probability of FI default: PD_B( z^*_{t+1} ) = Prob( z_{t+1} < z^*_{t+1} ) = G( z^*_{t+1}, σ_{z,t+1} ) (26)

### Productivity states and equilibrium implications for FIs
- Productivity for regions served by an FI has three realizations depending on z_{t+1} relative to z^*_{t+1}:
  - Case 1 (no default): with probability (1 − G(z^*_{t+1})), z_{t+1} = z_N, firms’ threshold ε_N defined by ε_N ≡ R_{b,t} b_t z_N / ( R_{k,t+1} k_t ) (27).
  - Case 2 (threshold): z_{t+1} = z^*_{t+1} → ε^*_{t+1} ≡ R_{b,t} b_t z^*_{t+1} / ( R_{k,t+1} k_t ) (29).
  - Case 3 (default): with probability G(z^*_{t+1}), z_{t+1} = z_D, firms’ threshold ε_D ≡ R_{b,t} b_t z_D / ( R_{k,t+1} k_t ) (30).
- Each period fraction G(z^*_t) of FIs default and productivity in that part of the economy is z_D, while 1 − G(z^*_t) survive with productivity z_N.
- FI objective in aggregate (31):
  - −n_{B,t} + E_t[ M_{t+1} ( z_N R_{k,t+1} k_t ( Γ(ε^N_{t+1}) − μ Δ(ε^N_{t+1}) ) − R_{d,t} d_t ) (1 − G(z^*_{t+1})) (1−τ) ].

### Mutual funds and intermediation returns
- Mutual funds lend d_t to FIs; in normal times receive principal + interest; if FI defaults they receive assets minus default cost fraction μ_B.
- Mutual fund expected payoff (32):
  - −d_t + E_t[ M_{t+1} ( R_{d,t} d_t (1 − G(z^*_{t+1})) + G(z^*_{t+1}) R_{k,t+1} k_t s^D_{B,t+1} (1 − μ_B) ) ]
- s^D_{B,t+1} defined (33): s^D_{B,t+1} = z_D ( Γ(ε^D_{t+1}) − μ Δ(ε^D_{t+1}) )
- Mutual funds are not taxed on gains, creating net benefit of financial intermediation equal to τ R_{d,t} d_t (1 − G(z^*_{t+1})), which offsets default costs.

### Optimization and equilibrium constraints
- Aggregate resource identity connecting firm, FI, and mutual fund balance sheets:
  - q_t k_t = n_{F,t} + n_{B,t} + d_t (34)
- Firms maximize expected cashflow subject to:
  - (i) zero-profit condition for FIs (38),
  - (ii) FI default threshold condition (40–41),
  - (iii) zero-profit condition for mutual funds (42–43).
- First-order conditions for d_t, Φ_{D,t}, z^*_{t+1}, and Φ_{B,t} derived; key expressions:
  - ∂d_t : λ_{I,t} = E_t[ M_{t+1} R_{k,t+1} q_t X_{t+1} ] (44)
  - X_{t+1} defined (45) and macro-financial feedback factor F_{t+1} ≡ X_{t+1} / λ_{I,t}.
- Interpretation: F_{t+1} depends on firm and intermediary leverage and on idiosyncratic and aggregate risk; firms choose variables to maximize F_{t+1}.

### Financial frictions and first-order conditions (excerpt)
- Two main frictions:
  - Payments of firms and FIs taxed at rate τ while mutual funds not taxed (can also represent discount factor differences); yields benefit τ R_{d,t} d_t (1 − G(z^*_{t+1})).
  - Firm default triggers cost fraction μ of firms’ earnings.
- FI default threshold z^*_{t+1} and funding R_{d,t} d_t chosen to trade off higher intermediation benefit vs. increased FI default likelihood G′(z^*_{t+1}) = g(z^*_{t+1}).
- First-order condition for z^*_{t+1} (47) simplified leads to:
  - −g(z^*_t) Z_t + λ_{V,t} ( Δ(ε^*_t) (1 − μ) + μ f(ε^*_t) ε^{*2}_t ) = 0 (52)
  - λ_{V,t} = g(z^*_t) Z_t / ( Δ(ε^*_t) (1 − μ) + μ f(ε^*_t) ε^{*2}_t ) (53)
- FIs’ leverage and default threshold conditions equate tax benefit associated with higher collateral value with expected productivity loss from more intermediary defaults (see 54–55).

### Capital accumulation, target levels, and capital gaps
- Firms and FIs accumulate capital out of retained earnings; laws of motion:
  - n_{F,t} = ( R_{k,t} q_{t−1} ( n_{F,t−1} + n_{B,t−1} + d_{t−1} ) ( s^N_{F,t} (1 − G(z^*_t)) + s^D_{F,t} G(z^*_t) ) (1−τ) (1 − γ_F) z_{u,t} ) + ω_F (56)
  - n_{B,t} = ( ( R_{k,t} q_{t−1} ( n_{F,t−1} + n_{B,t−1} + d_{t−1} ) s^N_{B,t} − R_{d,t−1} d_{t−1} ) (1 − G(z^*_t)) (1−τ) (1 − γ_{B,t}) z_{k,t} ) + ω_B (57)
- Aggregation: injections equal expected earnings minus equity payout plus constant household injections ω_F and ω_B.
- Potential (target) levels obtained by choosing n_{F,t} and n_{B,t} optimally in addition to d_t, R_{d,t}, z^*_{t+1}, R_{b,t}; optimality implies:
  - λ_{B,t} = λ_{I,t} = 1 (58)
- Capital gaps computed by comparing realized levels to these target levels (targets imply higher welfare than realized cases with λ ≠ 1).

### Estimation strategy and identification
- Estimation: standard Bayesian techniques per An and Schorfheide [2007].
- Model includes original RBC shocks plus four additional shock processes:
  - idiosyncratic risk shock,
  - aggregate risk shock,
  - shock to retained earnings/equity payout of firms,
  - shock to retained earnings/equity payout of FIs.
- Observables used: four macroeconomic variables and four financial variables to identify parameters and shocks.
  - Financial variables included: corporate credit spread (CCS), financial credit spread (FCS), bank Tier 1 capital, and stock prices.
- Identification logic:
  - Intermediary debt holders are sensitive to extreme events and aggregate risk because well diversified and not subject to idiosyncratic risk.
  - Conditional on intermediary default threshold z^*_t, financial spreads depend largely on aggregate risk; changes in aggregate risk inferred from FCS through pricing equation of intermediary debt.
  - Conditional on aggregate risk, changes in corporate credit risk inferred from CCS through pricing equation of corporate debt.

_Italic: Content based only on the supplied PDF excerpt._

### 4.1  Data

### 4.1  Data

### Data sources and construction
- Sample period: 2000Q1 to 2019Q2.
- Four macroeconomic variables: real quarter-on-quarter percentage changes of GDP, investment, employment, and consumption; controlled first for population growth, and then demeaned (as in Smets and Wouters [2007] and Christiano, Motto, and Rostagno [2014]).
- GDP (dY), investment (dI), and consumption (dC): Bureau of Economic Analysis.
  - Investment constructed as the sum of total fixed investment and durable consumption.
  - Consumption constructed as the sum of non-durable consumption and consumption services.
- Employment (dN) and population growth: Bureau of Labor Statistics.
- Corporate Credit Spread (CCS): percentage difference between the iTraxx Industrial Corporate Bond Yield Index and the constant 10-year treasury maturity bond yield; demeaned (years 2008 and 2009 excluded from demeaning).
- Financial Credit Spread (FCS): percentage difference between the iTraxx Financials Yield Index and the constant 10-year treasury maturity bond yield; demeaned (years 2008 and 2009 excluded from demeaning).
- Total bank Tier 1 capital growth (dnB): percentage change in Tier 1 capital of all FDIC insured commercial banks and savings institutions; data from FDIC; demeaned and adjusted for inflation.
- Firm net worth growth (dnF): percentage change in the Down Jones 30 Stock Market Index; adjusted for inflation and demeaned.

### Measurement equations linking data to model variables
- Sensitivity coefficients estimated to link observed FCS (FCSt), observed CCS (CCSt), observed firms’ equity growth (dnFt), and observed financial intermediary capital growth (dnBt) to model counterparts:
  - (E(Rb,t+1) − Rt − (Rb,ss − Rss)) * 100 = CCSt / κC
  - (E(Rd,t+1) − Rt − (Rd,ss − Rss)) * 100 = FCSt / κF
  - (log(nF,t / nF,t−1)) * 100 = dnF / κnF
  - (log(nB,t / nB,t−1)) * 100 = dnB / κnB
- Concern noted that observed spreads and capital may imperfectly reflect lending and funding conditions (e.g., deposit-funded intermediaries or government guarantees could cause observed funding costs to understate funding pressures).
- Years 2008 and 2009 excluded from demeaning procedure for spreads.

### Priors, calibration and estimated parameter sets
- Calibrated parameters (examples and rationale):
  - μ =  0.4 so that recovery value 1−μ is close to average recovery of senior secured bonds (50 percent) and senior secured loans (70 percent) per Moody’s.
  - Tax rate τ = 0.2 (same as for capital gains) to avoid tax distortion of results.
  - zD and zN set close to expected value of zt+1 in steady state conditional on being above or below z∗t+1 to limit dispersion impact.
  - Firms and financial intermediaries pay out fractions of earnings γF and γB as dividends; higher payout chosen for financial intermediaries.
- Four sets of parameters estimated:
  - Coefficients of autocorrelation.
  - Standard deviations of shocks.
  - Financial parameters (fit of model to financial data).
  - Parameters governing adjustment costs.

### Shock processes (exogenous shocks and estimation targets)
- Eight exogenous shocks defined:
  - total factor productivity zy,t
  - investment adjustment cost zi,t
  - FIs’ capital shock zk,t
  - labor market tightening zn,t
  - exogenous spending zg,t
  - credit risk σε,t
  - aggregate risk σz,t
  - firms’ income shock zu,t
- All shocks follow AR(1) logarithmic processes (examples preserved exactly):
  - ln(zy,t) = ρy ln(zy,t−1) + σy e y,t
  - ln(zi,t) = ρi ln(zi,t−1) + σi e i,t
  - ln(zk,t) = ρk ln(zk,t−1) + σk e k,t
  - ln(zn,t) = ρn ln(zn,t−1) + σn e n,t
  - ln(zg,t) = ρg ln(zg,t−1) + σg e g,t
  - ln(σε,t) = (1−ρε) ln(σε,ss) + ρε ln(σε,t−1) + σε e F,t
  - ln(σz,t) = (1−ρz) ln(σz,ss) + ρz ln(σz,t−1) + σε e B,t
  - ln(zu,t) = ρu ln(zu,t−1) + σu e u,t
- The shocks ej,t are normally distributed. Estimated: autocorrelation coefficients ρj and standard deviations σj.

### Financial and adjustment cost parameters estimated
- Estimated steady-state credit risk and aggregate risk: σε,ss and σz,ss.
- Estimated sensitivity coefficients: 1/κF, 1/κC, 1/κnF, and 1/κnB.
- Habit persistence parameter h and adjustment cost coefficients Φi, kF,p, and kB,p estimated.

### Key numeric priors and posterior notes
- Prior of σε,ss set equal to 0.23 so steady-state default probability of firms PD F(εNt+1) ≈ 1.5 percent; posterior mode of σε,ss is 0.245.
- Prior of σz,ss set equal to 0.12 so steady-state default probability of financial institutions PD B(z∗t+1) ≈ 0.75 percent; estimation brings it to 0.157.
- Interpretation: steady-state credit risk higher than steady-state aggregate risk implies lower probability of default, lower credit spreads, and higher leverage of FIs relative to firms near steady state.
- Sensitivity coefficients posterior modes: 1/κF =  0.12; 1/κC = 0.55; 1/κnF = 1.05; 1/κnB = 0.52.
  - Interpretation: observed intermediaries’ funding costs (FCS) map weakly to model funding-cost variable (model changes amount to less than 12% of observed FCS changes); CCS and capital of firms and banks correspond better to the model.

### Role of adjustment costs and habit formation
- Financial shocks have the largest standard deviations (financial variables more volatile than real variables).
- Adjustment costs and habit formation are important to fit the data.
- Cost of adjusting leverage larger for intermediaries than firms; identification of leverage adjustment costs is weak (posterior std dev close to prior std dev).

---

### Results — Estimated parameter values (summary)
- Posterior standard deviations mostly less than half of prior standard deviations, indicating data inform the posteriors.
- Posterior mode examples:
  - σε,ss posterior mode = 0.245.
  - σz,ss posterior mode = 0.157.
  - Sensitivity coefficients: 1/κF = 0.12; 1/κC = 0.55; 1/κnF = 1.05; 1/κnB = 0.52.

### Main findings (historical decomposition and impulse responses)
- Impulse responses to a 1% increase in probability of default of both FIs and firms:
  - Output, consumption, labor and investment decrease.
  - Financial and corporate spreads substantially increase.
  - Gaps in capital and probability of default for both FIs and firms rise (increased financial and corporate leverage relative to optimal values).
- Historical shock decomposition results:
  - Investment growth driven mainly by: equity price shock, idiosyncratic risk shock, aggregate risk shock, TFP shock, and investment efficiency shock.
  - CCS and FCS mainly driven by: aggregate risk shocks and firm capital shocks (confirmed by forecast error variance decomposition).
- Firm default probability:
  - Higher corporate risk associated with the 2001 Dot-com bubble and the GFC.
  - Private sector risk has recently driven up CCS.
- Are credit spreads and equity prices too high/low?
  - Corporate spreads are too low (Figure 7).
  - Equity prices were too high prior to COVID-19 outbreak; high equity prices attributed to corporate income shocks and risk shocks (Figure 8).
- Banking sector resilience:
  - Bank capital buffers have recently declined due to equity payout.
  - Three episodes with positive capital gaps (more capital needed): prior to Dot-com bubble, prior to the GFC, and the recent period.
  - Declining recent trend increases FIs vulnerability to adverse shocks.

### Counterfactual scenario (stress simulation)
- Stress test: apply GFC shocks from 2008–2011 to simulate a macro stress test over 2019–2022.
- Results:
  - Both banks and firms more resilient to adverse shocks today compared to pre-GFC period.
  - Financial intermediaries more vulnerable than firms at the current juncture.
  - Corporate and banking risk gaps elevated under adverse conditions.
  - Under adverse shocks similar to the GFC, deleveraging and severe recession would be expected.
  - There is room to activate the countercyclical capital buffer.

### Policy implications
- When risk and capital gaps are identified, capital should be increased in the respective sector by earnings retention or issuance of equity, or by any policy measure that closes capital and risk gaps while accounting for general equilibrium effects.
- The model can be used as:
  - Signaling device for macroprudential intervention (ex-ante and contemporaneous).
  - Gauge of whether macroprudential action was successful (ex-post), i.e., whether gaps were closed.
- Historical policy implications:
  - More capital or macroprudential policy was needed before and/or during: the Dot-com bubble (firm and FIs capital), the Global Financial Crisis (firm and FIs capital), and the recent period (FIs capital).
  - Countercyclical capital buffer could have been activated prior to the COVID-19 outbreak to guard against deleveraging.

### Lessons for the COVID-19 episode
- Under adverse conditions, default risk in both corporate and financial sectors increases while corporate equity capital and bank capital become scarce; already reflected in stock market losses after COVID-19 outbreak.
- Recommendations implied by the model:
  - Firms and banks should stop paying dividends and build/sustain capital levels to absorb expected losses.
  - This is especially important for banks that increased equity payout prior to COVID-19, to prevent a real-economy crisis turning into a financial crisis.
  - Capital in firms and banks addresses both solvency and liquidity concerns and strengthens liquidity in the system.

### Evaluation of model performance
- Model extension: adds default of FIs to the framework of Christiano, Motto, and Rostagno [2014].
- Comparison of two model-implied proxies (aggregate risk per equation 69 and FIs probability of default) with the financial uncertainty index of Jurado, Ludvigson, and Ng [2015] and Ludvigson, Ma, and Ng [2015]:
  - Both model-based measures move reasonably close to the financial uncertainty index despite that empirical measure played no role in estimation.
  - Additional comparisons reported (e.g., high correlation of model-based FIs probability of default with macroeconomic uncertainty measure).

*Source: IMF working paper chapter excerpt (sections 4.1–6).*

### References

### References

### Key findings from "Additional results"
- Credit risk and systemic risk shocks explain an important share of investment dynamics: "16.3% and 9.3%".
- The credit risk shock explains the largest share in the variation of the CCS and the firm’s probability of default (F).
- The aggregate risk shock is the main driver of the forecast variation in the FCS and the FIs’ probability of default (GZ).
- The shock to firm’s income explains most of the variation in the firm’s capital ("59.5%") and a substantial share in:
  - investment ("36%")
  - CCS ("16.12%")
  - FCS ("26.81%")
  - F ("18.62%")
  - GZ ("28.27%")
- The shock to FIs’ capital is the main driver of fluctuations in FIs’ capital ("40%") together with the credit risk shock ("20.26%") and firm’s capital shock ("20%").
- Risk shocks explain little in consumption, which is driven mainly by the TFP and labor market tightening shocks.
- The authors note that estimating a New-Keynesian framework with sticky prices and sticky wages may reduce residual shocks’ relevance and increase the importance of risk shocks; this is left for future research.
- Historical patterns:
  - Recently, firms have capital higher than optimal, similar to the period prior to the Dot-com bubble.
  - Banks display a low probability of default prior to the COVID-19 crisis.
- The model-based measure of FIs probability of default is shown to be highly correlated with the empirical macroeconomic uncertainty measure of Jurado, Ludvigson, and Ng [2015].

### Calibrated parameters (Table 1)
- α (power on capital in production function): 0.300
- β (discount factor): 0.990
- δ (depreciation rate of capital): 0.025
- μ (firm’s cost of default): 0.400
- μ_B (financial intermediary cost of default): 0.200
- τ (overall tax rate): 0.200
- γ_F (firm equity payout rate): 0.030
- γ_B (financial intermediary equity payout rate): 0.100
- z_D (productivity during a crisis): 0.700
- z_N (productivity in normal times): 1.010

### Selected estimated parameters (Table 2)
- ρ_y (autocorr. of TFP shock): Prior mean 0.9; Post. mode 0.9537; Post. SD 0.0238; Distr. beta; Prior SD 0.0500
- ρ_i (autocorr. of investment shock): Prior mean 0.9; Post. mode 0.6983; Post. SD 0.0637; Distr. beta; Prior SD 0.0500
- ρ_g (autocorr. of gov spending shock): Prior mean 0.9; Post. mode 0.9337; Post. SD 0.0393; Distr. beta; Prior SD 0.0500
- ρ_n (autocorr. of labor disutility shock): Prior mean 0.9; Post. mode 0.9643; Post. SD 0.0161; Distr. beta; Prior SD 0.0500
- ρ_u (autocorr. of firm’ capital shock): Prior mean 0.9; Post. mode 0.8575; Post. SD 0.0268; Distr. beta; Prior SD 0.0500
- ρ_k (autocorr. of FIs’ capital shock): Prior mean 0.5; Post. mode 0.2965; Post. SD 0.0947; Distr. beta; Prior SD 0.2000
- ρ_F (autocorr. of credit risk shock): Prior mean 0.5; Post. mode 0.9605; Post. SD 0.0150; Distr. beta; Prior SD 0.2000
- ρ_B (autocorr. of aggregate risk shock): Prior mean 0.5; Post. mode 0.9293; Post. SD 0.0311; Distr. beta; Prior SD 0.2000
- σ_y (SD of TFP shock): Prior mean 0.01; Post. mode 0.0080; Post. SD 0.0006; Distr. invg; Prior SD 0.0020
- σ_i (SD of investment shock): Prior mean 0.01; Post. mode 0.0269; Post. SD 0.0039; Distr. invg; Prior SD 0.0020
- σ_g (SD of gov. spending shock): Prior mean 0.01; Post. mode 0.0170; Post. SD 0.0013; Distr. invg; Prior SD 0.0020
- σ_n (SD of labor disutility): Prior mean 0.01; Post. mode 0.0096; Post. SD 0.0008; Distr. invg; Prior SD 0.0020
- σ_k (SD of FIs capital shock): Prior mean 0.01; Post. mode 0.0140; Post. SD 0.0019; Distr. invg; Prior SD 0.0020
- σ_u (SD of firms’ capital shock): Prior mean 0.01; Post. mode 0.0070; Post. SD 0.0007; Distr. invg; Prior SD 0.0020
- σ_F (SD of credit risk shock): Prior mean 0.01; Post. mode 0.0199; Post. SD 0.0026; Distr. invg; Prior SD 0.0020
- σ_B (SD of aggregate risk shock): Prior mean 0.01; Post. mode 0.0227; Post. SD 0.0028; Distr. invg; Prior SD 0.0020
- σ_F ss (steady state credit risk): Prior mean 0.23; Post. mode 0.2449; Post. SD 0.0122; Distr. norm; Prior SD 0.0200
- σ_Bss (steady state aggregate risk): Prior mean 0.12; Post. mode 0.1570; Post. SD 0.0111; Distr. norm; Prior SD 0.0200
- κ_C (inverse sensitivity to CCS): Prior mean 3.000; Post. mode 1.8201; Post. SD 0.1993; Distr. norm; Prior SD 1.0000
- κ_F (inverse sensitivity to FCS): Prior mean 6.000; Post. mode 8.4630; Post. SD 0.7039; Distr. norm; Prior SD 1.0000
- κ_nF (inverse sensitivity to dnF): Prior mean 1.000; Post. mode 0.9501; Post. SD 0.1297; Distr. norm; Prior SD 1.0000
- κ_nB (inverse sensitivity to dnB): Prior mean 4.000; Post. mode 1.9324; Post. SD 0.1838; Distr. norm; Prior SD 1.0000
- Φ_i (investment adjustment cost): Prior mean 1.000; Post. mode 3.1199; Post. SD 0.5663; Distr. norm; Prior SD 1.0000
- h (habit in consumption): Prior mean 0.500; Post. mode 0.4811; Post. SD 0.0537; Distr. norm; Prior SD 0.2000
- κ_F p (firm leverage adjustment cost): Prior mean 0.0500; Post. mode 0.0552; Post. SD 0.0214; Distr. norm; Prior SD 0.0250
- κ_Bp (FIs leverage adjustment cost): Prior mean 0.0500; Post. mode 0.0754; Post. SD 0.0200; Distr. norm; Prior SD 0.0250

### Forecast error variance decomposition (Table 3) — Variance decomp (%)
- Output: TFP 58.42; Investment 1.42; FIs capital 0.00; Labor mkt. 27.53; Gov. spending 2.69; Credit risk 3.95; Agg. risk 1.42; Firm’s capital 4.56
- Investment: TFP 16.32; Investment 15.46; FIs capital 0.04; Labor mkt. 6.38; Gov. spending 0.07; Credit risk 16.39; Agg. risk 0.29; Firm’s capital 36.13
- Consumption: TFP 57.63; Investment 1.41; FIs capital 0.00; Labor mkt. 27.69; Gov. spending 4.06; Credit risk 3.50; Agg. risk 1.33; Firm’s capital 4.37
- Labor: TFP 5.00; Investment 0.97; FIs capital 0.00; Labor mkt. 78.53; Gov. spending 12.09; Credit risk 0.84; Agg. risk 0.51; Firm’s capital 2.05
- CCS: TFP 8.28; Investment 4.40; FIs capital 7.08; Labor mkt. 3.07; Gov. spending 0.07; Credit risk 49.72; Agg. risk 11.26; Firm’s capital 16.12
- FCS: TFP 6.99; Investment 4.61; FIs capital 1.14; Labor mkt. 2.39; Gov. spending 0.07; Credit risk 23.83; Agg. risk 34.16; Firm’s capital 26.81
- Firm’ pr. of default (F): TFP 7.34; Investment 2.15; FIs capital 0.99; Labor mkt. 2.52; Gov. spending 0.08; Credit risk 63.11; Agg. risk 5.19; Firm’s capital 18.62
- FIs’ pr. of default (GZ): TFP 10.02; Investment 2.67; FIs capital 0.83; Labor mkt. 3.39; Gov. spending 0.13; Credit risk 19.07; Agg. risk 35.63; Firm’s capital 28.27
- FIs capital (dnB): TFP 11.26; Investment 0.18; FIs capital 39.90; Labor mkt. 3.76; Gov. spending 0.36; Credit risk 20.26; Agg. risk 4.23; Firm’s capital 20.05
- Firms capital (dnF): TFP 21.31; Investment 0.76; FIs capital 1.13; Labor mkt. 7.15; Gov. spending 0.54; Credit risk 4.74; Agg. risk 4.88; Firm’s capital 59.49

### Notes on figures referenced in the text
- Figure 12: Variables used in the estimation step. Shaded areas represent NBER recessions.
- Figure 13: Historical decomposition of gap in corporate capital. Gaps are computed as: Gap = (nF_optimal − nF)/nF where nF denotes the corporate capital. The x-axis denotes the year. The y-axis denotes percentages. 0.2 is equal to a 20 percent gap relative to the nominal amount of aggregate firm’s capital.
- Figure 14: Historical decomposition of FIs’ probability of default. The x-axis denotes the year. The y-axis denotes percentage points. A value of 0.04 indicates that FIs’ probability of default is 4%.
- Figure 15: FIs prob. of default (red line) vs. Jurado et al. 2015 Macroeconomic Uncertainty (blue line). All variables are standardized. The x-axis denotes the year. The y-axis denotes percentage points. A value of 4 corresponds to a level of uncertainty/prob. of default of 4%.

*wpiea2020209-print-pdf - References*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020209-print-pdf.pdf_