## _wp15153 - Section 6 studies the monetary policy trade-offs faced by the ECB. Section 7 concludes.

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### Overview of model structure
- Model: extension of Quint and Rabanal (2014) with sticky wages and price markup shocks; two-region, two-sector, two-agent general equilibrium model of a single currency area.
- Regions: home (size n) and rest-of-euro-area (size 1−n).
- Sectors: durables (non-tradable) and non-durables (traded).
- Agents: savers (size λ in each region) and borrowers (size 1−λ) who differ in discount factor and habit formation.
- Financial structure: domestic financial intermediaries take deposits and grant loans; international intermediaries trade bonds across regions and charge a risk premium that depends on the net foreign asset position.

### Financial accelerator and credit markets
- Mechanism adapted from Bernanke, Gertler and Gilchrist (1999) applied to housing and residential investment; shocks to housing valuation affect borrower balance sheets, default rates, and lending-deposit spreads.
- Key features and equations:
  - idiosyncratic borrower housing shock ω^j_t: log(ω^j_t) ∼ N(−σ^2_{ω,t}/2, σ^2_{ω,t}); σ_{ω,t} follows log(σ_{ω,t}) = (1−ρ_{σ_ω}) log(σ̄_ω) + ρ_{σ_ω} log(σ_{ω,t−1}) + u_{ω,t} with u_{ω,t} ∼ N(0, σ^2_{u_ω}); E[ω_t] = 1.
  - Ex-post default cutoff: ω̄^p_{t−1} = R^L_{t−1} S^B_{t−1} / (P^D_t D^B_t); default if ω < ω̄^p_{t−1}.
  - Debt-collection agency fee μ; banks receive (1−μ) of foreclosure proceeds; profits transferred to savers.
  - Participation/ex-ante threshold: ω̄^a_t determined by ω̄^a_t E_t[P^D_{t+1} D^B_{t+1}] = R^L_t S^B_t.
  - Deposit rate relation: R_t = E_t{ (1−μ) G(ω̄^a_t, σ_{ω,t}) P^D_{t+1} D^B_{t+1} / S^B_t + [1−F(ω̄^a_t, σ_{ω,t})] R^L_t }.
  - Lending-deposit ratio: R^L_t / R_t = E_t{ 1 / [ (1−μ) G(ω̄^a_t, σ_{ω,t}) / ω̄^a_t + [1−F(ω̄^a_t, σ_{ω,t}) ] ] }.
  - International intermediaries’ spread rule: R^*_t = R_t + { θ_t exp[ κ_B ( B_t / (P^C_t Y_C) ) ] − 1 }.
  - Spread depends on B_t / P^C_t relative to Y_C; κ_B elastic; θ_t region-wide risk premium shock. Profits (R^*_t − R_t) B_t split equally between savers of both countries.
  - Aggregate domestic intermediary balance sheet: n λ (S_t − B_t) = n (1−λ) S^B_t.
- Two departures from BGG: no agency problems/asymmetric information; one-period lending rate R^L_t is pre-determined at time t and does not depend on the state at t+1.

### Households, wages, and heterogeneity
- Savers (j ∈ [0, λ]) preferences:
  - E_0 ∑_{t=0}^∞ β^t [ γ ξ^C_t log(C_{j,t} − ε C_{t−1}) + (1−γ) ξ^D_t log(D_{j,t}) − (L_{j,t})^{1+φ}/(1+φ) ].
  - Two shocks to preferences: ξ^C_t (non-durable consumption) and ξ^D_t (housing demand); housing demand shock central to housing-credit cycles.
- Consumption and labor aggregation:
  - C_{j,t} = [ τ^{1/ι_C} (C^H_{j,t})^{(ι_C−1)/ι_C} + (1−τ)^{1/ι_C} (C^F_{j,t})^{(ι_C−1)/ι_C} ]^{ι_C/(ι_C−1)}.
  - L_{j,t} = [ α^{−ι_L} (L^C_{j,t})^{1+ι_L} + (1−α)^{−ι_L} (L^D_{j,t})^{1+ι_L} ]^{1/(1+ι_L)}.
- Savers’ nominal budget constraint:
  - P^C_t C_{j,t} + P^D_t I_{j,t} + S_{j,t} ≤ R_{t−1} S_{j,t−1} + W^C_t L^C_{j,t} + W^D_t L^D_{j,t} + Π_{j,t}.
- Housing law of motion:
  - D_{j,t} = (1−δ) D_{j,t−1} + [ 1 − z(I_{j,t−1} / I_{j,t−2}) ] I_{j,t−1}; z(·) convex with z̄ = z̄′ = 0 and z̄′′ > 0.
- Wage setting:
  - Nominal wages set by two unions under Calvo stickiness with readjust probabilities 1−θ_{C,W} and 1−θ_{D,W}; non-readjusted wages partially indexed with fractions φ_{C,W}, φ_{D,W}. Unions maximize savers’ utility; borrowers are union members but unions are run by savers.
- Borrowers (j ∈ [λ, 1]):
  - Lower discount factor β_B < β and possibly different habit ε_B.
  - Do not receive profits; subject to ω^j_t and default risk.
  - Aggregated borrower budget constraint provided in exact form in source.

### Firms, technology, and price dynamics
- Final goods aggregator for k = C, D with time-varying elasticity σ^k_t (iid price mark-up shocks).
- CPI aggregation for home region: P^C_t = [ τ (P^H_t)^{1−ι_C} + (1−τ) (P^F_t)^{1−ι_C} ]^{1/(1−ι_C)}.
- Intermediate production uses labor only:
  - Y^C_t(h) = A_t Z^C_t L^C_t(h), Y^D_t(h) = A_t Z^D_t L^D_t(h).
  - Non-stationary union-wide technology shock: log(A_t) = log(A_{t−1}) + ε^A_t (unit root).
- Real marginal costs:
  - MC^C_t = W^C_t / (P^H_t A_t Z^C_t), MC^D_t = W^D_t / (P^D_t A_t Z^D_t).
- Calvo price-setting with sector-specific indexation; non-durable price mark-up μ^C_t = σ^C_t / (σ^C_t − 1) = μ^C exp(ε^{μ^C}_t) with iid innovation ε^{μ^C}_t.

### Market clearing, net foreign assets, and closing conditions
- Goods market clearing (home non-durable sector):
  - n Y^C_t = n [ λ C^H_t + (1−λ) C^B,H_t ] + (1−n) [ λ^* C^{*H}_t + (1−λ^*) C^{B*H}_t ].
- Durable goods (domestic only):
  - n Y^D_t = n [ λ I_t + (1−λ) I^B_t ].
- Labor market clearing for k = C, D stated in aggregate integrals.
- Credit market clearing including international bond positions:
  - n λ B_t + (1−n) λ^* B^*_t = 0.
- Home NFA evolution (law of motion for home intermediaries’ bonds) provided in exact form in source.

### Monetary policy rule
- ECB (currency-union level) sets deposit rate R_t:
  - R_t = [ R̄ (P^{EMU}_t / P^{EMU}_{t−1} / Π̄_{EMU})^{γ_π} (Y^{EMU}_t / Y^{EMU}_{t−1})^{γ_y} ]^{1−γ_R} R^{γ_R}_{t−1} exp(ε^m_t).
- Union-wide aggregates:
  - P^{EMU}_t = (P^C_t)^n (P^{C*}_t)^{1−n}; Y^{EMU}_t = (Y_t)^n (Y^*_t)^{1−n}.
  - National real GDP in non-durables: Y_t = Y^C_t + Y^D_t P^D_t / P^C_t.

### Estimation approach and data
- Estimation: Bayesian methods (An and Schorfheide, 2007); log-linearization around steady state with zero inflation and stationary NFA positions; state-space representation; likelihood via Kalman filter; Metropolis-Hastings algorithm.
- Dynare version used: Dynare 4.3.2; posterior distributions based on 250,000 draws.
- Data:
  - Quarterly data from 2000q1–2013q4.
  - Core region: aggregated France and Germany.
  - HBS (rest-of-euro-area) region: Greece, Ireland, Italy, Portugal, Spain.
  - Observables: real private consumption spending; real residential investment; real GDP; HICP; housing prices; outstanding household debt; 3-month Euribor (policy rate counterpart).
  - Aggregation uses HICP household expenditure weights; series transformed to quarterly growth rates (prices/quantities); interest rates divided by 400; all series demeaned.
- Measurement mapping:
  - Model GDP = non-durable consumption + residential investment; business investment, government spending and net exports to third countries captured by an aggregate demand shock via gdp_t = (1− ̄g) y_t + ̄g (g_t) with ̄g steady-state ratio and g_t AR(1).

### Calibrated parameters (Table 1)
- Calibrated values preserved exactly:
  - β Discount factor savers = 0.99
  - ̄ω Loan-to-value ratio = 0.7
  - ̄F Default rate on loans = 0.025
  - ̄σ_ω Steady state risk = 0.1742
  - μ Proportion of housing value paid to debt-collection agency = 0.2
  - β_B Discount factor borrowers = 0.985
  - δ Depreciation rate = 0.0125
  - σ Elasticity of substitution between intermediate goods = 10
  - σ_L Elasticity of substitution between labor types = 10
  - n Size core economies = 0.6
  - ̄g Fraction of exogenous demand in GDP = 0.3
  - 1−τ Fraction of imported goods from HBS to core economies = 0.06
  - 1−τ* Fraction of imported goods from core to HBS economies = 0.09
  - α Size of non-durable sector in GDP = 0.94
- Additional calibrated implications:
  - δ assumed annual 5 percent and equal across countries (δ = δ* = 0.0125).
  - Mark-ups implied by σ and σ_L equal 10 percent.
  - Steady-state risk shock value ̄σ_ω = 0.1742; debt-collection fee μ = 0.2; β_B implied = 0.985.
  - Assumed symmetry α = α* to simplify steady state.

### Priors, posteriors and selected estimation results (Tables 2 and 3)
- Key posterior means and intervals (selected, exact numbers preserved):
  - λ Fraction of savers: prior Beta0.50.05; posterior Mean 0.57; 90% C.S. [0.50,0.64]
  - ε Habit formation savers: prior Beta0.50.15; posterior Mean 0.71; 90% C.S. [0.65,0.78]
  - εB Habit formation borrowers: prior Beta0.50.15; posterior Mean 0.63; 90% C.S. [0.52,0.73]
  - γπ Taylor rule reaction to inflation: prior Normal1.50.1; posterior Mean 1.34; 90% C.S. [1.16,1.50]
  - γy Taylor rule reaction to real growth: prior Gamma0.20.05; posterior Mean 0.29; 90% C.S. [0.19,0.41]
  - γr Interest rate smoothing: prior Beta0.66.0.15; posterior Mean 0.84; 90% C.S. [0.81,0.87]
  - κB International risk premium: prior Gamma0.0050.002; posterior Mean 0.006; 90% C.S. [0.002,0.009]
- Region-specific Calvo/posterior examples:
  - θC (price non-durables, core): posterior Mean 0.87; 90% C.S. [0.82,0.92]
  - θ∗C (price non-durables, HBS): posterior Mean 0.93; 90% C.S. [0.89,0.97]
  - θD (price durables, core): posterior Mean 0.50; 90% C.S. [0.39,0.61]
  - θ∗D (price durables, HBS): posterior Mean 0.43; 90% C.S. [0.31,0.54]
- AR(1) posterior means (selected):
  - ρZ,C Technology, non-durables: posterior Mean 0.76; 90% C.S. [0.67,0.84]
  - ρZ,D Technology, durables: posterior Mean 0.86; 90% C.S. [0.79,0.94]
  - ρξ,D Preference, durables: posterior Mean 0.96; 90% C.S. [0.94,0.98]
  - ρθ Risk premium, core-HBS: posterior Mean 0.87; 90% C.S. [0.82,0.92]
- Selected posterior shock standard deviations:
  - σA Technology, EMU-wide: posterior Mean 0.65; 90% C.S. [0.49,0.81]
  - σC,Z Technology, non-durables (core): posterior Mean 0.97; 90% C.S. [0.62,1.32]
  - σD,Z Technology, durables (core): posterior Mean 1.09; 90% C.S. [0.78,1.41]
  - σuω Risk shock (core): posterior Mean 12.9; 90% C.S. [9.74,15.88]
  - σu∗ω Risk shock (HBS): posterior Mean 33.47; 90% C.S. [27.13, 39.42]
- Additional notes:
  - Prior for fraction of savers centered at 0.5 (sd 0.05); posterior mean = 0.57.
  - Habit formation: borrowers ≈ 0.71; savers ≈ 0.63 (as reported in overview).
  - Elasticity of substitution between home and rest-of-euro-area non-durables posterior mean = 1.50.

### Variance decomposition — role of demand, financial and housing shocks (posterior mode, selected shares in percent)
- Core:
  - Output: Non-Durable Preference 38.4; Aggregate Demand 15.7; Technology 23.6; Monetary 8.9; Financial 1.6; Housing Pref. 9.9; Markups 1.9
  - Potential: Technology 75.9; Housing Pref. 5.1; Non-Durable Preference 1.1; Aggregate Demand 6.8; Monetary 0.0; Markups 0.0
  - Gap: Technology 49.6; Aggregate Demand 24.7; Monetary 9.3; Non-Durable Preference 2.5; Financial 6.0
  - Inflation: Technology 30.5; Markups 33.1; Monetary 17.8; Financial 8.4
  - Credit Growth: Housing Pref. 73.4; Technology 0.7; Financial 8.8; Monetary 0.1
  - House Prices: Housing Pref. 82.4; Technology 0.7; Monetary 1.7
- HBS:
  - Output: Technology 34.4; Financial 29.1; Housing Pref. 15.7; Aggregate Demand 7.2; Monetary 5.6
  - Potential: Technology 66.8; Housing Pref. 19.1; Financial 9.4
  - Gap: Financial 64.7; Technology 12.7; Housing Pref. 2.8; Aggregate Demand 1.6; Monetary 7.0
  - Inflation: Technology 3.9; Financial 19.9; Monetary 10.7; Markups 29.2
  - Credit Growth: Housing Pref. 90.8; Financial 1.8; Aggregate Demand 1.2
  - House Prices: Housing Pref. 88.3; Technology 0.2; Monetary 0.6
- EMU aggregates:
  - Gap: Technology 34.9; Financial 23.2; Aggregate Demand 20.1; Monetary 13.0
  - Inflation: Technology 35.1; Markups 23.7; Monetary 18.2; Financial 13.3

### Decomposing the business cycle and potential output
- Potential output defined as output with flexible prices and wages but with financial frictions, monopolistic competition and other real frictions in place; price markup shocks removed from potential.
- Potential output estimated unconditionally as counterfactual under flexible prices/wages; depends on counterfactual housing stock independent from past policy.
- Posterior modes numerically very close to posterior means in Tables 2 and 3.

### HBS boom-and-bust narrative and quantitative findings
- Aggregate HBS experienced a large housing- and credit-fueled boom and bust:
  - Housing demand shocks accounted for between a third and a half of the contribution to the boom (2002-2008) despite the housing sector size of 6 percent of GDP (long-run calibration).
  - Collapse of output in 2012-2013 mostly attributed to financial shocks (intra-european financial tensions and sudden stop).
- Output gap:
  - Estimated output gap about −4 percent of GDP at end of 2013.
  - Mid-2000s boom: gap mainly driven by financial shocks (region-wide and housing risk).
- Credit and house prices:
  - Virtually all credit deviations from trend driven by housing preference shock in aggregate HBS.
  - Initial credit boom (2001-2004) included financial shock contributions; later boom dominated by housing demand shocks.
  - Monetary policy had a small effect on credit.

### Core: different drivers and smaller fluctuations
- Core determinants:
  - Output largely driven by non-durable preference shocks (38.4 percent) and aggregate demand (15.7 percent).
  - Output gap driven by technology (49.6 percent) and aggregate demand (24.7 percent).
  - Monetary shocks explain 8.9–9.3 percent of output/gap fluctuations.
- Credit and house prices:
  - No large credit boom; house prices less volatile and driven by housing preference, productivity and monetary shocks.
- Output gap at end-2013: model implies gap close to zero for the core.

### Role of financial frictions (counterfactual λ = 1 vs baseline)
- In the core: output gaps with and without financial frictions are very similar; HP filter measure also similar to model gaps.
- In the HBS: model with financial frictions produces a more volatile output gap than without frictions; financial and housing preference shocks amplify through the financial accelerator.
- Financial wedge defined as FWt = gapFFt − gapNOFFt = ỹNOFFt − ỹFFt (difference between potential outputs).
  - Boom period: financial shocks explain larger gap early on; from 2003 housing preference shocks become main shock amplified by financial friction.
  - Crisis: negative housing demand shocks amplified by accelerator deepen negative gap.
- HP filter contrasts with model-based evidence: HP gives near-zero gap by 2013, whereas model gap is −4 percent and consistent with high unemployment.

### Monetary policy trade-offs and empirical stance
- Natural real interest rates (NRIR) computed as level of real rates consistent with flexible prices/wages and excluding inefficient (mark-up) shocks:
  - Core NRIR declined over time and has been below historical mean since the global financial crisis; it tracks the output gap—declined 2000-2006 then jumped in 2007.
  - HBS NRIR was small but positive during 2000-2007 overheating period.
  - Tension 2003-2006: core needed lower rates while HBS needed higher rates; later synchronization during bust made natural rates converge in sign.
- Regional deviations from Taylor-rule prescription measured as Devit = rt − rT,i t (rt = 3-month Euribor; rT,i t uses estimated Taylor-rule parameters with region-specific CPI and output growth). Positive deviation implies contractionary stance.
- Empirical stance over sample:
  - Core: contractionary 2000-2003; about right 2004-2006; too contractionary 2008-2009 after ECB tightening; expansionary from 2010 onward.
  - HBS: initially contractionary; largely expansionary 2002-2006 (procyclical for HBS); contractionary end of boom around 2007-2008; expansionary during crisis.
- Conclusion: pre-crisis ECB faced trade-offs across regions; post-crisis synchronization eased common policy implementation but monetary stimulus insufficient to fully close HBS output gap. Region-specific macroeconomic policies (including macroprudential and fiscal tools) remain necessary.

### Impulse response analysis (IRFs) — magnitudes and channels (selected qualitative magnitudes)
- Financial shocks:
  - Real quantities and prices decline via financial accelerator; impact larger in HBS; spillovers to core sizable.
  - Core housing sector risk shock: output decline of about 0.06 percent below steady state; CPI inflation initially falls about 0.025 percent.
  - HBS housing sector risk shock: output contraction of almost 0.2 percent; larger declines in house prices and CPI inflation.
  - HBS risk premium shock: largest macro impact — contraction in output, CPI inflation and house prices close to three times larger than other financial shocks; ECB cuts rates more forcefully.
- Housing demand shocks:
  - Declines in residential investment and house prices transmit to nondurable sector via balance sheet effects, yielding long-lasting declines in CPI inflation and output.
  - Spillovers differ by region: after a housing bust in the core, HBS output may decline because trade channel outweighs interest rate cut; core may experience positive spillovers if monetary policy effect dominates trade effect.
- Monetary policy shocks:
  - Transmission mechanically similar across regions; real effects similar and hump-shaped.
  - House price response to tightening larger in HBS; CPI inflation response somewhat larger in core.

### Key policy-relevant conclusions
- Inclusion of financial variables, frictions and housing matters especially for countries with large housing and credit fluctuations (aggregate HBS during 2000s); it materially changes assessment of cyclical position relative to HP filter.
- In the euro area core, where there was no credit boom, including financial variables does not materially change assessments relative to the HP filter.
- The DSGE model yields an estimated negative output gap of 4 percent of GDP by end-2013 for HBS aggregate, in contrast to a much smaller HP-filter gap.
- Monetary policy faced a trade-off pre-crisis due to differing regional cyclical positions and natural rates; post-crisis synchronization eased a common policy stance but did not obviate the need for region-specific macroeconomic policies (including macroprudential and fiscal tools).
- Model uncertainty remains important: different modeling choices produce different output gap estimates and further work on model uncertainty is acknowledged.

### Appendix B — linearized equilibrium conditions and shock processes (overview)
- Appendix B provides the full set of normalized, log-linear equilibrium conditions and measurement equations, including:
  - Notation and normalization for steady state and log-deviations; definitions for Q_t, real wage deviations, normalized real debt ˜S^B_t, b_t deviations of foreign assets, etc.
  - Representative log-linearized conditions labeled (B.1)–(B.78) covering savers’, borrowers’, intermediaries’, firms’, wage and price Phillips curves, housing dynamics, market clearing, aggregation, and ECB policy rule.
  - Cross-region interest rate relation (B.71) and NFA evolution (B.72) preserved in exact form.
  - Measurement linking model output to data: gdp_t = (1− ̄g) y_t + ̄g (g_t) (B.77).
  - Shock process system for stationary and non-stationary shocks (B.79)–(B.91) with σ_{ω,t} dynamics (B.87)–(B.88) and union-wide iid innovations for ε^A_t, ε^m_t, and ε^{μ}_t.
- Estimation implementation details: Dynare 4.3.2; Metropolis-Hastings with 250,000 draws; data transformations and aggregation procedures described in Appendix A.

*Source: _wp15153 - Section 6 studies the monetary policy trade-offs faced by the ECB. Section 7 concludes. (Appendix B and related appendices summarized exactly as in source content.)*

### Section 6 studies the monetary policy trade-offs faced by the ECB. Section 7 concludes.

### _wp15153 - Section 6 studies the monetary policy trade-offs faced by the ECB. Section 7 concludes.

### Overview of the model structure
- Model: extension of Quint and Rabanal (2014) with sticky wages and price markup shocks; two-region, two-sector, two-agent general equilibrium model of a single currency area.
- Regions: home (size n) and rest-of-euro-area (size 1−n).
- Sectors: durables (non-tradable) and non-durables (traded).
- Agents: savers (size λ in each region) and borrowers (size 1−λ) who differ in discount factor and habit formation.
- Financial structure: domestic financial intermediaries take deposits and grant loans; international intermediaries trade bonds across regions and charge a risk premium that depends on the net foreign asset position.

### Financial accelerator and credit markets
- Mechanism adapted from Bernanke, Gertler and Gilchrist (1999) (BGG) applied to housing and residential investment; shocks to housing valuation affect borrower balance sheets, default rates, and lending-deposit spreads.
- Two differences from BGG:
  - No agency problems or asymmetric information; borrowers default only when underwater (value of outstanding debt > value of house).
  - One-period lending rate R^L_t is pre-determined at time t and does not depend on the state at t+1.
- Idiosyncratic borrower housing shock ω^j_t:
  - log(ω^j_t) ∼ N(−σ^2_{ω,t}/2, σ^2_{ω,t}).
  - σ_{ω,t} is time-varying and follows log(σ_{ω,t}) = (1−ρ_{σ_ω}) log(σ̄_ω) + ρ_{σ_ω} log(σ_{ω,t−1}) + u_{ω,t} with u_{ω,t} ∼ N(0, σ^2_{u_ω}).
  - E[ω_t] = 1 (idiosyncratic risk but no aggregate risk in housing market).
- Default and foreclosure:
  - Ex-post default cutoff: ω̄^p_{t−1} = R^L_{t−1} S^B_{t−1} / (P^D_t D^B_t). Households default when ω realization < ω̄^p_{t−1}.
  - Debt-collection agencies charge fraction μ of house value; banks receive (1−μ) of foreclosure proceeds; profits transferred to savers.
- Participation constraint and lending spread:
  - Ex-ante threshold ω̄^a_t determined by ω̄^a_t E_t[P^D_{t+1} D^B_{t+1}] = R^L_t S^B_t.
  - Deposit rate R_t equals expected return to lending: R_t = E_t{ (1−μ) G(ω̄^a_t, σ_{ω,t}) P^D_{t+1} D^B_{t+1} / S^B_t + [1−F(ω̄^a_t, σ_{ω,t})] R^L_t }.
  - Relationship: R^L_t / R_t = E_t{ 1 / [ (1−μ) G(ω̄^a_t, σ_{ω,t}) / ω̄^a_t + [1−F(ω̄^a_t, σ_{ω,t}) ] ] }.
  - For given credit demand, σ_{ω,t}, and E_t[P^D_{t+1} D^B_{t+1}], intermediaries set R^L_t and ω̄^a_t to satisfy constraints; participation constraint delivers ex-ante zero profits but allows ex-post profits/losses (savers recapitalize as needed).
- International intermediaries’ spread rule:
  - R^*_t = R_t + { θ_t exp[ κ_B ( B_t / (P^C_t Y_C) ) ] − 1 }.
  - Spread depends on real net foreign assets B_t / P^C_t relative to steady-state non-durable GDP Y_C; κ_B is risk premium elasticity; θ_t is a region-wide risk premium shock.
  - Profits (R^*_t − R_t) B_t split equally between savers of both countries.
- Aggregate domestic intermediary balance sheet:
  - n λ (S_t − B_t) = n (1−λ) S^B_t.

### Households, wages, and heterogeneity
- Savers (j ∈ [0, λ]) maximize:
  - E_0 ∑_{t=0}^∞ β^t [ γ ξ^C_t log(C_{j,t} − ε C_{t−1}) + (1−γ) ξ^D_t log(D_{j,t}) − (L_{j,t})^{1+φ}/(1+φ) ].
  - Preferences hit by two shocks: ξ^C_t (non-durable consumption) and ξ^D_t (housing demand). Housing demand shock is key to explaining housing and credit boom-bust cycles.
- Non-durable consumption aggregator:
  - C_{j,t} = [ τ^{1/ι_C} (C^H_{j,t})^{(ι_C−1)/ι_C} + (1−τ)^{1/ι_C} (C^F_{j,t})^{(ι_C−1)/ι_C} ]^{ι_C/(ι_C−1)}.
- Labor aggregator with imperfect substitutability across sectors:
  - L_{j,t} = [ α^{−ι_L} (L^C_{j,t})^{1+ι_L} + (1−α)^{−ι_L} (L^D_{j,t})^{1+ι_L} ]^{1/(1+ι_L)}.
- Savers’ nominal budget constraint:
  - P^C_t C_{j,t} + P^D_t I_{j,t} + S_{j,t} ≤ R_{t−1} S_{j,t−1} + W^C_t L^C_{j,t} + W^D_t L^D_{j,t} + Π_{j,t}.
- Housing law of motion:
  - D_{j,t} = (1−δ) D_{j,t−1} + [ 1 − z(I_{j,t−1} / I_{j,t−2}) ] I_{j,t−1}; z(·) convex with z̄ = z̄′ = 0 and z̄′′ > 0.
- Wage setting:
  - Nominal wages set by two unions (one per sector) under Calvo stickiness with readjust probabilities 1−θ_{C,W} and 1−θ_{D,W}; non-readjusted wages partially indexed to past CPI inflation with fractions φ_{C,W}, φ_{D,W}. Unions maximize savers’ utility; borrowers are union members but unions are run by savers.
- Borrowers (j ∈ [λ, 1]):
  - Lower discount factor β_B < β and possibly different habit ε_B.
  - Do not receive profits from firms/intermediaries; subject to idiosyncratic housing-quality shock ω^j_t and default risk.
  - Aggregated borrower budget constraint:
    - P^C_t C^B_t + P^D_t [ I^B_t + G(ω̄^p_{t−1}, σ_{ω,t−1}) D^B_t ] + [1−F(ω̄^p_{t−1}, σ_{ω,t−1})] R^L_{t−1} S^B_{t−1} ≤ S^B_t + W^C_t L^C,B_t + W^D_t L^D,B_t.

### Firms, technology, and price dynamics
- Final goods (for k = C, D):
  - Y^k_t ≡ [ (1/n)^{1/σ^k_t} ∫_0^n Y^k_t(h)^{(σ^k_t−1)/σ^k_t} dh ]^{σ^k_t/(σ^k_t−1)} with time-varying elasticity σ^k_t (iid price mark-up shocks).
- CPI for home region:
  - P^C_t = [ τ (P^H_t)^{1−ι_C} + (1−τ) (P^F_t)^{1−ι_C} ]^{1/(1−ι_C)}.
- Intermediate goods production (labor only):
  - Y^C_t(h) = A_t Z^C_t L^C_t(h), Y^D_t(h) = A_t Z^D_t L^D_t(h).
  - Non-stationary union-wide technology shock: log(A_t) = log(A_{t−1}) + ε^A_t (unit root).
- Real marginal costs:
  - MC^C_t = W^C_t / (P^H_t A_t Z^C_t), MC^D_t = W^D_t / (P^D_t A_t Z^D_t).
- Calvo price-setting with sector-specific indexation; sectoral inflation depends on one lead and one lag of inflation and real marginal cost; non-durable price mark-up μ^C_t = σ^C_t / (σ^C_t − 1) = μ^C exp(ε^{μ^C}_t) with iid innovation ε^{μ^C}_t.

### Market clearing, net foreign assets, and closing conditions
- Goods market clearing (home non-durable sector):
  - n Y^C_t = n [ λ C^H_t + (1−λ) C^B,H_t ] + (1−n) [ λ^* C^{*H}_t + (1−λ^*) C^{B*H}_t ].
- Durable goods (domestic only):
  - n Y^D_t = n [ λ I_t + (1−λ) I^B_t ].
- Labor market clearing for each sector k = C, D:
  - ∫_0^n L^k_t(h) dh = λ ∫_0^n L^k,j_t dj + (1−λ) ∫_0^n L^{k,B}_j_t dj.
- Credit market clearing including international bond positions:
  - n λ B_t + (1−n) λ^* B^*_t = 0.
- Law of motion for home international intermediaries’ bonds (home NFA evolution):
  - n λ B_t = n λ R_{t−1} B_{t−1} + { (1−n) P^H_t [ λ^* C^{*H}_t + (1−λ^*) C^{B*H}_t ] − n P^F_t [ λ C^F_t + (1−λ) C^{B F}_t ] }.

### Monetary policy
- Central bank at currency-union level sets deposit rate R_t according to rule:
  - R_t = [ R̄ (P^{EMU}_t / P^{EMU}_{t−1} / Π̄_{EMU})^{γ_π} (Y^{EMU}_t / Y^{EMU}_{t−1})^{γ_y} ]^{1−γ_R} R^{γ_R}_{t−1} exp(ε^m_t).
- Union-wide aggregates:
  - P^{EMU}_t = (P^C_t)^n (P^{C*}_t)^{1−n}; Y^{EMU}_t = (Y_t)^n (Y^*_t)^{1−n}.
  - National real GDP expressed in non-durables: Y_t = Y^C_t + Y^D_t P^D_t / P^C_t; analogous for Y^*_t.

### Estimation approach and data
- Estimation: Bayesian methods (An and Schorfheide, 2007); log-linearization around steady state with zero inflation and stationary NFA positions; state-space representation and likelihood via Kalman filter; Metropolis-Hastings algorithm for posteriors.
- Data:
  - Quarterly data from 2000q1–2013q4.
  - Core (home) region: aggregated France and Germany.
  - HBS (rest-of-euro-area) region: Greece, Ireland, Italy, Portugal, Spain.
  - Thirteen macroeconomic time series used (listing of series omitted in source content).

*Source: _wp15153 - Section 6 studies the monetary policy trade-offs faced by the ECB. Section 7 concludes.*

### Appendix B details the full set of normalized, linearized equilibrium conditions of the model.

### _wp15153 - Appendix B details the full set of normalized, linearized equilibrium conditions of the model.

### Estimation approach and data
- Estimation is done using Dynare 4.3.2.
- The posterior distributions are based on 250,000 draws of the Metropolis-Hastings algorithm.
- Observables (six per region plus a policy rate):
  - real private consumption spending
  - real residential investment
  - real gross domestic product (GDP)
  - the harmonized index of consumer prices (HICP)
  - housing prices
  - outstanding debt for households
  - 3-month Euribor rate (used as counterpart of the deposit rate in the core)
- Aggregation:
  - Data is aggregated taking the economic size of the countries into account, using the household expenditure weights used by the Harmonised Index of Consumer Prices (HICP) for euro area countries.
- Data transformations:
  - Quarterly growth rates of all price and quantity (seasonally adjusted) data are used.
  - Interest rates are divided by 400 to obtain a quarterly and logged equivalent variable to the model.
  - All series are demeaned.
- GDP mapping between model and data:
  - The model’s measure of GDP includes only non-durable consumption and residential investment; it leaves out business investment, government spending and net exports with third countries.
  - An aggregate demand shock collects these omitted components. In log-linear form:
    - gdp_t = (1− ̄g) y_t + ̄g (g_t)
    - where ̄g is the steady state ratio of exogenous demand to GDP, and g_t is an exogenous AR(1) process.

### Calibrated parameters (Table 1)
- Parameters calibrated because observables do not identify them:
  - β Discount factor savers = 0.99
  - ̄ω Loan-to-value ratio = 0.7
  - ̄F Default rate on loans = 0.025
  - ̄σ_ω Steady state risk = 0.1742
  - μ Proportion of housing value paid to debt-collection agency = 0.2
  - β_B Discount factor borrowers = 0.985
  - δ Depreciation rate = 0.0125
  - σ Elasticity of substitution between intermediate goods = 10
  - σ_L Elasticity of substitution between labor types = 10
  - n Size core economies = 0.6
  - ̄g Fraction of exogenous demand in GDP = 0.3
  - 1−τ Fraction of imported goods from HBS to core economies = 0.06
  - 1−τ* Fraction of imported goods from core to HBS economies = 0.09
  - α Size of non-durable sector in GDP = 0.94
- Additional calibrated assumptions and implications:
  - The steady state value of the risk shock is ̄σ_ω = 0.1742.
  - Debt-collection agency fee μ = 0.2.
  - Using these values, the zero-profit condition for financial intermediaries and the consumption Euler equation for borrowers imply β_B = 0.985.
  - Depreciation rate assumed annual 5 percent and equal across countries (δ = δ* = 0.0125).
  - Degree of monopolistic competition in goods and labor markets (σ and σ_L) implies mark-ups of 10 percent.
  - Size of the core countries in the euro area set to n = 0.6.
  - Steady-state ratio of exogenous demand to GDP set to ̄g = 0.3 (̄g = ̄g*).
  - Bilateral trade parameters 1−τ and 1−τ* calibrated to match weighted average imports and to ensure trade balance and zero net foreign asset position in steady state.
  - Assumed symmetry: α = α* (size of durable and non-durable sectors same across core and HBS countries), simplifying steady state with all relative prices equal to one and per capita quantities equal.

### Prior and posterior distributions (Tables 2 and 3) — main findings
- Estimation strategy notes:
  - Given the short sample, some parameters are calibrated and others restricted to be the same across countries.
  - Parameters related to nominal rigidities and shocks are allowed to differ across sectors and countries.
  - Parameters relating to preferences, adjustment costs, and the fraction of savers are assumed the same in both countries.
  - Housing demand shock and TFP shock in non-durables include a common component across countries. Example for housing demand shock:
    - log(ξ_Dt) = ρ_{ξ,D} log(ξ_Dt−1) + ε_{ξ,Dt} + ε_{ξ,D,COMt}
    - log(ξ_D* t) = ρ*_{ξ,D} log(ξ_D* t−1) + ε_{ξ,D* t} + ε_{ξ,D,COMt}
    - Region-specific (ε_{ξ,Dt} and ε_{ξ,D* t}) and common (ε_{ξ,D,COMt}) innovations are Normal iid with mean zero.
- Preferences and household parameters:
  - Prior for fraction of savers centered at 0.5 with standard deviation 0.05 (highly informative).
  - Posterior mean for fraction of savers = 0.57.
  - Assumption: number of savers in each region (λ) is the same in both regions.
  - Habit formation coefficients:
    - Borrowers ≈ 0.71
    - Savers ≈ 0.63
  - Elasticity of substitution between home and rest-of-euro-area non-durables: posterior mean = 1.50 (equal to prior mean).
  - Labor parameters:
    - Labor disutility coefficient φ posterior mean = 1.14
    - Degree of costly labor reallocation ≈ 0.63
- Taylor rule and international risk premium:
  - Taylor rule coefficients:
    - Response to inflation = 1.34 (below prior mean)
    - Response to real GDP growth = 0.31
    - Interest rate inertia = 0.84
  - Risk premia elasticity κ_B prior: gamma with mean 0.01.
  - Empirical finding: the risk premium elasticity between countries moves about 0.6 basis points with a one percent increase in the external debt-to-GDP ratio.
- Nominal rigidities (Calvo and indexation):
  - Beta priors for all Calvo probabilities with mean 0.75 (average duration 4 quarters).
  - More informative priors for wage-setting (prior sd = 0.05) than price-setting (prior sd = 0.15).
  - Priors for indexation parameters mean = 0.33.
  - Findings on price and wage rigidity:
    - More price rigidity in the non-durable sector (higher Calvo probabilities) than in the durable sector; similar across countries.
    - Prices are reset about every 10 quarters in the non-durable sector and about 2 quarters in the durable sector.
    - Wage rigidity similar across countries and sectors, averaging wage durations between roughly 4 and 6 quarters.
    - Both price and wage indexation is low in all prices and sectors.
- Shocks and common components:
  - Common innovations to non-durable technology shocks and durable preference shocks are important.
  - Mean of the (log) risk shock: log(0.1742) = −1.74.
  - Prior standard deviation for the innovation to the housing risk shock = 0.25 (that is, 25 percent).

*Source: _wp15153 - Appendix B details the full set of normalized, linearized equilibrium conditions of the model.*

### 1.25 and -2.25.  Given the properties of the log-normal distribution, this means that the default

### _wp15153 - 1.25 and -2.25.  Given the properties of the log-normal distribution, this means that the default

### Parameter estimation and priors/posteriors (high-level)
- Default mortgage rate implied by log-normal assumption ranges between 0.04 and 13.6 percent with 95 percent probability.
- Selected posterior estimates (Table 2, common parameters):
  - λ Fraction of savers: prior Beta0.50.05; posterior Mean 0.57; 90% C.S. [0.50,0.64]
  - ε Habit formation savers: prior Beta0.50.15; posterior Mean 0.71; 90% C.S. [0.65,0.78]
  - εB Habit formation borrowers: prior Beta0.50.15; posterior Mean 0.63; 90% C.S. [0.52,0.73]
  - γπ Taylor rule reaction to inflation: prior Normal1.50.1; posterior Mean 1.34; 90% C.S. [1.16,1.50]
  - γy Taylor rule reaction to real growth: prior Gamma0.20.05; posterior Mean 0.29; 90% C.S. [0.19,0.41]
  - γr Interest rate smoothing: prior Beta0.66.0.15; posterior Mean 0.84; 90% C.S. [0.81,0.87]
  - κB International risk premium: prior Gamma0.0050.002; posterior Mean 0.006; 90% C.S. [0.002,0.009]
- Region-specific rigidities (selected):
  - θC Calvo lottery, price non-durables (core): prior Beta0.75.0.15; posterior Mean 0.87; 90% C.S. [0.82,0.92]
  - θ∗C Calvo lottery, price non-durables (HBS): prior Beta0.75.0.15; posterior Mean 0.93; 90% C.S. [0.89,0.97]
  - θD Calvo lottery, price durables (core): posterior Mean 0.50; 90% C.S. [0.39,0.61]
  - θ∗D Calvo lottery, price durables (HBS): posterior Mean 0.43; 90% C.S. [0.31,0.54]

### Shock process estimates (AR(1) coefficients and shock standard deviations)
- AR(1) posterior means (Table 3, selected):
  - ρZ,C Technology, non-durables: prior Beta0.70.10; posterior Mean 0.76; 90% C.S. [0.67,0.84]
  - ρZ,D Technology, durables: posterior Mean 0.86; 90% C.S. [0.79,0.94]
  - ρξ,D Preference, durables: posterior Mean 0.96; 90% C.S. [0.94,0.98]
  - ρθ Risk premium, core-HBS: posterior Mean 0.87; 90% C.S. [0.82,0.92]
- Selected posterior shock standard deviations (Table 3, Gamma priors):
  - σA Technology, EMU-wide: prior Gamma0.7 0.2; posterior Mean 0.65; 90% C.S. [0.49,0.81]
  - σC,Z Technology, non-durables (core): posterior Mean 0.97; 90% C.S. [0.62,1.32]
  - σD,Z Technology, durables (core): posterior Mean 1.09; 90% C.S. [0.78,1.41]
  - σuω Risk shock (core): prior Gamma25 12.5; posterior Mean 12.9; 90% C.S. [9.74,15.88]
  - σu∗ω Risk shock (HBS): prior Gamma25 12.5; posterior Mean 33.47; 90% C.S. [27.13, 39.42]

### Variance decomposition — Role of demand and financial shocks (Table 4, posterior mode)
- Core (shares of variance in percent):
  - Output: Non-Durable Preference 38.4; Aggregate Demand 15.7; Technology 23.6; Monetary 8.9; Financial 1.6; Housing Pref. 9.9; Markups 1.9
  - Potential: Technology 75.9; Housing Pref. 5.1; Non-Durable Preference 1.1; Aggregate Demand 6.8; Monetary 0.0; Markups 0.0
  - Gap: Technology 49.6; Aggregate Demand 24.7; Monetary 9.3; Non-Durable Preference 2.5; Financial 6.0
  - Inflation: Technology 30.5; Markups 33.1; Monetary 17.8; Financial 8.4
  - Credit Growth: Housing Pref. 73.4; Technology 0.7; Financial 8.8; Monetary 0.1
  - House Prices: Housing Pref. 82.4; Technology 0.7; Monetary 1.7
- HBS (shares of variance in percent):
  - Output: Technology 34.4; Financial 29.1; Housing Pref. 15.7; Aggregate Demand 7.2; Monetary 5.6
  - Potential: Technology 66.8; Housing Pref. 19.1; Financial 9.4
  - Gap: Financial 64.7; Technology 12.7; Housing Pref. 2.8; Aggregate Demand 1.6; Monetary 7.0
  - Inflation: Technology 3.9; Financial 19.9; Monetary 10.7; Markups 29.2
  - Credit Growth: Housing Pref. 90.8; Financial 1.8; Aggregate Demand 1.2
  - House Prices: Housing Pref. 88.3; Technology 0.2; Monetary 0.6
- EMU aggregates:
  - Gap: Technology 34.9; Financial 23.2; Aggregate Demand 20.1; Monetary 13.0
  - Inflation: Technology 35.1; Markups 23.7; Monetary 18.2; Financial 13.3

### Decomposing the business cycle and potential output
- Potential output definition: the level of output consistent with flexible prices and wages but with financial frictions, monopolistic competition and other real frictions in place; price markup shocks removed from potential.
- The paper estimates unconditional potential output (counterfactual had prices and wages always been flexible), with potential depending on counterfactual housing stock independent from past policy.
- Posterior modes are numerically very close to posterior means presented in Tables 2 and 3.

### HBS countries: boom-and-bust narrative and quantitative findings
- Aggregate HBS experienced a large housing- and credit-fueled boom and bust:
  - Housing demand shocks represented between a third and a half of the contribution to the boom (2002-2008) despite housing sector size of 6 percent of GDP (long-run average calibration).
  - Collapse of output in 2012-2013 mostly attributed to financial shocks (intra-european financial tensions and sudden stop).
- Output gap:
  - Estimated output gap about -4 percent of GDP at end of 2013.
  - During mid-2000s boom, output gap mainly driven by financial shocks (both region-wide and housing risk).
- Credit and house prices:
  - Virtually all credit deviations from trend are driven by the housing preference shock in the aggregate HBS.
  - Initial credit boom phase (2001-2004) had contributions from financial shocks; later boom dominated by housing demand shocks.
  - Monetary policy had a small effect on credit.

### Core: different drivers and smaller fluctuations
- Core determinants:
  - Output largely driven by non-durable preference shocks (38 percent) and aggregate demand (24 percent) for output behavior.
  - Output gap driven by technology (50 percent) and aggregate demand (25 percent).
  - Monetary shocks explain 9 percent of both output and gap fluctuations.
- Credit and house prices:
  - No large credit boom; house prices less volatile and driven by a mix of housing preference, productivity and monetary shocks.
- Output gap at end-2013: model implies gap close to zero for the core.

### Role of financial frictions
- Counterfactual comparison: model with financial frictions versus model without financial frictions (λ = 1) fed the same shocks and parameters to isolate effect of financial accelerator.
- Findings:
  - In the core, output gaps with and without financial frictions are very similar; HP filter measure is also similar to model gaps.
  - In the HBS countries, output gap from model with financial frictions is more volatile than without frictions; financial and housing demand shocks set the financial accelerator in motion.
  - The HP filter gives a different picture (negative output gap for mid-2000s) and implies a close-to-zero gap by 2013, which contrasts with model-based evidence (e.g., high unemployment and model gap of -4 percent).
- Financial wedge definition and behavior:
  - Financial wedge FWt = gapFFt − gapNOFFt = ỹNOFFt − ỹFFt (difference between potential outputs).
  - During the boom (post euro), financial shocks explain the larger gap early on; from 2003 housing preference shocks become the main shock amplified by financial friction.
  - When crisis hits, negative housing demand shocks are amplified by accelerator mechanism, deepening the negative gap. Core shows much smaller financial wedge movements.

### Does one monetary policy fit all?
- Natural real interest rates (NRIR) computed as level of real rates consistent with flexible prices/wages and excluding inefficient (mark-up) shocks:
  - Core NRIR declined overtime and has been below historical mean since the global financial crisis; tracks the output gap—declined 2000-2006 then jumped in 2007.
  - HBS NRIR was small but positive during 2000-2007 overheating period.
  - Tension 2003-2006: core needed lower rates while HBS needed higher rates; later synchronization during bust made natural rates converge in sign.
- Regional deviations from a Taylor-rule prescription:
  - Devit = rt − rT,i t where rt is 3-month Euribor and rT,i t uses estimated Taylor-rule parameters with region-specific CPI and output growth.
  - Interpretation: positive deviation implies contractionary stance.
- Empirical stance:
  - Core: contractionary 2000-2003; about right 2004-2006; too contractionary 2008-2009 after ECB tightening; expansionary from 2010 onward.
  - HBS: initially contractionary; largely expansionary 2002-2006 (procyclical for HBS); contractionary end of boom around 2007-2008; expansionary during crisis.
- Conclusion on one-size-fits-all: before the crisis ECB faced trade-offs to satisfy both regions; post-crisis synchronization eased common policy implementation, but monetary stimulus was insufficient to fully close HBS output gap.

### Impulse response analysis (IRFs) — magnitudes and channels
- Financial shocks (Figure 7):
  - Common features: real quantities and prices decline via financial accelerator; impact larger in HBS; spillovers more pronounced in core; house price responses larger than nondurable (CPI) inflation response.
  - Core housing sector risk shock: output decline of about 0.06 percent below steady state; CPI inflation initially falls about 0.025 percent.
  - HBS housing sector risk shock: output contraction of almost 0.2 percent; larger declines in house prices and CPI inflation.
  - Risk premium shock (HBS): largest macroeconomic impact — contraction in output, CPI inflation and house prices close to three times larger than other financial shocks; ECB cuts rates more forcefully.
- Housing demand shocks (Figure 8):
  - Both regions: decline in residential investment and house prices transmitted to nondurable sector via balance sheet effects, leading to a long-lasting decline in CPI inflation and output.
  - ECB cuts rates after housing bust; spillovers differ:
    - HBS: output declines after a housing bust in the core because trade channel outweighs interest rate cut.
    - Core: spillovers positive because monetary policy effect dominates trade effect.
- Monetary policy shocks (Figure 9):
  - Transmission mechanically similar across regions with similar estimated parameters.
  - Real effects: similar impact and hump-shaped responses in both regions.
  - Price effects: house price response to a tightening larger in HBS; CPI inflation response somewhat larger in the core.

### Key policy-relevant conclusions
- Inclusion of financial variables, frictions and housing matters especially for countries with large housing and credit fluctuations (aggregate HBS during 2000s); it materially changes assessment of cyclical position relative to HP filter.
- In the euro area core, where there was no credit boom, including financial variables does not materially change assessments relative to the HP filter.
- The DSGE model yields an estimated negative output gap of 4 percent of GDP by end-2013 for HBS aggregate, in contrast to a much smaller HP-filter gap.
- Monetary policy faced a trade-off pre-crisis due to differing regional cyclical positions and natural rates; post-crisis synchronization eased a common policy stance but did not obviate the need for region-specific macroeconomic policies (including macroprudential and fiscal tools).
- Model uncertainty remains important: different modeling choices produce different output gap estimates; the authors acknowledge more work is needed on model uncertainty.

*Source: _wp15153 - 1.25 and -2.25.  Given the properties of the log-normal distribution, this means that the default*

### References

### _wp15153 - References

### References overview
- Bibliographic list of cited works spanning DSGE modeling, housing and financial frictions, monetary policy, potential output estimation, and empirical macroeconomic methods.
- Selected authors and themes include:
  - Adam, Kuang, and A. Marcet (2011): House Price Booms and the Current Account.
  - Adolfson, Laseen, Lindé, Svensson (2011) and Adolfson et al. (2007): DSGE models and Bayesian estimation.
  - Bernanke, Gertler, Gilchrist (1999): The Financial Accelerator.
  - Borio, Disyatat, Juselius (2014): Measures of potential output.
  - Iacoviello (2005) and Iacoviello & Neri (2010): Housing collateral in business cycle models.
  - Smets & Wouters (2003, 2012): Estimated DSGE models and unemployment in New Keynesian frameworks.
  - IMF chapters: International Monetary Fund (2012, 2013) World Economic Outlook chapters on household debt and inflation.
- Working papers, ECB papers, BIS working papers, NBER Macroeconomics Annual entries, and central-bank and research-discussion papers are cited.

### Data and sources (Appendix A)
- Regional aggregation:
  - Core: aggregate data for France and Germany.
  - HBS countries: combine Greece, Ireland, Italy, Portugal, and Spain.
  - Aggregation uses household expenditure weights from the Harmonised Index of Consumer Prices (HICP) for euro area countries; weights recomputed to sum to one for the sample.
  - When some series start later than 2000q1, aggregation for those quarters uses available data with weights adjusted accordingly.
  - All data are seasonally adjusted if not already seasonally adjusted by the original source.
- Series definitions and sources (exact phrasing preserved):
  - HICP Inflation: Quarter on quarter log differences in the Harmonized Index of Consumer Prices (HICP), not seasonally adjusted by the source. Source: ECB.
  - Change in Real House Price Data: Quarter on quarter log differences in real housing prices. All data is provided by the OECD.
  - Real Private Consumption: Quarter on quarter log differences of final consumption of households and nonprofit institutions serving households (NPISH), seasonally adjusted by the source. Source: Eurostat.
  - Real Residential Investment: Quarter on quarter log differences of gross fixed capital formation in construction work for housing, seasonally adjusted by the source. Data for Greece, Ireland and Spain are seasonally adjusted using the X-12 ARIMA function in DMX. Source: Eurostat.
  - Real GDP: Quarter on quarter log differences of the real gross domestic product, seasonally adjusted by the source. Source: Eurostat.
  - ECB Interest Rate: 3-month Euribor, divided by 400. Source: ECB.
  - Household Outstanding Debt: Quarter on quarter log differences in household debt. The data are seasonally adjusted by the source only for France. For all other countries the data has been seasonally adjusted using the X-12 ARIMA function in DMX. Data for Ireland starts in 2002q1. Source: Eurostat.
- Calibration and auxiliary data:
  - Import data: Source: IMF Direction of Trade Statistics.
  - Nominal household consumption: Source: IFS.
  - Size of the non-durable sector: ratio of gross value added by the construction sector to that of all branches (Source: Eurostat).
  - Steady state ratio of defaults: calculated using non-performing loans as percent of total loans for the euro area between 2000-2011 (Source: World Bank World Development Indicators Database).

### Linearized model structure (Appendix B)
- Notation and normalization:
  - Upper case variables denote steady state values; lower case denote log-linear deviations from steady state.
  - Rest of the euro area variables indicated with asterisks.
  - Q_t denotes relative price of durables in terms of non-durables: Q_t ≡ P^D_t / P^C_t.
  - ω^i_t denotes deviation of real wages (nominal wages W^i_t divided by CPI index P^C_t, for i={C,D}) from steady state.
  - ˜S^B_t denotes real domestic debt in terms of non-durable goods (˜S^B_t ≡ S^B_t / P^C_t).
  - b_t denotes deviations of foreign assets as percent of steady state non-durable output from its steady state value of zero (b_t ≡ B_t / (P^C_t Y^C)).
  - ˆ̄ω^i_t and ˆσ_ω,t denote deviations for threshold ̄ω^i_t and variance ̄σ_ω,t (for i={a,p}).
  - Terms of trade: T_t = P_{F,t} / P_{H,t}.
  - Average interest rate of those who default: R^D_t = G(̄ω^P_{t−1}, σ_ω,t−1) P^D_t D^B_t / S^B_{t-1}.
  - Aggregate non-durable consumption: C^{TOT}_t = λ C_t + (1−λ) C^B_t.
  - Variables with unit root shock are normalized by EMU-level technology A_t; lower case denotes deviations from steady state of normalized variables (e.g., c_t = log(C_t/A_t) − log(C/A)).

- Representative set of log-linearized conditions (exact equation labels preserved):
  - Savers’ optimal decision (home region): equation (B.1).
  - Housing-law Lagrange multiplier relations: equations (B.2), (B.12), (B.13).
  - Consumption Euler and dynamics with habit and investment adjustment terms: equations (B.3), (B.14).
  - Marginal rates of substitution (MRS) for savers and borrowers in sectors C and D: (B.4)–(B.7).
  - Wage Phillips Curves for sectors C and D with parameters κ_{C,W}, κ_{D,W}: equations (B.8)–(B.9).
  - Wage equalization between types: ω^C_t = mrs^C,B_t (B.10); ω^D_t = mrs^D,B_t (B.11).
  - Borrowers’ q and housing conditions: (B.12)–(B.15).
  - Borrowers’ budget constraint (home): (B.16).
  - Financial intermediaries’ participation / lending rate condition: (B.17).
  - Ex-ante and ex-post default thresholds: (B.18) and (B.19).
  - Domestic and imported non-durable consumption decomposition and aggregate consumption identity: (B.20)–(B.22).
  - Production functions and total hours definitions: (B.23)–(B.26).
  - CPI aggregation: ∆p^C_t = τ ∆p^H_t + (1−τ) ∆p^F_t (B.27).
  - Relative price of housing dynamics: q_t = q_{t-1} + ∆p^D_t − ∆p^C_t (B.28).
  - Pricing equations (home region): (B.29) and (B.30) with κ_C and κ_D defined.
  - Market clearing (non-durable goods): y^C_t = τ c^H,t + (1−n)(1−τ^∗)/n c^{∗}_H,t (B.31).
  - Aggregate investment equality: (B.32).
  - Laws of motion for housing stocks: (B.33)–(B.34).
  - Aggregated output: y_t = α y^C_t + (1−α)(y^D_t + q_t) (B.35).

### Rest-of-euro-area region conditions
- Mirrored set of log-linearized conditions for rest of euro area with asterisks:
  - Savers’ and borrowers’ first-order conditions and MRS: (B.36)–(B.42).
  - Wage Phillips Curves and wage equalization: (B.43)–(B.46).
  - Borrowers’ q, housing, budget and lending conditions: (B.47)–(B.52).
  - Ex-ante and ex-post default thresholds (rest): (B.53)–(B.54).
  - Consumption decompositions and aggregate consumption identity (rest): (B.55)–(B.57).
  - Production functions, total hours definitions, CPI, relative price of housing, pricing equations, market clearing, investment, housing laws of motion, and aggregated output for the rest region: (B.58)–(B.70).

### Euro area aggregation, policy rule, and measurement equations
- Cross-region interest rate relation:
  - r^∗_t = r_t + β(κ^b b_t + θ_t). (B.71)
- Net foreign assets evolution:
  - λ b_t = λ 1/β b_{t−1} + (1−n)(1−τ^∗)/n (c^∗_{H,t} − t_t) − (1−τ) c_{F,t}. (B.72)
  - Note: uses identity t_t = − t^∗_t.
- Evolution of terms of trade:
  - t_t = t_{t−1} + ∆p^F_t − ∆p^H_t. (B.73)
- ECB Taylor rule (monetary policy):
  - r_t = γ_R r_{t−1} + (1−γ_R)[γ_π ∆p^{EMU}_t + γ_y (y^{EMU}_t − y^{EMU}_{t−1} − ε^A_t)] + ε^m_t. (B.74)
  - Euro area CPI and output definitions:
    - ∆p^{EMU}_t = n ∆p^C_t + (1−n) ∆p^{C∗}_t. (B.75)
    - y^{EMU}_t = n y_t + (1−n) y^{∗}_t. (B.76)
- Measurement linking model output to data:
  - gdp_t = (1− ̄g) y_t + ̄g (g_t). (B.77)
  - gdp^∗_t = (1− ̄g^∗) y^∗_t + ̄g^∗ (g^∗_t). (B.78)

### Shock processes
- Stationary and non-stationary shock dynamics (equations preserved):
  - ξ^C_t = ρ_{ξ,H} ξ^C_{t−1} + ε_{ξ,C,t}. (B.79)
  - ξ^{C∗}_t = ρ_{ξ,H} ξ^{C∗}_{t−1} + ε_{ξ,C∗,t}. (B.80)
  - ξ^D_t = ρ_{ξ,D} ξ^D_{t−1} + ε_{ξ,D,t} + ε_{ξ,D,COM,t}. (B.81)
  - ξ^{D∗}_t = ρ_{ξ,D} ξ^{D∗}_{t−1} + ε_{ξ,D∗,t} + ε_{ξ,D,COM,t}. (B.82)
  - z^C_t = ρ_{Z,C} z^C_{t−1} + ε_{Z,C,t} + ε_{Z,C,COM,t}. (B.83)
  - z^{C∗}_t = ρ_{Z,C} z^{C∗}_{t−1} + ε_{Z,C∗,t} + ε_{Z,C,COM,t}. (B.84)
  - z^D_t = ρ_{Z,D} z^D_{t−1} + ε_{Z,D,t}. (B.85)
  - z^{D∗}_t = ρ_{Z,D} z^{D∗}_{t−1} + ε_{Z,D∗,t}. (B.86)
  - σ_{ω,t} = (1−ρ_{σω}) ̄σ_ω + ρ_{σω} σ_{ω,t−1} + u_{ω,t}. (B.87)
  - σ^{∗}_{ω,t} = (1−ρ_{σω}) ̄σ_ω + ρ_{σω} σ^{∗}_{ω,t−1} + u^{∗}_{ω,t}. (B.88)
  - g_t = ρ_g g_{t−1} + ε_{g,t}. (B.89)
  - g^{∗}_t = ρ^{∗}_g g^{∗}_{t−1} + ε_{g∗,t}. (B.90)
  - θ_t = ρ_θ θ_{t−1} + ε_{θ,t}. (B.91)
- Non-stationary innovations:
  - Union-wide technology shock ε^A_t, monetary policy shock ε^m_t, and price markup shocks ε^{μ}_C_t and ε^{μ}_{C∗}_t are iid.

*Source: _wp15153 - References (Appendix A and Appendix B content).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15153.pdf_