## _wp14194

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### Introduction and key contributions
- Objective:
  - (i) assess how well DSGE models that endogenize financial frictions fit US post-war macroeconomic data, and (ii) identify the driving forces of the real business cycle in the US.
- Key contributions:
  - Introduces a New Keynesian DSGE model with an explicit Financial Intermediary (FI) sector that includes (i) the spread between lending and borrowing as a measure of financial stress and (ii) financial shocks closely related to countercyclical spreads.
  - Augments Gertler and Karadi (2011) (GK (2011)) by adding a marginal efficiency of investment (MEI) wedge and uses Bayesian estimation to evaluate model fit via marginal data density (MDD).
  - Performs variance decomposition to compare roles of financial shocks and MEI shocks in explaining macroeconomic volatility.
- Principal findings:
  - The model fits US post-war macroeconomic data well.
  - Financial shocks play a greater role in explaining the volatility of macroeconomic variables than marginal efficiency of investment (MEI) shocks.
  - Policy implication: policies affecting the credit spread are likely to be among the most effective instruments for recovery during financial crises.

### Model overview and financial frictions
- Structure:
  - Standard DSGE New Keynesian framework based on GK (2011) with a simplified, homogeneous FI sector and seven sectors: Households (HH), Financial intermediaries (FI), Central bank (CB), Government (Gov), Non-financial goods producer firm (GPF), Capital producer firms (CPF), Monopolistically competitive retail firms (RF).
- Two distinguishing features:
  - Inclusion of the external financing spread (lending minus borrowing) as a measure of financial stress.
  - Introduction of an investment shock (MEI) determining the efficiency of newly produced investment goods.
- Financial frictions specification:
  - Moral hazard incentive constraint limits intermediaries' ability to obtain funds from households and generates an external financing premium (spread).
  - Spread defined as (R_{k,t+1} − R_{t+1}) and enters bankers’ net worth dynamics and value functions.

### Households — preferences, budget, and first-order conditions
- Preferences and shock process:
  - Maximize E_t ∑_{i=0}^∞ β^i b_{t+i} [ log(C_{t+i} − h C_{t+i−1}) − χ (1−φ)^{-1} L_{t+i}^{1+φ} ].
  - Intertemporal preference shock: log b_t = ρ_b log b_{t−1} + σ_b ε_{b,t}, ε_{b,t} ∼ i.i.d. N(0,1).
- Budget constraint:
  - C_t + B_{t+1} = W_t L_t + R_t B_t + Π_t − T_t.
- Key first-order conditions:
  - Labor supply: ρ_t W_t = χ_t L_t^φ.
  - Euler equation: β E_t Λ_{t,t+1} R_{t+1} = 1, where Λ_{t,t+1} ≡ ρ_{t+1} / ρ_t.
- Note:
  - Intertemporal preference shock b_t affects marginal utility of consumption and marginal disutility of labor; reported to be unimportant in variance decomposition (unreported exercise).

### Financial Intermediaries — balance sheets, constraints, and value
- Balance sheet and net worth evolution:
  - Net worth N_{j,t} = Q_t S_{j,t} − B_{j,t}.
  - N_{j,t+1} = R_{t+1} N_{j,t} + (R_{k,t+1} − R_{t+1}) Q_t S_{j,t}.
- Bankers’ objective (value function):
  - V_{j,t} ≡ (1−θ) E_t ∑_{i=0}^∞ θ^i β^i Λ_{t,t+1+i} N_{j,t+1+i}.
  - Value linear form: V_{j,t} = ν_t Q_t S_{j,t} + η_t N_{j,t}.
- Recursions:
  - ν_t = E_t{ (1−θ)(R_{k,t+1} − R_{t+1}) + β Λ_{t,t+1} θ x_{t,t+1} ν_{t+1} }.
  - η_t = E_t{ (1−θ) R_{t+1} + β Λ_{t,t+1} θ z_{t,t+1} η_{t+1} }.
- Moral hazard and borrowing limits:
  - Banker can divert fraction λ of assets Q_t S_{j,t}; incentive constraint when binding leads to leverage relation QtSj,t = φt Njt with φt = ηtλ−νt.
  - Participation condition: β^i Λ_{t,t+1+i} (R_{k,t+1+i} − R_{t+1+i}) ≥ 0, ∀ i ≥ 0.

### Moral hazard, leverage, and net worth dynamics (aggregate)
- Incentive constraint (prevent diversion):
  - Vt(Sj,t,Bj,t) ≥ λQtSj,t . (16)
- Leverage when constraint binds:
  - QtSj,t = ηtλ−νt Nj,t . (17)
  - QtSj,t = φt Njt , (18)
  - φt = ηtλ−νt.
- Net worth evolution for existing bankers:
  - Nt = θ[(Rk,t − Rt)φt + Rt] Nt−1 .
- Entry and transfers:
  - New bankers’ start-up net worth: Nnt = ω1−θ (1−θ) QtSt−1 = ω QtSt−1 . (19)
- Aggregate net worth:
  - Nt = θ[(Rk,t − Rt)φt + Rt] Nt−1 + ω QtSt−1 . (21)

### Production, capital accumulation, and MEI shock
- Intermediate good producer:
  - Yt = At (Ut ξt Kt)α Lt1−α . (22)
  - log At = ρA log At−1 + σA εA,t . (23)
  - log ξt = (1−ρξ) log ξss + ρ log ξt−1 + σξ εξ,t . (24)
- Capital law of motion and MEI:
  - Kt+1 = (1−δ(Ut)) ξt Kt + γt Φ(It / Kt) Kt . (30)
  - log γt = ρi log γt−1 + σγ εγ,t . (31)
  - γt is marginal efficiency of investment (MEI) shock.
- Adjustment costs and price of capital:
  - Φ(It / Kt) = [1 − S(It / It−1)] It / Kt . (33)
  - S(It / It−1) = ηi 2 (It / It−1 − 1)2 . (34)
  - Qt = (Φ′(It / Kt))−1 . (32)
  - Tobin’s Q condition: Qt γt [1 − S(It / It−1) − S′(It / It−1) (It / It−1)] + β Et { Qt+1 Λt,t+1 γt+1 S′(It+1 / It) (It+1 / It)2 } = 1 . (39)

### Retail firms, monetary policy, and government
- Retail/price rigidity:
  - Yt = Ymt Dt . (40) with Calvo pricing; Calvo probability is $; indexation γP; elasticity εP.
  - Inflation dynamics summarized by equations (42)–(45), including: π1−εP t = $ πγP (1−εP) t−1 + (1−$) π∗(1−εP) t . (45)
- Central bank rule:
  - (it / i) = (it−1 / i) ρr ( (πt / π∗) κπ (Yt / Y∗t) κy ) (1−ρr) eσi εi,t . (46)
  - Fisher equation: it = Rt+1 Et πt+1 . (47)
- Government:
  - Gt = Tt . (48)
  - Gt = (1 − 1 / gt) Yt . (49)
  - log gt = (1 − ρg) log gss + ρg log gt−1 + σg εg,t . (50)

### Resource constraint and model closure
- Aggregate resource constraint:
  - Yt = Ct + It + S(It / It−1) It + Gt . (52)
- Model dimensionality:
  - Equations (1) to (52) determine twelve endogenous variables.
  - Stochastic drivers: six exogenous disturbances — bt, At, γt, ξt, gt, and εi.

### Data, observables, and measurement
- Observable quarterly US series (sample 1962Q1 to 2004Q4):
  - ∆log(GDPt) — log difference of real GDP per capita.
  - ∆log(INVt) — log difference of real investment per capita.
  - ∆log(Pt) — log difference of the GDP deflator (inflation).
  - FedFundt — Federal Funds Rate.
- Measurement equation:
  - yd = [ ∆log(GDPt) ; ∆log(INVt) ; ∆log(Pt) ; FedFundt ] = [ Y(t)−Y(t−1) ; I(t)−I(t−1) ; π(t) ; i(t) ].
- Appendix data transformations (exact formulas):
  - output = LN( GDPC96 / LNSindex ) * 100
  - investment = LN( ( FPI / GDPDEF ) / LNSindex ) * 100
  - inflation = LN( GDPDEF / GDPDEF(-1) ) * 100
  - interest rate = Federal Funds Rate / 4

### Bayesian estimation: priors, algorithm, and posterior summaries
- Priors and fixed parameter choices:
  - Standard errors of innovations: inverse-gamma with mean 0.10 and standard deviation 2.0; covariance diagonal.
  - AR(1) persistence priors: beta with mean 0.5 and standard deviation 0.2.
  - Fixed parameters:
    - Steady state depreciation rate = 0.02.
    - Exogenous spending-GDP ratio = 0.2.
    - Discount factor β = 0.99.
    - Inverse Frisch elasticity of labor = 0.275.
    - Banker survival rate θ = 0.971.
    - Proportional transfer to entering bankers ω = 2×10^5 (reported) and in fixed parameters ω = 0.00002 (model lists ω = 0.00002 under fixed parameters).
- MCMC:
  - Metropolis-Hastings algorithm; 20,000 iterations simulated.
  - Convergence observed after 10,000 draws (Brooks and Gelman diagnostics).
- Posterior persistence estimates (AR(1) means):
  - Government spending ρ̂ = 0.98.
  - Productivity ρ̂ = 0.96.
  - Capital quality ρ̂ = 0.89.
- Selected posterior means (behavioral and shock std.):
  - σa (Std Tech. Inv.): 0.68
  - σg (Std Gov. Spending): 0.65
  - σs (Std Capital quality): 0.37
  - σi (Std MP): 0.38
  - ρa (Auto. Tech.): 0.96
  - ρg (Auto. Gov. Spending): 0.98
  - ρs (Auto. Capital quality): 0.89
  - ρi (Auto. MP): 0.58
  - α (Capital Share): 0.49
  - ζ (Elasticity Util. Cost): 7.13
  - ηi (Inv. Adju. Cost Coefficient): 1.74
  - λ (Fraction of Asset Diverted): 0.46
  - χ (Coefficient of Leisure): 3.40
  - P (Elasticity of substitution): 4.16
  - $ (Calvo Probability): 0.56
  - γP (Price indexation): 0.31
  - κπ (Reaction Inflation): 2.19
  - κy (Reaction Output-Gap): 0.20
  - ρi (Degree IR Smoothing): 0.39
  - h (Habit Formation): 0.95
- Fixed parameters reiterated:
  - β (Discount Factor): 0.99
  - θ (Bankers Survival Rate): 0.971
  - δss (SS Depreciation): 0.02
  - ω (Transfer to new Bankers): 0.00002
  - g/y (SS spending over GDP): 0.2
  - φ (Inverse Frisch): 0.275

### Model fit — marginal data density (MDD) and moments
- Laplace approximation of log data density (MDD):
  - my model: -629.58
  - SW (2007) model (same observables): -676.02
- Extended sample (1962Q1-2010Q4) MDDs:
  - my model’s MDD = -709.80
  - SW model’s MDD = -778.21
- Model dimensionality and estimation counts:
  - Number of shocks: 67
  - Number of observables: 4
  - Number of Estimated Parameters: 25 (my model) and 36 (SW (2007))
  - Number of Fixed Parameters: 6
- Matching of moments (selected):
  - Output Growth: Model STD 0.98, Data STD 0.86; Auto Corr 1: Model 0.35, Data 0.23; Auto Corr 2: Model 0.19, Data 0.21; Auto Corr 3: Model 0.09, Data 0.08
  - Investment Growth: Model STD 2.47, Data STD 2.26; Auto Corr 1: Model 0.59, Data 0.53; Auto Corr 2: Model 0.32, Data 0.37; Auto Corr 3: Model 0.15, Data 0.20
  - Inflation Rate: Model STD 1.06, Data STD 0.83; Auto Corr 1: Model 0.84, Data 0.87; Auto Corr 2: Model 0.67, Data 0.82; Auto Corr 3: Model 0.55, Data 0.79
  - Interest Rate: Model STD 1.06, Data STD 0.83; Auto Corr 1: Model 0.95, Data 0.94; Auto Corr 2: Model 0.87, Data 0.86; Auto Corr 3: Model 0.77, Data 0.79

### Drivers of the business cycle — variance decomposition (business cycle frequencies)
- Main conclusions:
  - MEI processes play virtually no role in business cycles once financial shocks are introduced.
  - Capital quality shocks are the primary drivers across most key macro variables.
- Selected percent contributions (as reported):
  - Output: Tech 35.71, Cap. Qual. 48.69, Gov. Spen. 14.75, Monetary Policy 0.84, MEI 0.22, Preference 0.00
  - Consumption: Tech 17.21, Cap. Qual. 69.75, Gov. Spen. 8.79, Monetary Policy 5.67, MEI 10.17, Preference 0.03
  - Investment: Tech 58.55, Cap. Qual. 72.36, Gov. Spen. 36.48, Monetary Policy 12.04, MEI 22.91, Preference 1.20
  - Capital: Tech 9.92, Cap. Qual. 2.89, Gov. Spen. 9.41, Monetary Policy 0.70, MEI 3.65, Preference 80.30
  - Gov’t Spending: Tech 4.20, Cap. Qual. 5.45, Gov. Spen. 73.42, Monetary Policy 8.99, MEI 1.20, Preference 0.09
  - Asset Prices: Tech 34.02, Cap. Qual. 44.53, Gov. Spen. 3.37, Monetary Policy 4.30, MEI 7.71, Preference 11.81
  - Welfare: Tech 11.81, Cap. Qual. 52.59, Gov. Spen. 9.89, Monetary Policy 4.92, MEI 8.15, Preference 0.13
  - Inflation: Tech 6.95, Cap. Qual. 8.95, Gov. Spen. 0.94, Monetary Policy 60.99, MEI 14.11, Preference 2.24
- Mechanism:
  - Negative capital quality shock (financial shock) causes capital losses for intermediaries, tightens credit via deteriorated net worth, raises spread, and amplifies downturns beyond direct capital accumulation effects.

### Impulse responses and recovery dynamics
- IRF computation:
  - IRFs are expected future paths conditional on a one standard deviation shock in period 1, evaluated at estimated parameter modes.
- Technology shock (negative one standard deviation; ρ = 0.96):
  - Amplified investment and spread responses due to bank balance sheet channel; investment declines sharply, intermediaries’ asset side deteriorates, spread rises, funding cost increases, and output/investment decline in hump-shaped manner.
- Monetary policy shock (one standard deviation increase in short-term nominal rate):
  - Interest rate smoothing lower than SW (2007) (ρi = 0.39 vs SW), peak effects occur earlier; financial frictions amplify responses via countercyclical premium.
- Capital quality shock (negative one standard deviation financial shock):
  - Quantitative IRF magnitudes reported:
    - Asset price drops 1.1% below steady state on impact.
    - Output initially falls about 0.5 percent and decreases further to about 0.97 percent relative to zero within five quarters.
    - Investment drops to 1.4 percent below zero on impact.
    - Hours of work initially drop to 0.16 percent below steady state but increase to a maximum of 1 percent after 20 quarters; then decline slowly while remaining positive relative to steady state.
  - Recovery dynamics:
    - Output recovery occurs only after 6 quarters and takes 5 years to return to the pre-crisis steady state.
    - Slow recovery driven by prolonged deleveraging as intermediaries build equity relative to assets.
- Empirical point:
  - Average deleveraging period is 5 to 6 years and typically starts 2 years after onset of financial crisis; model matches slow deleveraging dynamics observed in banking crises preceded by credit bubbles.

### Policy implications and recommendations
- Key policy conclusion:
  - Policies that reduce the costs associated with financial transactions and policies that affect the credit spread are likely to be most effective in facilitating economic recovery during financial crises.
- Modeling recommendation:
  - Analyses of economic policy impacts on business cycles should use macroeconomic models that explicitly incorporate a financial sector and endogenous financial frictions.

*Source: IMF Working Paper WP/14/194 — Financial Frictions and Sources of Business Cycle, Marzie Taheri Sanjani (Sections 1–4).*

### Section 1

### _wp14194 - Section 1

### Introduction
- Objective:
  - Twofold: (i) assess how well DSGE models that endogenize financial frictions fit US post-war macroeconomic data, and (ii) identify the driving forces of the real business cycle in the US.
- Motivation:
  - August 2007 onset of the subprime mortgage crisis led to soaring money market spreads, impaired financial activity, and disrupted term maturities due to counterparty risk and adverse selection.
  - Financial accelerator mechanisms amplify the deterioration in real economic activity during financial stress.
- Key contributions:
  - Introduces a New Keynesian DSGE model with an explicit Financial Intermediary (FI) sector that includes (i) the spread between lending and borrowing as a measure of financial stress and (ii) financial shocks closely related to countercyclical spreads.
  - Augments Gertler and Karadi (2011) (GK (2011)) by adding a marginal efficiency of investment (MEI) wedge and uses Bayesian estimation to evaluate model fit via marginal data density (MDD).
  - Performs variance decomposition to compare the roles of financial shocks and MEI shocks in explaining macroeconomic volatility.
- Principal findings summarized in Section 1:
  - The model fits US post-war macroeconomic data well.
  - Financial shocks play a greater role in explaining the volatility of macroeconomic variables than marginal efficiency of investment (MEI) shocks.
  - Policy implication: policies affecting the credit spread are likely to be among the most effective instruments for recovery during financial crises.
- Literature context:
  - Builds on GK (2011) and nests Smets and Wouters (2007) (SW (2007)) in the frictionless limit.
  - Aligns with and extends Justiniano, Primiceri, and Tambalotti (2010) (JPT (2010)) by decomposing financial shocks from MEI shocks.
  - Relates to financial accelerator literature (Bernanke and Gertler (1989), Kiyotaki and Moore (1997), BGG (1999)) and to recent work emphasizing financial shocks (Christiano, Motto, and Rostagno (2013); Mandelman (2010); Nolan and Thoenissen (2009); Ajello (2010); Hirakata, Sudo, and Ueda (2011); Liu, Waggoner, and Zha (2011)).

### Model (overview)
- Model structure:
  - Standard DSGE New Keynesian framework based on GK (2011) with a simplified, homogeneous financial intermediary sector and seven sectors: Households (HH), Financial intermediaries (FI), Central bank (CB), Government (Gov), Non-financial goods producer firm (GPF), Capital producer firms (CPF), Monopolistically competitive retail firms (RF).
  - Retail firms included solely to introduce nominal price rigidities.
- Two distinguishing features:
  - Inclusion of the external financing spread (lending minus borrowing) as a measure of financial stress.
  - Introduction of an investment shock determining the efficiency of newly produced investment goods (MEI shock), following Greenwood, Hercowitz, and Huffman (1988).
- Rationale for using GK (2011) baseline:
  - Provides a minimal, uniform framework that meaningfully includes FIs while remaining nested with standard NK models (CEE (2005), SW (2007)).
  - Inclusion of financial intermediaries alters model dynamics and policy implications without excessive complexity.
- Financial frictions:
  - Source: moral hazard incentive constraint that limits intermediaries' ability to obtain funds from households.
  - Role: allows the model to capture crisis dynamics through bank balance sheet channels in addition to standard capital accumulation effects.

### Households (key definitions and first-order conditions)
- Household setup:
  - Continuum of identical households with intra-household occupational split: at any time, 1−f fraction are “workers” and f fraction are “bankers”; agents switch occupations with a history-independent probability.
  - Workers supply labor and return wages to households; bankers manage financial intermediaries and transfer earnings to households.
  - Bankers exit with i.i.d probability 1−θ each period; upon exit bankers transfer retained earnings to households and new bankers receive startup funds from the household.
  - Households choose consumption C_t, one-period real bond B_t, supply labor L_t, pay lump-sum tax T_t, and receive net transfer Π_t (funds from existing bankers minus funds to new bankers).
- Household preferences and budget constraint:
  - Maximize E_t ∑_{i=0}^∞ β^i b_{t+i} [ log(C_{t+i} − h C_{t+i−1}) − χ (1−φ)^{-1} L_{t+i}^{1+φ} ]
  - Subject to: C_t + B_{t+1} = W_t L_t + R_t B_t + Π_t − T_t
  - Parameters/variables explicitly used:
    - χ: coefficient of leisure
    - h: habit persistence parameter
    - φ: inverse Frisch parameter
    - b_t: intertemporal preference shock; follows log b_t = ρ_b log b_{t−1} + σ_b ε_{b,t}, ε_{b,t} ∼ i.i.d. N(0,1)
    - β: time discount factor
- First-order conditions:
  - Labor supply: ρ_t W_t = χ_t L_t^φ
  - Euler equation: β E_t Λ_{t,t+1} R_{t+1} = 1
  - Marginal utility and stochastic discount factor:
    - ρ_t = b_t (C_t − h C_{t−1})^{−1} − β h E_t b_{t+1} (C_{t+1} − h C_t)^{−1}
    - Λ_{t,t+1} ≡ ρ_{t+1} / ρ_t
- Note:
  - The intertemporal preference shock b_t affects both marginal utility of consumption and marginal disutility of labor; the author reports that, in an unreported exercise, this shock is not important in variance decomposition of macro volatility.

### Financial Intermediaries (structure, constraints, and value)
- Role:
  - Bankers raise deposits B_{j,t} from households at the risk-free rate R_{t+1} and purchase financial claims S_t from goods producer firms at relative price Q_t.
- Balance sheet and net worth:
  - Net worth N_{j,t} = Q_t S_{j,t} − B_{j,t}
  - At t+1 claims pay stochastic return R_{k,t+1}; banker net worth evolves as:
    - N_{j,t+1} = R_{k,t+1} Q_t S_{j,t} − R_{t+1} B_{j,t}
    - Which can be written N_{j,t+1} = R_{t+1} N_{j,t} + (R_{k,t+1} − R_{t+1}) Q_t S_{j,t}
  - The spread (R_{k,t+1} − R_{t+1}) is the interest rate spread or external financing premium and is related to financial frictions.
- Participation and profitability condition:
  - Necessary condition for bankers to stay in business: discounted expected return on non-financial claims ≥ discounted cost of borrowing from households:
    - β^i Λ_{t,t+1+i} (R_{k,t+1+i} − R_{t+1+i}) ≥ 0, ∀ i ≥ 0
- Bankers’ objective and value function:
  - Bankers maximize expected discounted terminal wealth:
    - V_{j,t} ≡ (1−θ) E_t ∑_{i=0}^∞ θ^i β^i Λ_{t,t+1+i} N_{j,t+1+i}
    - Rewritten Bellman form: V_{t−1}(S_{j,t−1}, B_{j,t−1}) = E_{t−1}{Λ_{t−1,t} N_{j,t}} + max_{S_{j,t}} V_t(S_{j,t}, B_{j,t})
  - Value function is linear: V_{j,t} = ν_t Q_t S_{j,t} + η_t N_{j,t}
    - ν_t: marginal value of assets at end of period t
    - η_t: marginal cost of deposits
  - Recursions for ν_t and η_t:
    - ν_t = E_t{ (1−θ)(R_{k,t+1} − R_{t+1}) + β Λ_{t,t+1} θ x_{t,t+1} ν_{t+1} }
    - η_t = E_t{ (1−θ) R_{t+1} + β Λ_{t,t+1} θ z_{t,t+1} η_{t+1} }
    - Definitions:
      - x_{t,t+1} ≡ (Q_{t+i} S_{j,t+i}) / (Q_t S_{j,t})
      - z_{t,t+1} ≡ N_{j,t+i} / N_{j,t}
- Moral hazard and borrowing limits:
  - To prevent unbounded borrowing in the frictionless case, a moral hazard problem is introduced: each period the banker can divert a fraction λ of her total assets Q_t S_{j,t} back to her family. Diversion implies default on debt.
  - The moral hazard constraint limits bankers’ borrowing ability and generates financial frictions that connect bank balance sheets to real activity.

*Source: IMF Working Paper WP/14/194 — Financial Frictions and Sources of Business Cycle, Marzie Taheri Sanjani (Section 1).*

### Section 2

### _wp14194 - Section 2

### Moral hazard, leverage, and lending constraint
- Creditors can reclaim the remaining fraction (1−λ) of funds; moral hazard leads creditors to restrict lending.
- Incentive constraint to prevent diversion:
  - Vt(Sj,t,Bj,t) ≥ λQtSj,t . (16)
- When the constraint binds, assets are a levered ratio (φt) of net worth:
  - QtSj,t = ηtλ−νt Nj,t (17)
  - QtSj,t = φt Njt , (18)
  - φt = ηtλ−νt is the leverage ratio of financial intermediaries.

### Net worth dynamics of bankers
- Net worth evolution for existing bankers:
  - Nt = θ[(Rk,t − Rt)φt + Rt] Nt−1 .
- Wealth shocks to existing bankers are omitted (variance decomposition shows negligible role for the business cycle in this specification).
- Entry and transfers:
  - A fraction (1−θ) bankers become workers and transfer accumulated net worth QtSt to households.
  - A fraction ω1−θ of this transfer is given to new bankers entering at t+1 as start-up net worth Nnt:
    - Nnt = ω1−θ (1−θ) QtSt−1 = ω QtSt−1 . (19)
- Aggregate net worth:
  - Nt = Net + Nnt (20)
  - Nt = θ[(Rk,t − Rt)φt + Rt] Nt−1 + ω QtSt−1 . (21)

### Role of financial intermediaries in production
- Intermediaries issue loans to non-financial goods producers by purchasing financial claims at unit price of capital; loans fund production of intermediate goods.
- Arbitrage condition linking units of capital and stock issuance:
  - Qt Kt+1 = Qt St .

### Intermediate good producer firm (non-financial goods firm)
- Technology and production:
  - Yt = At (Ut ξt Kt)α Lt1−α . (22)
- Exogenous processes:
  - log At = ρA log At−1 + σA εA,t . (23)
  - log ξt = (1−ρξ) log ξss + ρ log ξt−1 + σξ εξ,t . (24)
  - εA,t ∼ i.i.d. N(0,1); εξ,t ∼ i.i.d. N(0,1).
- Interpretation:
  - α is capital share; ξt captures exogenous disturbances to capital quality and microeconomic frictions affecting intertemporal capital accumulation.
- Profit (expected) at t+1:
  - Υt+1 = Et β Λt,t+1 [ Pm,t+1 Yt+1 + (Qt+1 − δ(Ut+1)) ξt+1 Kt+1 − Rk,t+1 Qt Kt+1 − Wt+1 Lt+1 ]. (25)
- Depreciation functional form:
  - δ(Ut) = δss + b 1 + ζ U1+ζ t . (26)
- First-order conditions (firm optimization):
  - Labor: Pm,t (1−α) Yt / Lt = Wt . (27)
  - Utilization: Pm,t α Yt / Ut = δ′(Ut) ξt Kt . (28)
  - Capital rental/return:
    - Rk,t+1 = Pm,t+1 α Yt+1 / Kt+1 + (Qt+1 − δ(Ut+1)) ξt+1 / Qt . (29)
- Key implication:
  - Negative shocks to ξt raise rental cost Rk,t and deteriorate bankers’ net worth, linking capital quality shocks to spread dynamics.

### Capital producer firm and capital accumulation
- Capital law of motion:
  - Kt+1 = (1−δ(Ut)) ξt Kt + γt Φ(It / Kt) Kt . (30)
- MEI shock:
  - log γt = ρi log γt−1 + σγ εγ,t . (31)
  - εγ,t ∼ i.i.d. N(0,1).
- Interpretation:
  - γt is marginal efficiency of investment (MEI) shock representing inefficiencies in transforming investment into installed capital.
  - The model posits MEI shocks capture randomness, uncertainty, and waste in the investment process.
- Adjustment cost and price of capital:
  - Φ(It / Kt) = [1 − S(It / It−1)] It / Kt . (33)
  - S(It / It−1) = ηi 2 (It / It−1 − 1)2 . (34)
  - Kt+1 = ξt (1−δ(Ut)) Kt + γt [1 − S(It / It−1)] It . (35)
  - Price of capital:
    - Qt = (Φ′(It / Kt))−1 . (32)
- Tobin’s Q first-order condition (shadow price condition):
  - Qt γt [1 − S(It / It−1) − S′(It / It−1) (It / It−1)] + β Et { Qt+1 Λt,t+1 γt+1 S′(It+1 / It) (It+1 / It)2 } = 1 . (39)
- Capital producer profit per unit:
  - Ψt = Qt γt Φ(It / Kt) − It / Kt − ( Q̄t − Qt ) . (36)

### Retail firms and price rigidity
- Retail output and price dispersion:
  - Yt = Ymt Dt . (40)
  - Recursive price dispersion D t expression (Calvo framework) given in text (41).
- Optimal price recursion and inflation dynamics summarized:
  - Equations (42)–(45) underpin standard sticky-price inflation dynamics; Calvo probability is $; γP is indexation; εP is elasticity of substitution.
  - Inflation dynamic:
    - π1−εP t = $ πγP (1−εP) t−1 + (1−$) π∗(1−εP) t . (45)

### Central bank (monetary policy)
- Interest rate feedback rule with smoothing:
  - (it / i) = (it−1 / i) ρr ( (πt / π∗) κπ (Yt / Y∗t) κy ) (1−ρr) eσi εi,t . (46)
  - ρr degree of smoothing; i steady state gross nominal interest rate; κπ and κy policy parameters; εi,t ∼ i.i.d. N(0,1).
- Fisher equation linking nominal and real rates:
  - it = Rt+1 Et πt+1 . (47)

### Government budget and spending shocks
- Government budget constraint:
  - Gt = Tt . (48)
- Government spending parameterization:
  - Gt = (1 − 1 / gt) Yt . (49)
  - log gt = (1 − ρg) log gss + ρg log gt−1 + σg εg,t . (50)
  - εg,t ∼ i.i.d. N(0,1). (51)
  - gss is steady state government spending.

### Resource constraint and model closure
- Aggregate resource constraint:
  - Yt = Ct + It + S(It / It−1) It + Gt . (52)
- Model dimensionality:
  - Equations (1) to (52) determine twelve endogenous variables.
  - Stochastic drivers: six exogenous disturbances — bt, At, γt, ξt, gt, and εi.

### Empirical evaluation — data and measurement
- Observable variables used for estimation (quarterly US series):
  - ∆log(GDPt) — log difference of real GDP per capita (real GDP divided by civilian non-institutional population 16+).
  - ∆log(INVt) — log difference of real investment per capita (nominal private investment divided by civilian non-institutional population and GDP deflator).
  - ∆log(Pt) — log difference of the GDP deflator (inflation).
  - FedFundt — Federal Funds Rate.
- Series details:
  - GDP and investment expressed in 100 times log.
  - Inflation rate and interest rate expressed on a quarterly basis.
  - Sample: quarterly data from 1962Q1 to 2004Q4.
- Measurement equation:
  - yd = [ ∆log(GDPt) ; ∆log(INVt) ; ∆log(Pt) ; FedFundt ] = [ Y(t)−Y(t−1) ; I(t)−I(t−1) ; π(t) ; i(t) ] .

### Bayesian estimation: priors and algorithm
- Priors:
  - Standard errors of innovations follow an inverse-gamma distribution with mean 0.10 and standard deviation 2.0; covariance matrix is diagonal.
  - Persistence of AR(1) processes: beta distributed with mean 0.5 and standard deviation 0.2.
  - Structural parameter priors follow standard measures in the literature.
- Fixed parameters in estimation:
  - Steady state depreciation rate = 0.02.
  - Exogenous spending-GDP ratio = 0.2.
  - Discount factor β = 0.99.
  - Inverse Frisch elasticity of labor = 0.275.
  - Banker survival rate θ = 0.971 (following GK (2011)).
  - Proportional transfer to entering bankers ω = 2×10^5.
- MCMC and convergence:
  - Metropolis-Hastings algorithm used; 20,000 iterations simulated.
  - Convergence (Brooks and Gelman diagnostics) observed after 10,000 draws.
- Smoothed shocks and spread:
  - Figure 4 (described) shows normalized financial shocks and default risk spread (BAA − AAA yields); historical recessions coincide with rises in spread and financial shocks; correlation between financial shock and spread is small but positive.
- Posterior estimates summary:
  - Government spending, productivity, and capital quality processes estimated as most persistent with AR(1) means:
    - Government spending ρ̂ = 0.98.
    - Productivity ρ̂ = 0.96.
    - Capital quality ρ̂ = 0.89.

*Source: _wp14194 - Section 2*

### Section 3

### _wp14194 - Section 3

### Parameter estimates and priors
- Estimated means of the standard error of the shocks:
  - government spending: 0.65
  - productivity (Std Tech.): 0.68
  - capital quality: 0.37
- High persistence of these processes implies importance in explaining forecast error variance of real variables.
- Behavioral parameter highlights:
  - habit parameter (h): 0.95
- Monetary policy reaction function estimates:
  - long-run reaction coefficient to inflation (κπ): 2.19
  - coefficient on lagged interest rate (degree of interest rate smoothing, ρi): 0.39
  - reaction to output gap (κy): 0.20
- Select posterior vs prior table entries (means reported as in section):
  - σa (Std Tech. Inv.): Posterior mean 0.68
  - σg (Std Gov. Spending): Posterior mean 0.65
  - σs (Std Capital quality): Posterior mean 0.37
  - σi (Std MP): Posterior mean 0.38
  - ρa (Auto. Tech.): Posterior mean 0.96
  - ρg (Auto. Gov. Spending): Posterior mean 0.98
  - ρs (Auto. Capital quality): Posterior mean 0.89
  - ρi (Auto. MP): Posterior mean 0.58
  - α (Capital Share): Posterior mean 0.49
  - ζ (Elasticity Util. Cost): Posterior mean 7.13
  - ηi (Inv. Adju. Cost Coefficient): Posterior mean 1.74
  - λ (Fraction of Asset Diverted): Posterior mean 0.46
  - χ (Coefficient of Leisure): Posterior mean 3.40
  - P (Elasticity of substitution): Posterior mean 4.16
  - $ (Calvo Probability): Posterior mean 0.56
  - γP (Price indexation): Posterior mean 0.31
  - κπ (Reaction Inflation): Posterior mean 2.19
  - κy (Reaction Output-Gap): Posterior mean 0.20
  - ρi (Degree IR Smoothing): Posterior mean 0.39
  - h (Habit Formation): Posterior mean 0.95
- Fixed parameters listed in the model:
  - β (Discount Factor): 0.99
  - θ (Bankers Survival Rate): 0.971
  - δss (SS Depreciation): 0.02
  - ω (Transfer to new Bankers): 0.00002
  - g/y (SS spending over GDP): 0.2
  - φ (Inverse Frisch): 0.275

### Model fit (Bayesian evidence and moments)
- Laplace approximation of log data density (MDD):
  - my model: -629.58
  - SW (2007) model (same observables): -676.02
- Model dimensionality and diagnostics:
  - Number of shocks: 67
  - Number of observables: 4
  - Number of Estimated Parameters: 25 (my model) and 36 (SW (2007))
  - Number of Fixed Parameters: 6
- Additional sample MDD note:
  - For an extended sample (1962Q1-2010Q4) reported MDDs: my model’s MDD is -709.80 and the SW model’s MDD is -778.21.
- Interpretation:
  - Smaller absolute value of MDD for the model with financial frictions implies a better fit.
  - The GK model (with financial frictions) is simpler (one-third fewer estimated parameters than SW) and demonstrates reasonable out-of-sample predictive power.
  - Financial frictions and financial shocks enable more direct capture of financial crisis impacts via countercyclical spread.
- Theoretical vs data moments (standard deviation and 3 lags autocorrelation) — model matches observed moments relatively well for:
  - Output Growth: Model STD 0.98, Data STD 0.86; Auto Corr 1: Model 0.35, Data 0.23; Auto Corr 2: Model 0.19, Data 0.21; Auto Corr 3: Model 0.09, Data 0.08
  - Investment Growth: Model STD 2.47, Data STD 2.26; Auto Corr 1: Model 0.59, Data 0.53; Auto Corr 2: Model 0.32, Data 0.37; Auto Corr 3: Model 0.15, Data 0.20
  - Inflation Rate: Model STD 1.06, Data STD 0.83; Auto Corr 1: Model 0.84, Data 0.87; Auto Corr 2: Model 0.67, Data 0.82; Auto Corr 3: Model 0.55, Data 0.79
  - Interest Rate: Model STD 1.06, Data STD 0.83; Auto Corr 1: Model 0.95, Data 0.94; Auto Corr 2: Model 0.87, Data 0.86; Auto Corr 3: Model 0.77, Data 0.79

### Drivers of the business cycle (variance decomposition)
- Main findings from Table 4 (variance decompositions, percent contribution at business cycle frequencies):
  - MEI processes play virtually no role in business cycles once financial shocks are introduced.
  - Capital quality shocks are the primary drivers:
    - Output: capital quality contributes 48.69 percent
    - Consumption: capital quality contributes 75.87 percent
    - Investment: capital quality contributes 36.48 percent
    - Spread: capital quality contributes 62.04 percent
    - Welfare: capital quality contributes 60.0 percent (reported as 60)
  - Selected variance decomposition percentages by variable and shock (as reported):
    - Output: Tech 35.71, Cap. Qual. 48.69, Gov. Spen. 14.75, Monetary Policy 0.84, MEI 0.22, Preference 0.00
    - Consumption: Tech 17.21, Cap. Qual. 69.75, Gov. Spen. 8.79, Monetary Policy 5.67, MEI 10.17, Preference 0.03
    - Investment: Tech 58.55, Cap. Qual. 72.36, Gov. Spen. 36.48, Monetary Policy 12.04, MEI 22.91, Preference 1.20 (table formatting in source lists these values per column)
    - Capital: Tech 9.92, Cap. Qual. 2.89, Gov. Spen. 9.41, Monetary Policy 0.70, MEI 3.65, Preference 80.30
    - Gov’t Spending: Tech 4.20, Cap. Qual. 5.45, Gov. Spen. 73.42, Monetary Policy 8.99, MEI 1.20, Preference 0.09
    - Asset Prices: Tech 34.02, Cap. Qual. 44.53, Gov. Spen. 3.37, Monetary Policy 4.30, MEI 7.71, Preference 11.81
    - Welfare: Tech 11.81, Cap. Qual. 52.59, Gov. Spen. 9.89, Monetary Policy 4.92, MEI 8.15, Preference 0.13
    - Inflation: Tech 6.95, Cap. Qual. 8.95, Gov. Spen. 0.94, Monetary Policy 60.99, MEI 14.11, Preference 2.24
- Mechanisms and interpretation:
  - Financial shock defined as a shock affecting the quality of productive capital stock and closely correlated with the spread (external financing premium).
  - In presence of financial frictions, a negative capital quality shock:
    - creates capital losses in the financial sector,
    - reduces intermediaries’ net asset positions,
    - tightens credit and amplifies downturns beyond the direct capital accumulation channel.
  - Three transmission channels for financial shocks:
    - capital stock de facto depreciates more rapidly,
    - output of intermediate producers falls,
    - financial balance sheets deteriorate as stock valuations fall.
  - Contrast with models without financial frictions: capital quality shocks mainly affect time-to-build capital accumulation; with frictions, bank balance sheet channel amplifies effects.

### Impulse response functions (IRFs)
- Computation method:
  - IRFs computed as expected future path of endogenous variables conditional on a one standard deviation shock in period 1, evaluated at estimated parameter modes.
- Technology shock (negative one standard deviation TFP shock; quarterly autoregressive factor 0.96):
  - Amplified responses relative to standard New Keynesian models due to bank balance sheet channel.
  - Dynamics:
    - Investment declines sharply, lowering shadow price of capital.
    - Intermediaries’ asset side deteriorates, limiting lending and raising spread.
    - Higher cost of funding reduces demand for capital, further reducing output, investment, shadow price.
    - Inflation and interest rate rise but return to steady state relatively quickly.
    - Household consumption falls; hours of work increase.
    - Output and investment decline in a hump-shaped manner.
- Monetary policy shock (unanticipated one standard deviation increase in short-term nominal interest rate):
  - Degree of interest rate smoothing is less than in SW (2007), so peak effects occur at earlier horizons.
  - Financial frictions amplify responses because of strongly countercyclical premium.
  - Peak effect of policy shock on inflation occurs before peak effect on output (consistent with SW (2007)).
- Capital quality shock (negative one standard deviation financial shock):
  - Generates a recession with two notable features:
    - Severe declines in output, investment, consumption, interest rate, and capital stock.
    - Slow recovery driven by prolonged deleveraging.
  - Quantitative IRF magnitudes reported:
    - Asset price drops 1.1% below steady state on impact.
    - Output initially falls about 0.5 percent and decreases further to about 0.97 percent relative to zero within five quarters.
    - Investment drops to 1.4 percent below zero on impact.
    - Hours of work initially drop to 0.16 percent below steady state but increase to a maximum of 1 percent after 20 quarters; then decline slowly while remaining positive relative to steady state.
  - Mechanism: asset price decline shocks intermediaries’ balance sheets → spread soars → cost of funding increases → banks’ high leverage amplifies capital stock decline and slows recovery.

*Source: _wp14194 - Section 3*

### Section 4

### _wp14194 - Section 4

### Impulse responses to capital quality shock
- Figure 7: Impulse responses to capital quality shock.
- Impulse responses show:
  - Output: recovery occurs only after 6 quarters and it takes 5 years for it to return to the pre-crisis steady state.
  - Hours: model lacks wage rigidities, which further affects the response of hours; to get a more accurate response of hours, one would need to include appropriate labor market imperfections in the model.
  - During the recovery period, the intermediary sector deleverages by building up equity relative to assets.
- Chart axes and labels (as shown in source):
  - Output % ∆ SS (vertical scale includes values such as −0.9, −0.8, −0.7, −0.6, −0.5)
  - Investment % ∆ SS (vertical scale includes values such as −2, −1, 0)
  - Hours % ∆ SS (vertical scale includes values such as −0.2, 0, 0.2, 0.4, 0.6, 0.8)
  - Capital % ∆ SS (vertical scale includes values such as −2.5, −2, −1.5, −1, −0.5)
  - Inflation % ∆ SS (vertical scale includes values such as −0.15, −0.1, −0.05, 0)
  - Interest Rate % ∆ SS (vertical scale includes values such as −0.3, −0.2, −0.1, 0)
  - Spread % ∆ SS (vertical scale includes values such as 0, 0.05, 0.1)
  - Asset Price % ∆ SS (vertical scale includes values such as −1, −0.5, 0)

### Recovery dynamics and deleveraging
- Empirical point: banking crises preceded by credit bubbles are typically followed by prolonged deleveraging.
- Data summary: the average deleveraging period is 5 to 6 years and typically starts 2 years after the onset of a financial crisis.
- Model implication: slow recovery consistent with evidence from the recent financial crisis.

### Model, estimation, and sources of fluctuations
- Model developed:
  - A New Keynesian DSGE model featuring endogenous market friction, based on the model of Gertler and Karadi (2011), with financial shocks and marginal efficiency of investment shocks.
- Estimation:
  - Model parameters are estimated using a Bayesian technique.
  - Fit assessed by comparing the marginal data density (MDD) of the model with the frictionless economy described in Smets and Wouters (2007).
  - Finding: a Bayesian model comparison would pick the model with financial frictions over the baseline frictionless model.
- Sources of business cycle fluctuations:
  - Analysis in the spirit of Justiniano, Primiceri, and Tambalotti (2010).
  - Presence of financial frictions provides a framework emphasizing the role of the credit spread and the wedge between marginal efficiency of investment shocks and financial shocks in explaining the US business cycle.
  - Variance decomposition result: with the presence of financial intermediaries explicitly in the model, financial shocks are the predominant source of variability in key macroeconomic variables at business cycle frequencies.

### Policy implications and recommendations
- Key conclusion: policies that reduce the costs associated with financial transactions would be most effective in facilitating economic recovery.
- Modeling recommendation: when analyzing the impact of economic policies on business cycle developments, it is essential to use macroeconomic models that explicitly incorporate a financial sector.

### Appendix - Data
- Transformations and definitions (exact formulas as in source):
  - output = LN( GDPC96 / LNSindex ) * 100
  - investment = LN( ( FPI / GDPDEF ) / LNSindex ) * 100
  - inflation = LN( GDPDEF / GDPDEF(-1) ) * 100
  - interest rate = Federal Funds Rate / 4
- Data series definitions and sources (verbatim):
  - GDPC96:  Real Gross Domestic Product - Billions of Chained 1996 Dollars, Seasonally Adjusted Annual Rate.  Source:  US Department of Commerce, Bureau of Economic Analysis
  - GDPDEF: Gross Domestic Product - Implicit Price Deflator - 1996=100, Seasonally Adjusted. Source:  US Department of Commerce, Bureau of Economic Analysis
  - FPI: Fixed Private Investment - Billions of Dollars, Seasonally Adjusted Annual Rate.  Source: US Department of Commerce, Bureau of Economic Analysis
  - Federal Funds Rate:  Averages of Daily Figures - Percent.  Source:  Board of Governors of the Federal Reserve System.
  - LFU800000000:  Population level - 16 Years and Older - Not Seasonally Adjusted.  Source:  US Bureau of Labor Statistics
  - LNS10000000:  Labor Force Status :  Civilian noninstitutional population - Age :  16 years and over- Seasonally Adjusted - Number in thousands.  Source:  US Bureau of Labor Statistics.  (Before 1976 : LFU800000000 :  Population level - 16 Years and Older)

*Source: _wp14194 - Section 4*

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