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---

### Model overview and key features
- Model variants:
  - Closed-economy Euro Area ("EA model") and 2-bloc currency-union (CU) version estimated three times treating respectively France, Germany, and Italy as the domestic country (CUF, CUG, CUI).
- Agents and frictions:
  - Optimizing households (fraction ω), rule-of-thumb households (fraction 1−ω), labor unions, financial intermediaries (FIs), non-financial firms, capital producers, fiscal authority, central bank.
  - Frictions: price stickiness parameter θp, wage stickiness parameter θw, investment adjustment costs ψi, habit formation h, FI net-worth constraint with adjustment cost ψn ∈[0,∞), long-term government debt.
  - Unconventional monetary policy captured via the Wu and Xia (2017) shadow rate.
- Role of financial intermediaries:
  - Sole buyers of long-term investment bonds ̄F t and long-term government bonds ̄B FI t.
  - Finance via deposits D t from optimizers; constrained by net worth N t and portfolio adjustment cost parameter ψn.
  - Tighter borrowing constraints raise term premia and lower activity.
- Monetary and fiscal policymaking:
  - Union-wide Taylor-type rule: log(R t /R ) = ρ r log(R t−1 /R ) + (1−ρ r ){ρ π log(Π u t /Π u ) + ρ y log(Y u t ̃/Y u t )} + e m t .
  - Government budget constraint: ̄B t = R L t /Π t ̄B t−1 + ̄G t − T t and fiscal rules log(τ t /τ ) and log(g t /g ) per equations (7) and (8) with fiscal foresight parameters θ τ , θ g ∈[0,1].

### Bayesian estimation, data, and calibration
- Sample and observables:
  - Sample period: 1999Q1 until 2018Q3.
  - EA model observables (10 variables): Real GDP, real private consumption, real private investment, real wage, real government spending (consumption, investment, transfers), real total revenues, inflation, the term premium, the ratio of the primary fiscal balance to GDP, and the nominal interest rate (EA).
  - CU model observables (19 variables): EA model’s 10 variables plus corresponding first 9 variables for the foreign bloc.
- Data and lower-bound treatment:
  - Sample start dates: "1999Q2 for the EA and France, 2001Q1 for Italy, and 2002Q2 for Germany."
  - EONIA approached the ZLB in "2012Q1", turned negative in "2014Q4", and remained at the ELB; policy-rate proxy replaced by the Wu and Xia (2017) Eonia shadow rate, extended back to 1999Q1 using Eonia.
- Estimation and computation:
  - Likelihood via Kalman filter.
  - Posterior sampling: MCMC-MH with two parallel chains of500,000draws each.
  - Stochastic simulations: draw "1,000 sets of stochastic realizations of all shocks" for T = 1,500 quarters; retain 1,000 time series each 1,000 periods long after discarding first 500 periods as burn-in; simulations performed by Dynare.
- Key calibration highlights (selected exact values preserved):
  - Discount factor β 0.99; δ 0.025; α 0.33; εp 6; εw 6.
  - Government spending to GDP g∗y 0.20; government debt to GDP b∗y 0.78.
  - Steady-state tax rates: τc,∗ 0.20; τk,∗ 0.30; τw,∗ 0.38.
  - Duration of long-term bonds (1−κ)−1 40 (calibrated κ to set duration to 10 years).
  - Country-specific calibrations (France / Italy / Germany):
    - Country size n: 0.22 / 0.18 / 0.29.
    - Government spending to GDP gy: 0.23 / 0.19 / 0.19.
    - Government debt to GDP by: 0.79 / 1.14 / 0.69.
    - Steady state tax rates τc: 0.20 / 0.17 / 0.20; τk: 0.47 / 0.30 / 0.22; τw: 0.40 / 0.43 / 0.38.
    - Exports to GDP xy: 0.27 / 0.26 / 0.40; Imports to GDP my: 0.28 / 0.25 / 0.35.

### DMFCI construction and implementation
- Purpose and novelty:
  - Dynamic Monetary-Fiscal Condition Indices (DMFCIs) measure the combined effect on output of monetary and fiscal instruments, including unconventional monetary policy via a shadow rate, and produce decompositions into monetary, domestic fiscal, and rest-of-EA fiscal components.
- Computation steps (preserved formula structure and procedural specifics):
  1. Generate stochastic simulations (1,000 simulated datasets per model; see Estimation and Simulations).
  2. On each artificial dataset, run regression (equation (11)) of detrended output ˆyt on lags of the real shadow rate ˆrt, lags of CAPB capbt, lags of rest-of-EA CAPB capb∗t, and exogenous regressors eκt and eκ∗t where κ={b,μ,φ}; choose lag structure to maximize median adjusted R-squared, ̄R2.
  3. Use median (and distribution) of regression coefficients to build DMFCI as DMFCI t = DMFCI M t + DMFCI F t + DMFCI F∗ t with components per equations (12)–(15):
     - DMFCI M t = Σ_{j=1}^{N r} ̃α r j (ˆr s t−j − ˆr s b−j+1).
     - DMFCI F t = Σ_{j=1}^{N capb} ̃α capb j (capb s t−j − capb s b−j+1).
     - DMFCI F∗ t = Σ_{j=1}^{N capb∗} ̃α capb∗ j [(capb∗ t−j) s − (capb∗ b−j+1) s].
- Implementation choices and interpretation:
  - Potential output ̃Yt defined as level prevailing "in the absence of price and wage stickiness as well as price and wage mark-up shocks."
  - CAPB defined as capbt ≡ CAPB t ̃Yt; elasticities set ηG = 0 and ηT = 1.
  - Base quarter chosen as "2005q1"; indices reported starting from 2007; positive index = looser stance, negative = tighter stance.
  - Distributions: report median with "5th and 95th percentiles."

### Key empirical findings and evolution of DMFCIs (2007–2018)
- Four headline results (preserved wording and chronology):
  - 1) The EA’s overall policy became looser after the GFC, with most loosening occurring between 2009 and 2011; stance tightened during the sovereign debt crisis and loosened again around 2014 with ECB accommodative actions.
  - 2) Aggregate EA DMFCI patterns do not map one-to-one to national patterns; fiscal stances were heterogeneous:
    - France: fiscal policy strongly expansionary during the GFC, then restrictive after the sovereign debt crisis.
    - Germany: restrictive fiscal policy except for a short period after the GFC.
    - Italy: fiscal stance found always tighter than the pre-crisis period.
    - Monetary policy was the dominant accommodative force after 2013-14.
  - 3) Historical shock decomposition: monetary policy component is sizable and aligns with the DMFCI; fiscal policy component is small and correlates poorly with the DMFCI—reflecting that shock decompositions capture mainly unexpected policy innovations while DMFCI captures both expected and unexpected policy.
  - 4) Dynamics: monetary policy historically leads GDP changes; fiscal policy generally did not lead GDP based on dynamic cross-correlations and impulse response analysis.
- Evolution and country heterogeneity (summary):
  - Overall stance loosened in all countries from 2008 onward; loosening halted during 2011-14 and became accommodative again thereafter driven mainly by ECB actions.
  - Fiscal loosening: more pronounced in France, less strong in Germany, absent in Italy.
  - Fiscal policies of the rest of the EA played a marginal role in the three economies.
  - Monetary policy dominated the EA’s policy mix since the GFC.

### Comparisons with alternative indices and shock decompositions
- Correlations with alternative indices (exact values preserved):
  - Correlation DMFCI_M (EA) vs EC MCI (2008Q1-2018Q3): -0.26.
  - Correlation DMFCI_M (EA) vs EC MCI (2008Q1-2013Q4): -0.58.
  - Correlations DMFCI_M vs IMF FCI: EA -0.59; France -0.55; Germany -0.52; Italy -0.49.
  - Correlations DMFCI_F vs IMF CAPB (annualized, percent of potential output): EA -0.92; France -0.96; Germany -0.89; Italy -0.73.
- Interpretation:
  - Weaker correlation with EC MCI over 2008Q1-2018Q3 reflects DMFCI including unconventional monetary policy; restricting to pre-PSPP raises correlation.
  - High negative correlations with IMF FCI reflect common capture of unconventional monetary measures.
  - Strong negative correlations between DMFCI_F and IMF CAPB indicate close comovement of fiscal component and cyclically-adjusted primary balances.
- Historical decomposition vs DMFCI (exact variance attributions and findings):
  - Unexpected monetary policy shocks explain approximately between 30 and 40 percent of the variance of the real shadow rate (depending on the country).
  - Unexpected fiscal shocks explain between less than 1 percent (Italy and EA overall) and 15 percent (Germany) of the variance of the CAPB.
  - Much fiscal support in 2008-2010 occurred through automatic stabilizers rather than discretionary fiscal decisions.
  - Consequence: historical shock decompositions underrepresent fiscal stance changes that are anticipated or driven by non-discretionary factors; DMFCI captures actual outcomes irrespective of anticipation.

### Dynamic properties and impulse-response evidence
- Leading/lag relationships (sample 2008Q1-2018Q3; correlations reported at horizons):
  - EA aggregate: DMFCI_M leads GDP; looser monetary policy associated with higher GDP growth after three quarters with correlation 0.66. DMFCI_F is negatively and significantly associated with the cycle contemporaneously.
  - France and Germany: monetary policy leads the cycle by two quarters; fiscal stance is coincident with the cycle.
  - Italy: monetary policy leads GDP growth by three quarters with correlation 0.71; fiscal policy associated with larger GDP growth after four quarters with correlation 0.42 (suggesting contractionary fiscal effect over sample).
- DSGE impulse responses:
  - IRFs set shocks equal to estimated standard deviations (Table E.6).
  - Monetary policy shock: boost to GDP is more accentuated, delayed and persistent than boost from government spending shock; similar pronounced effects on inflation.
  - Fiscal shock (government spending): smaller and less persistent GDP effects; fall in CAPB after spending shock explained by simulated rise in government spending.
  - Implication: larger, delayed and persistent monetary responses explain why DMFCI_M leads GDP growth while DMFCI_F does not.

### Estimation and posterior highlights (selected exact posterior means and diagnostics)
- Selected posterior structural parameters (posterior means with brackets preserved as in source where ranges provided):
  - Inv. Frisch elasticity η:
    - France: 0.53 [0.46;0.59]; Italy: 0.51 [0.47;0.54]; Germany: 0.46 [0.42;0.50]; Euro Area: 0.50 [0.42;0.58].
  - Habits h:
    - France: 0.73 [0.68;0.79]; Italy: 0.77 [0.73;0.82]; Germany: 0.79 [0.74;0.84]; Euro Area: 0.79 [0.74;0.85].
  - Fraction optimizing ω:
    - France: 0.74 [0.68;0.80]; Italy: 0.90 [0.87;0.94]; Germany: 0.85 [0.80;0.91]; Euro Area: 0.85 [0.79;0.90].
  - Investment adjustment cost ψ_i:
    - France: 6.58 [5.66;7.49]; Italy: 4.29 [3.63;4.93]; Germany: 5.38 [4.19;6.56]; Euro Area: 4.97 [3.81;6.12].
  - Net worth adjustment costs ψ_n:
    - France: 0.70 [0.60;0.81]; Italy: 0.97 [0.87;1.08]; Germany: 0.89 [0.80;0.98]; Euro Area: 0.75 [0.59;0.91].
  - Price stickiness θ_p:
    - France: 65.9 [49.6;81.4]; Italy: 52.1 [44.8;58.3]; Germany: 66.3 [51.3;80.7]; Euro Area: 64.9 [47.0;82.6].
  - Wage stickiness θ_w:
    - France: 69.0 [56.3;81.1]; Italy: 51.4 [45.6;57.0]; Germany: 60.4 [50.5;71.6]; Euro Area: 69.4 [51.6;86.3].
  - Inflation Taylor rule coefficient ρ_π:
    - France: 1.94 [1.84;2.05]; Italy: 1.94 [1.84;2.07]; Germany: 1.70 [1.56;1.81]; Euro Area: 1.88 [1.74;2.02].
  - Interest rate smoothing ρ_r:
    - France: 0.80 [0.76;0.84]; Italy: 0.85 [0.81;0.88]; Germany: 0.85 [0.80;0.89]; Euro Area: 0.82 [0.78;0.86].
- Selected constants and diagnostics (posterior means):
  - Trend γ: France 0.46 [0.42;0.50]; Italy 0.35 [0.31;0.40]; Germany 0.27 [0.22;0.33]; Euro Area 0.44 [0.39;0.49].
  - Inflation constant ̄Π: France 0.15 [0.09;0.21]; Italy 0.18 [0.14;0.23]; Germany 0.08 [0.04;0.11]; Euro Area 0.26 [0.17;0.35].
  - Term premium constant ̄TP: France 0.29 [0.18;0.39]; Italy 0.39 [0.32;0.46]; Germany 0.22 [0.11;0.32]; Euro Area 0.39 [0.28;0.50].
  - Log-likelihood: France -1198.73; Italy -11583.87; Germany -1251.46; Euro Area -595.95.

### DMFCI regression performance and robustness
- Regression fit (median ̄R^2 from 1,000 regressions):
  - Euro Area median ̄R^2 = 0.64 (Table F.1).
  - France median ̄R^2 = 0.77 (Table F.2).
  - Germany median ̄R^2 = 0.87 (Table F.3).
  - Italy median ̄R^2 = 0.81 (Table F.4).
- Cross-country DMFCI correlations (selected exact values):
  - Correlation DMFCI^F: EA vs France = 0.78; EA vs Germany = 0.72; EA vs Italy = 0.57.
  - France vs Germany = 0.40; France vs Italy = 0.42; Germany vs Italy = 0.32.
- Robustness to alternative potential output definitions (correlations of DMFCI changes):
  - Total DMFCI correlations: EA 0.92; France 0.91; Germany 0.95; Italy 0.85.
  - Monetary component DMFCI^M correlations: EA 0.98; France 1.00; Germany 1.00; Italy 0.98.
  - Fiscal component DMFCI^F correlations: EA 0.87; France 0.90; Germany 0.94; Italy 0.90.

### Practical implications and policy recommendations (from analysis)
- Measurement and communication:
  - DMFCIs provide a synthetic, communicable measure of overall policy stance that captures both anticipated and unanticipated policy changes and decomposes into monetary and fiscal contributions.
- Policy mix implications:
  - Monetary policy delivered the lion’s share of economic stimulus post-GFC; a more expansionary fiscal policy could play an important role in boosting EA economic activity, especially relevant for the COVID-19 pandemic response.
  - Given monetary policy became "the only game in town" after 2014, a more balanced policy mix with fiscal expansion would be a useful complement to monetary accommodation.
- Modeling implications:
  - Limited asset-market participation, financial intermediaries, long-term bonds and shadow-rate treatment are essential to capture term-premium and QE channels and to properly assess the combined monetary-fiscal stance.

*Source: wpiea2020086-print-pdf, canonical URL https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020086-print-pdf.pdf*

### 4.1  Relation with the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . .   17

### 4.1  Relation with the Literature . . . . . . . . . . . . . . . . . . . . . . . . . .   17

### Contained sections and structure
- 4.1  Relation with the Literature . . . . . . . . . . . . . . . . . . . . . . . . .   17
- 4.2  Computation of the DMFCI . . . . . . . . . . . . . . . . . . . . . . . . . . .   18
- 4.3  Evolution of DMFCIs in the Euro Area . . . . . . . . . . . . . . . . . . . .   21
- 4.4  Comparison with Alternative Available Indices . . . . . . . . . . . . . . .   22
- 4.5  Comparison between DMFCIs and Historical Contribution of Shocks in the DSGE Model  . . . . . . . . . . . . . . . . . . . . . . . . . .   25
- 4.6  Dynamic Properties of the DMFCI in the Light of the DSGE Model . . . .   29
- 5  Concluding Remarks31

*Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020086-print-pdf.pdf*

### References33

### wpiea2020086-print-pdf - References33

### Appendix structure and contents
- Appendix A: Detailed description of the model
  - A.1 Households
    - A.1.1 Optimizing Households
    - A.1.2 Rule-of-Thumb Households
    - A.1.3 Wage Setting
    - A.1.4 Aggregation
  - A.2 Financial Intermediaries
  - A.3 Non-Financial Firms
  - A.4 New Capital Producers
  - A.5 Trade and Market Clearing
  - A.6 Equilibrium and Exogenous Processes
- Appendix B: Equilibrium Conditions of the Detrended System
  - B.1 Domestic Country
  - B.2 Foreign Country
  - B.3 Market Clearing and Trade
  - B.4 Central Bank
  - B.5 Exogenous Processes
- Appendix C: Steady State
- Appendix D: Detailed Derivation of the Wage Setting Equation
- Appendix E: Data, Measurement Equations, and Estimates
- Appendix F: DMFCI – Regression Coefficients
- Appendix G: DMFCI – Additional Results

### Model overview and key features
- Model types:
  - Closed-economy version estimated on aggregate Euro Area (EA) data ("EA model").
  - 2-bloc currency-union (CU) version with a home country of size n and a foreign bloc of size 1−n; estimated three times (CUF, CUG, CUI) treating respectively France, Germany, and Italy as the domestic country.
- Frictions and mechanisms included:
  - Price stickiness parameter θp and wage stickiness parameter θw (à la Rotemberg, 1982).
  - Investment adjustment costs ψi (IAC) as in Christiano et al. (2005a).
  - Habit formation in consumption h.
  - Mix of ω optimizers (Ricardian) and 1−ω rule-of-thumb households (non-Ricardian), following Galí et al. (2007).
  - Financial intermediaries (FIs) that purchase long-term private and government bonds and hold short-term liabilities; FIs face a net-worth-based constraint with adjustment cost parameter ψn ∈[0,∞).
  - Long-term government debt and a detailed fiscal sector with distortionary taxes and fiscal rules allowing automatic stabilizers.
  - Unconventional monetary policy captured via the shadow monetary policy rate of Wu and Xia (2017).
- Agents within each country:
  - Optimizing households, rule-of-thumb households, labor unions, financial intermediaries, non-financial firms, capital producers, the fiscal authority, and the central bank.
- Role of financial intermediaries:
  - Sole buyers of long-term investment bonds ( ̄F t ) and long-term government bonds ( ̄B FI t ).
  - Finance purchases via deposits D t from optimizing households, constrained by net worth N t.
  - Friction severity determined by ψn; tighter borrowing constraints raise term premia and lower activity.
- Rationale:
  - Inclusion of limited asset market participation enables New-Keynesian fiscal effects.
  - FIs and long-term bonds introduce term-premium and credit shock channels relevant for the GFC.
  - Fiscal block and shadow rate enable analysis of QE and anticipated fiscal effects.

### Policymakers: central bank and fiscal authority
- Central bank (union-wide Taylor-type rule):
  - log(R t /R ) = ρ r log(R t−1 /R ) + (1−ρ r ){ρ π log(Π u t /Π u ) + ρ y log(Y u t ̃/Y u t )} + e m t . (1)
  - Union-wide inflation and output definitions:
    - Π u t = (Π t ) n (Π ∗ t ) 1−n . (2)
    - Y u t = nY t + (1−n)Y ∗ t . (3)
  - Parameters: ρ r (interest rate smoothing), ρ π (response to inflation deviations), ρ y (response to output gap).
  - Monetary policy shock: e m t (AR(1)).
- Fiscal authority (country-level budget and fiscal rules):
  - Government budget constraint:
    - ̄B t = R L t /Π t ̄B t−1 + ̄G t − T t . (4)
    - ̄G t = G t + τ l t denotes total government expenditure.
  - Tax revenue decomposition:
    - T t = τ C t C t + τ W t W t H t + τ k t (R k t − δP k t ) K t . (5)
  - Primary balance to GDP:
    - PB Y t = T t − ̄G t /Y t . (6)
  - Tax and expenditure proportionality assumption:
    - τ C t = τ t τ C , τ W t = τ t τ W , τ k t = τ t τ k , G t = g t G, τ l t = g t τ l .
  - Fiscal rules (stabilize debt and react to output deviations):
    - log(τ t /τ ) = ρ τ log(τ t−1 /τ ) + ρ τb log( ̄B t−1 / ̄B ) + ρ τy log(Y t /Y ) + (1−θ τ ) ε τ t + θ τ ε τ t−1 . (7)
    - log(g t /g ) = ρ g log(g t−1 /g ) − ρ gb log( ̄B t−1 / ̄B ) − ρ gy log(Y t /Y ) + (1−θ g ) ε g t + θ g ε g t−1 . (8)
  - Parameters: ρ g , ρ τ (persistence of instruments), ρ τb , ρ gb (response to debt deviations), ρ τy , ρ gy (response to output deviations).
  - Fiscal shocks: ε g t (government spending shock), ε τ t (tax shock), with pre-announcement/foresight parameters θ τ , θ g ∈[0,1].
    - θ i = 1 implies perfect foresight of fiscal policies; θ i = 0 implies no foresight; values in (0,1) imply limited fiscal foresight.
- Currency union foreign-block fiscal rules mirror domestic rules with “∗” notation:
  - log(τ ∗ t /τ ∗) = ρ ∗ τ log(τ ∗ t−1 /τ ∗) + ρ ∗ τb log( ̄B ∗ t−1 / ̄B ∗) + ρ ∗ τy log(Y ∗ t /Y ∗) + (1−θ ∗ τ ) ε τ,∗ t + θ ∗ τ ε τ,∗ t−1 . (9)
  - log(g ∗ t /g ∗) = ρ ∗ g log(g ∗ t−1 /g ∗) − ρ ∗ gb log( ̄B ∗ t−1 / ̄B ∗) − ρ ∗ gy log(Y ∗ t /Y ∗) + (1−θ ∗ g ) ε g,∗ t + θ ∗ g ε g,∗ t−1 . (10)

### Bayesian estimation and data
- Estimation approach:
  - Bayesian methods used to estimate four DSGE models (EA model and three CU variants).
- Sample and observables:
  - Sample period: 1999Q1 until 2018Q3.
  - EA model observables (10 variables):
    - Real GDP, real private consumption, real private investment, real wage, real government spending (consumption, investment, transfers), real total revenues, inflation, the term premium, the ratio of the primary fiscal balance to GDP, and the nominal interest rate (EA).
  - CU model observables (19 variables):
    - The EA model’s 10 variables plus the corresponding first 9 variables (all except the nominal interest rate) for the rest of the EA (foreign bloc).
- Motivation for sample choice:
  - Encompasses much of the history of the EMU and key turning points: the GFC, the sovereign debt crisis, and the ECB Public Sector Purchase Programme (PSPP) started in March 2015.

### DMFCI construction and usage (from paper overview)
- Dynamic Monetary-Fiscal Condition Indices (DMFCIs):
  - Built from regression estimates of monetary and fiscal policy impacts on output using data simulated from the estimated DSGE models.
  - Four DMFCIs constructed for 2007-2018: EA aggregate and three country-level indices (France, Germany, Italy).
- Advantages of DMFCIs:
  - Provide a synthetic, communicable measure of overall policy stance compared to full DSGE outputs.
  - Capture both expected (anticipated) and unexpected (unanticipated) components of policy—particularly important for fiscal stance.
  - Decompose area-wide monetary policy vs country-level fiscal policies.
  - Incorporate dynamic lags to capture the time profile of policy effects.
- Relation to literature:
  - Connects to Monetary Condition Indices (MCIs) tradition (Bernanke and Mihov, 1998; Gerlach and Smets, 2000; Osborne-Kinch and Holton, 2010) but innovates by:
    - Simulating from a fully-fledged structural DSGE model.
    - Embedding lags (per Batini and Turnbull, 2002).
    - Incorporating fiscal policy (CAPB) as a second instrument.
- Model features supporting DMFCI construction:
  - Limited asset-market participation, financial intermediaries with maturity transformation, long-term government debt, and use of shadow policy rate to capture unconventional monetary policy.

### Key empirical findings highlighted in the introduction
- Four key results reported:
  - 1) The EA’s overall policy became looser after the GFC, with most loosening occurring between 2009 and 2011; stance tightened during the sovereign debt crisis and loosened again around 2014 with ECB accommodative actions.
  - 2) Aggregate EA DMFCI patterns do not map one-to-one to national patterns; fiscal stances were heterogeneous:
    - France: fiscal policy strongly expansionary during the GFC, then restrictive after the sovereign debt crisis.
    - Germany: restrictive fiscal policy except for a short period after the GFC.
    - Italy: fiscal stance found always tighter than the pre-crisis period.
    - Monetary policy was the dominant accommodative force after 2013-14.
  - 3) Historical shock decomposition: monetary policy component is sizable and aligns with the DMFCI; fiscal policy component is small and correlates poorly with the DMFCI—reflecting that shock decompositions capture mainly unexpected policy innovations while DMFCI captures both expected and unexpected policy.
  - 4) Dynamics: monetary policy historically leads GDP changes; fiscal policy generally did not lead GDP based on dynamic cross-correlations and impulse response analysis.

*Italic line: Source: wpiea2020086-print-pdf - References33, canonical URL https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020086-print-pdf.pdf*

### Appendix E discusses data transformations and

### Appendix E discusses data transformations and

### Data, sample coverage, and treatment of the ZLB/ELB
- Sample start dates: "1999Q2 for the EA and France, 2001Q1 for Italy, and 2002Q2 for Germany."
- EA composition for aggregation: "eleven countries of the EA, namely Austria, Belgium, Finland, France, Germany, Ireland, Italy, Luxembourg, the Netherlands, Portugal and Spain" aggregated "weighting by nominal GDP."
- Greece excluded because "quarterly Greek fiscal data is not fully available."
- Policy-rate proxy and lower-bound issues:
  - EONIA approached the ZLB in "2012Q1", turned negative in "2014Q4", and remained at the effective lower bound (ELB).
  - Chosen solution: replace policy rate with a shadow rate and use "the Eonia shadow rate constructed by Wu and Xia (2017)." The Wu and Xia series "starts in 2004Q4 hence we extend it back to 1999Q1 using the Eonia rate given that the two coincide in normal times."
  - Alternative approaches noted but not used: estimating up to pre-ELB and using non-linear methods, with references to literature and caveats about capturing unconventional monetary policy.

### Estimation method and computational details
- Likelihood evaluation: "The Kalman filter is used to evaluate the likelihood function."
- Posterior sampling: "the Monte-Carlo-Markov-Chain Metropolis-Hastings (MCMC-MH) algorithm with two parallel chains of500,000draws each" to generate posterior samples.
- Structural parameters and steady states calibrated at a quarterly frequency.
- Stochastic simulations for DMFCI construction:
  - Draw "1,000 sets of stochastic realizations of all shocks for a number of quarters equal toT= 1,500."
  - Save "the 1,000 time series (each 1,000 periods long, after discarding the first 500 periods as burn-in)" for specified variables.
  - Simulations performed by Dynare using Taylor approximation and perturbation methods.

### Calibration (EA and country-specific)
- Table 1 calibrated Euro Area parameters (values/steady-state targets preserved):
  - Discount factor β 0.99
  - Capital depreciation rate δ 0.025
  - Capital share of income α 0.33
  - Elasticity of substitution goods εp 6
  - Elasticity of substitution labor εw 6
  - Government spending to GDP g∗y 0.20
  - Government debt to GDP b∗y 0.78
  - Steady state Tax rate consumption τc,∗ 0.20
  - Steady state Tax rate capital τk,∗ 0.30
  - Steady state Tax rate labor income τw,∗ 0.38
  - Duration of long-term bonds (1−κ)−1 40
  - Disutility of labor B,B∗H= 1
  - FI additional discount ζ,ζ∗L= 6
- Table 2 calibrated country-specific parameters (France / Italy / Germany):
  - Country size n 0.22 0.18 0.29
  - Government spending to GDP gy 0.23 0.19 0.19
  - Government debt to GDP by 0.79 1.14 0.69
  - Steady state Tax rate consumption τc 0.20 0.17 0.20
  - Steady state Tax rate capital τk 0.47 0.30 0.22
  - Steady state Tax rate labor income τw 0.40 0.43 0.38
  - Exports to GDP xy 0.27 0.26 0.40
  - Imports to GDP my 0.28 0.25 0.35
- Additional calibration choices:
  - "We calibrate κ such that the duration of the long-term bonds is set to 10 years."
  - Disutility of labor B set to match "steady state hours equal to 1."
  - FI additional discount set to match "a steady state leverage of 6."
  - Country size n ∈ [0,1] "is set as the share of nominal GDP among the EA countries considered."
  - Ratios of government spending, government debt, exports and imports to GDP "are set in line with the data from Eurostat" (averages over "2000-2018").
  - Steady-state tax rates calculated from "European Commission’s Taxation Trends Report 2018" (averages over "2003-2016").

### Priors, posterior findings, and model features
- Priors largely follow Smets and Wouters (2003; 2005) and related EA literature.
- Specific prior choices:
  - Prior mean of inverse Frisch elasticity η set to 0.5.
  - Prior mean of habit parameter h set to 0.70.
  - Share of optimizing households initialized to equal share of rule-of-thumbers.
  - Prices assumed a priori to last "3.7 quarters"; wages "2.5 quarters."
  - Prior for FIs’ net worth adjustment parameter ψn: Normal(mean 0.785, standard deviation 0.10).
  - Priors for tax-rule coefficients from Zubairy (2014).
  - Prior for government-spending cyclicality ρgy, ρ∗gy ~ Normal(mean 0.10, standard deviation 0.05).
  - Prior mean for interest-rate smoothing parameter ρr set to 0.80.
  - Beta (B) distribution used for parameters bounded between 0 and 1; Inverse Gamma (IG) distribution used for standard deviations of shocks with "2 degrees of freedom" (and elsewhere an inverse gamma with "shape parameter 0.1 and scale parameter 2" for loose priors).
- Posterior highlights:
  - Fraction of optimizing households ω ranges from "0.74 in France to 0.90 in Italy."
  - Habit and IACs: estimated ψi "generally close in size to what found by Smets and Wouters (2005) and Forni et al. (2009)."
  - Posterior ψn indicates "a non-negligible degree of financial frictions."
  - Detection of "a substantial degree of nominal rigidities" in line with multiple EA studies cited.
  - Fiscal-rule estimates: slightly stronger responses of government spending than taxes to government debt and output deviations in the EA; responses "essentially equal in France, Italy and Germany."
  - Government spending estimated to be countercyclical given positive values of ρgy and ρ∗gy.
  - Fiscal foresight parameters: "agents foresee these shocks at least in part, with stronger pre-announcement effects of government spending than taxes (θg > θτ, θ∗g > θ∗τ)."
  - Taylor rule parameters and shock-process parameters "take standard values."

### Dynamic Monetary-Fiscal Condition Indices (DMFCIs) — rationale and construction
- Purpose and novelty:
  - DMFCIs measure the "combined effect on output of multiple macroeconomic policy levers (monetary and fiscal)" and can be decomposed into individual contributions.
  - Use of the shadow monetary policy rate incorporates "the effects of unconventional monetary policy."
  - Dynamic by construction: DMFCIs consider the impact over time of interest-rate and fiscal instruments, making them "contemporaneous" indicators of stance and allowing forward projection.
  - Weights are derived from a DSGE system rather than a single equation.
- Computation steps summary:
  1. Generate stochastic simulations (see Estimation and Simulations section above).
  2. On each artificial dataset, run regression (equation (11)) of detrended output ˆyt on lags of the real shadow rate ˆrt, lags of the CAPB capbt, lags of rest-of-EA CAPB capb∗t, and exogenous regressors (domestic and rest-of-EA non-policy demand shocks eκt and eκ∗t, where κ={b,μ,φ}).
     - Regression specification (equation (11)) preserved as in source.
     - Lag structure chosen to maximize the median adjusted R-squared, ̄R2, across the 1,000 regressions.
     - Expected signs: negative coefficients on the shadow rate and CAPB (αrj and αcapbj).
     - "Selected quantiles from the distribution of regression coefficients and the median ̄R2 are reported in Appendix F (Tables F.1-F.4)." (listed in source; not reproduced here.)
  3. Use median (and distribution) of regression coefficients to build DMFCI as algebraic sum of monetary, domestic fiscal, and rest-of-EA fiscal components via equations (12)-(15):
     - DMFCI t = DMFCI M t + DMFCI F t + DMFCI F∗ t (equation (12))
     - DMFCI M t = Σ_{j=1}^{N r} ̃α r j (ˆr s t−j − ˆr s b−j+1) (equation (13))
     - DMFCI F t = Σ_{j=1}^{N capb} ̃α capb j (capb s t−j − capb s b−j+1) (equation (14))
     - DMFCI F∗ t = Σ_{j=1}^{N capb∗} ̃α capb∗ j [(capb∗ t−j) s − (capb∗ b−j+1) s] (equation (15))
     - Definitions preserved: ̃α coefficients are coefficients on lags; ˆr s t, capb s t, and (capb∗ t) s are smoothed series; b is chosen base period.
- CAPB and potential output definitions and implementation details:
  - Potential output ̃Yt defined as the level prevailing "in the absence of price and wage stickiness as well as price and wage mark-up shocks."
  - CAPB as fraction of potential output: capbt ≡ CAPB t ̃Yt.
  - Cyclically-adjusted revenue and expenditure definitions given, with elasticities ηT and ηG and relation:
    - capbt = Tt/Yt (1 + gapt)−(ηT−1) − Ḡt/Yt (1 + gapt)−(ηG−1)
  - Elasticity choices following Fedelino et al. (2009): "ηG = 0 and ηT = 1."
- Implementation choices:
  - Base quarter chosen as "2005q1."
    - Rationale: "the difference between the steady-state interest rate and its actual value is the lowest precisely in 2005q1 and the value of the model-implied CAPB in 2005 is close to zero (0.2) in the EA."
    - The base allows indices to be computed from "some quarter between 2006 and 2007 for all countries" and indices are reported "starting from 2007."
  - Index sign interpretation: "a positive value of the index represents a looser policy stance while a negative value represents a tighter one with respect to the base year."
  - Distributions: given coefficients across 1,000 regressions, build distributions of indices and report median with "5th and 95th percentiles."
- Country-specific dynamics of coefficients:
  - In cumulative terms, signs align with theory.
  - Country differences in persistence:
    - "In France the effects of monetary and fiscal policies are rather short-lived."
    - "In Italy, these effects are protracted."
    - "Germany showing a lag structure in between those reported for France and Italy."

### Model usage and output applications
- DMFCIs constructed for four units: "the EA as a whole, as well as and for France, Germany and Italy taken individually."
- DMFCIs enable:
  - Decomposition of overall policy stance into monetary and fiscal contributions.
  - Comparison with alternative available indices and historical contribution of policy shocks to GDP (sections indicated in main text).
  - Analysis of dynamic properties combined with impulse response functions from estimated DSGE models.

*Source: wpiea2020086-print-pdf - Appendix E discusses data transformations and (canonical source PDF as provided).*

### 4.3  Evolution of DMFCIs in the Euro Area

### 4.3 Evolution of DMFCIs in the Euro Area

### Evolution and main results
- Three informational dimensions provided by Figure 1:
  - Whether policy has become looser (a positive DMFCI) or tighter (a negative DMFCI) with respect to the base year.
  - Which component of the DMFCI (monetary or fiscal) is quantitatively more important within each country.
  - Whether the stance is different from the base year with high probability (median and the 5th and 95th percentiles are reported).
- Three main empirical results:
  - The overall stance of demand policies was loosened in all countries from 2008 onward; this loosening halted during 2011-14 (mostly reflecting a reversal of fiscal loosening) and became accommodative again thereafter, mostly reflecting monetary accommodation by the ECB.
  - Heterogeneity across countries:
    - Fiscal loosening was more pronounced in France, less strong in Germany, and absent in Italy.
    - In France and Germany the initial accommodation was strengthened further by ECB loosening starting in 2014; in Italy the overall stance remained tighter than pre-crisis until later and became even tighter in 2011 due to fiscal consolidation measures.
    - Fiscal policies of the rest of the EA played a marginal role in the three EA economies.
  - The euro area’s policy mix since the GFC has been dominated by monetary policy; fiscal policy has somewhat rowed against the cycle, especially in Italy and Germany, making monetary easing the “only game in town.”
    - Rostagno et al. (2019) estimate that the policy package implemented by the ECB contributed almost 3 percentage points to euro-area real GDP growth between 2015 and 2018 and is responsible for part of the job creation observed in the EA; in their analysis, 3/4 million people found a job thanks to ECB measures since 2014.

### Quantitative presentation details (Figure 1 notes)
- Solid lines represent median DMFCI while dotted lines represent the 5th and 95th percentiles.
- For each country DMFCI are rescaled by the standard deviation of the respective total DMFCI.

### Temporal patterns and component roles
- Monetary policy component differences across countries reflect different inflation expectations that make real interest rates country-specific.
- Indices start in 2007q1 for the EA, 2006q1 for France, 2006q2 for Germany, and 2006q3 for Italy (base date constraints described in the source).
- Monetary accommodation after 2014 is a key driver of the renewed accommodative stance across the EA and major economies.

---

### Comparison with alternative available indices (summary of correlations)
- The DMFCI is the first index combining monetary and fiscal stances and accounting for unconventional monetary policy.
- Correlations reported in Table 3 (note: alternative indices interpret sign oppositely; positive in alternatives = tightening):
  - Correlation between DMFCI_M (EA) and EC MCI (2008Q1-2018Q3): -0.26
  - Correlation between DMFCI_M (EA) and EC MCI (2008Q1-2013Q4): -0.58
  - Correlations between DMFCI_M and IMF FCI:
    - EA: -0.59
    - France: -0.55
    - Germany: -0.52
    - Italy: -0.49
  - Correlations between DMFCI_F and IMF CAPB (annualized, percent of potential output):
    - EA: -0.92
    - France: -0.96
    - Germany: -0.89
    - Italy: -0.73
- Interpretation notes:
  - The weaker correlation with the EC MCI over 2008Q1-2018Q3 mainly reflects that DMFCI includes unconventional monetary policy whereas the EC MCI does not; restricting the sample to pre-PSPP increases correlation from -0.26 to -0.58.
  - High negative correlations with IMF FCI reflect that both DMFCI_M and FCI capture unconventional monetary measures affecting long-term and market interest rates.
  - Strong correlations between DMFCI_F and IMF CAPB reflect close comovement between the DMFCI fiscal component and cyclically-adjusted primary balances.
- Overall assessment:
  - The DMFCI subindices comove with existing indices when comparable; where they differ it is often because DMFCI captures unconventional monetary policies and combines monetary and fiscal stances into a single aggregate measure.

---

### Comparison with historical contribution of shocks in the DSGE model
- Conceptual distinction:
  - Historical decomposition of output in the DSGE captures unexpected discretionary policy innovations (residual policy shocks).
  - The DMFCI captures all policy changes (both expected/anticipated and unexpected/unanticipated), by tracking actual outcomes (shadow rate and CAPB).
- Key empirical findings (from Figure 2 and text):
  - Three common findings across EA, France, Germany, Italy from historical shock decomposition:
    - Discretionary monetary policy played a more prominent role in supporting output growth than discretionary fiscal policy.
    - Monetary policy was generally countercyclical during the financial crisis and the sovereign debt crisis and continued to contribute positively to output growth from 2009 to 2018, with the exception of a few episodes in 2010.
    - Discretionary fiscal policy played a minor role in supporting output; its counter/cyclical nature is less clear-cut.
    - Fiscal policies of the rest of the EA played a very limited role, even more limited than domestic fiscal policy.
  - Yearly changes in DMFCI components:
    - Monetary component: generally positive over 2007-2018, indicating progressively more accommodative monetary policy.
    - Fiscal component: negative in all countries during 2011-2013, oscillating around zero afterwards (consistent with literature finding restrictive fiscal policies in 2011-13 and neutral average stance in 2014-2016).
- Mismatch between DMFCI and historical decomposition for fiscal stance:
  - The contribution of fiscal policy shocks to GDP is very limited for all four economies, contrasting with changes in the fiscal stance captured by the DMFCI.
  - Quantification of variance attribution (Figure 3):
    - Unexpected monetary policy shocks explain approximately between 30 and 40 percent of the variance of the real shadow rate (depending on the country).
    - Unexpected fiscal shocks explain between less than 1 percent (Italy and EA overall) and 15 percent (Germany) of the variance of the CAPB.
  - Interpretation:
    - The bulk of fiscal outcomes is driven by shocks other than unexpected fiscal shocks (i.e., anticipated policy changes, automatic stabilizers, or non-policy shocks), making historical contributions of fiscal shocks an unsatisfactory representation of the fiscal stance.
    - Anticipation effects are strong; the DSGE includes a MA component in fiscal rules to capture anticipation (see model details in source).
    - Much fiscal support in 2008-2010 occurred through automatic stabilizers rather than discretionary fiscal decisions.
- Consequence for indicator choice:
  - By using actual monetary and fiscal outcomes (real shadow rate and CAPB), irrespective of whether changes are discretionary/anticipated, the DMFCIs more thoroughly capture the evolution of the actual monetary and fiscal stance.
  - The discrepancy between DMFCI and historical policy-shock contributions is larger before 2014; after 2014 the increasing role of monetary policy in both measures improves comovement.
- Robustness:
  - Using an alternative definition of the output gap does not change the general picture (reference to robustness in Appendix G).

*Source: wpiea2020086-print-pdf — 4.3 Evolution of DMFCIs in the Euro Area*

### 4.6  Dynamic Properties of the DMFCI in the Light of the DSGE

### 4.6  Dynamic Properties of the DMFCI in the Light of the DSGE Model

### DMFCI as leading / coincident / lagging indicator of GDP
- Sample and timing:
  - Correlations computed over the sample 2008Q1-2018Q3.
  - Correlations report the DMFCI at time t−i, with i = [−6, −5, ..., 0], and GDP growth at time t.
- Euro Area (EA) aggregate:
  - The monetary-stance component (DMFCI_M) leads the business cycle.
  - A looser monetary policy is associated with higher GDP growth after three quarters, with a correlation coefficient of 0.66.
  - The fiscal-stance component (DMFCI_F) is negatively and significantly associated with the cycle contemporaneously, consistent with a countercyclical fiscal stance.
- Country evidence:
  - France: monetary policy stance leads the cycle by two quarters; fiscal stance is coincident with the cycle.
  - Germany: monetary policy stance leads the cycle by two quarters; fiscal stance is coincident with the cycle.
  - Italy:
    - Monetary policy stance leads GDP growth by three quarters, with correlation coefficient equal to 0.71.
    - A looser (tighter) fiscal policy is associated with larger (smaller) GDP growth after four quarters, with a correlation coefficient equal to 0.42, suggesting a contractionary effect of fiscal policy over the sample.
- Additional note:
  - The component of the fiscal policy of the rest of the EA, DMFCI_F*, is not reported since its role is very limited.

### Interpretation using DSGE impulse response functions (IRFs)
- Setup and scope:
  - Figure 5 reports estimated IRFs to monetary policy shocks and government spending shocks in the EA and its three major economies.
  - For presentation, all shocks are set such that the policy changes are expansionary; results are symmetric.
  - The size of monetary and fiscal policy shocks is set equal to the estimated standard deviation for each country (reported in Table E.6).
  - The model features two fiscal policy shocks, government spending and tax rate; government spending shocks are quantitatively more important and are reported.
- Main IRF findings:
  - The boost in GDP in response to a monetary policy shock is much more accentuated, delayed and persistent than the boost from a government spending shock.
  - Monetary policy shocks have similar pronounced effects on inflation.
  - The fall in the CAPB following a monetary policy shock is explained by the countercyclical response of fiscal instruments to the monetary-policy-induced expansion.
  - The fall in the CAPB after a government spending shock is explained by the simulated rise in government spending.
  - Interpreting IRFs as responses to “typical” historical shocks (given shock sizes equal to estimated standard deviations) implies that typical monetary policy shocks have a greater effect on output than typical fiscal policy shocks.
- Implication for DMFCI dynamics:
  - The larger, delayed and persistent output response to monetary shocks helps explain why the monetary component of the DMFCI leads GDP growth, whereas the fiscal component does not exhibit the same leading property.

### Key findings and policy implications (summary)
- Historical policy stance evolution:
  - The EA’s overall policy became looser in the aftermath of the crisis, with most loosening manifesting itself between 2009 and 2011 following a short-lived fiscal expansion.
  - The overall policy stance of the EA was then tightened before being loosened again around 2014, when the ECB embraced more drastic accommodative policy actions and monetary policy became the “only game in town.”
- Cross-country heterogeneity:
  - Loosening of fiscal policy during the global financial crisis (GFC) was bold in France, less strong in Germany, and absent in Italy.
- Comparison of DMFCI and DSGE historical shock decompositions:
  - The DMFCI provides a more comprehensive measure of fiscal stance than the historical contribution of policy shocks to GDP growth from the estimated DSGE model.
  - Both the DMFCI and the DSGE historical decomposition convey a similar message concerning the monetary policy stance.
  - The DMFCI is simpler to communicate and shows information on the policy stance and its components in a straightforward manner.
- Policy implication:
  - Having the ECB deliver the lion’s share of economic stimulus for several years, a more expansionary fiscal policy could play an important role in boosting economic activity in the EA.
  - This consideration is particularly relevant for dealing with the economic consequences of the COVID-19 pandemic.
  - At least at the initial stages of the latest crisis, EA policymakers appear to be deploying a more balanced policy mix.

*Source: 4.6 Dynamic Properties of the DMFCI in the Light of the DSGE Model (wpiea2020086-print-pdf).*

### References

### wpiea2020086-print-pdf - References (Appendices A–G)

### A. Model structure — main features
- Two symmetric countries (home and foreign) within a currency union; full equilibrium conditions and steady state reported in Sections B and C.
- Households:
  - Continuum i∈[0,1]; fraction ω are optimizers, fraction 1−ω are rule-of-thumb.
  - Optimizing households: utility E_t{∑_{s=0}^∞ e_{b_t} β^{t+s} [ ln(C^o_{t+s} − h C^o_{t+s−1}) − B/(1+η) (H^o_{t+s})^{1+η} ] } with β∈(0,1), h∈(0,1), B>0, η>0.
  - Assets for optimizers: short-term deposits D^o_t, physical capital K^o_t, foreign assets NFA^o_t (gross foreign rate R^*_t), short-term government bonds and central bank short-term debt (treated as deposits).
  - Rule-of-thumb households consume entire disposable income each period; same instantaneous utility as optimizers but no access to financial markets.
- Habit formation and preference shock: internal habit h and preference shock e_{b_t} enter marginal utility (equation (A.8)).
- Investment financing:
  - Investment financed via perpetual long-term bonds (perpetuities) with parameter κ; new issue price Q_t and bond perpetuity structure F^o_t = CI_t + κ CI_{t−1} + κ^2 CI_{t−2} + ... .
  - Loan-in-advance constraint on investment: P^k_t I^o_t ≤ Q_t CI_t P_t (equation (A.7)).
- Net foreign asset premium: Ψ_t ≡ exp{ ψ_1 (NFA^o_t/Y_t − NFA^o/Y) } − 1 with ψ_1>0 (equation (A.2) and (A.12) uncovered interest parity R_t = Ψ_t R^*_t).
- Aggregation: C_t = ω C^o_t + (1−ω) C^r_t; D_t = ω D^o_t; I_t = ω I^o_t; K_t = ω K^o_t.

### A. Wage-setting and price rigidities
- Labor: unions z∈[0,1] set wages W_{z,t} for members; demand H_{z,t} = (W_{z,t}/W_t)^{-ε_w e_{w,t}} H_t with wage markup shock e_{w,t}.
- Unions maximize weighted utility of members: max_{W_{z,t}} E_t ∑ β^{t+k} [ ω U^o_{t+k} + (1−ω) U^r_{t+k} ] subject to demand and budget constraints (equation (A.16)).
- Quadratic wage adjustment costs per Rotemberg (1982) generate nominal stickiness: Φ_t = θ_w/2 [ (W_{z,t} Π_{ι_w,t−1}^−1 / (W_{z,t−1} Π^{1−ι_w}) − Π^{1−ι_w} )^2 ] W_t H_t (equation (A.17) and derivation in Appendix D).
- Wage schedule (A.18) summarized:
  - ̄Λ_t = ω Λ^o_t + (1−ω) Λ^r_t;
  - Π_{w,t} = γ W_t / W_{t−1} Π_t;
  - Wage equation includes (1−τ_{w,t})(1 − ε_w e_{w,t}) term, wage stickiness θ_w and indexation ι_w, and forward-looking term with β E_t[...].

- Price setting for firms:
  - Monopolistic competition, Dixit-Stiglitz demand (A.34), Rotemberg price adjustment costs θ_p/2 (...) Y_t.
  - New Keynesian price relation (A.39) with price-markup shock e_{P,t} and price indexation ι_p.

### A. Financial intermediaries and frictions
- Financial intermediaries (FIs) are sole buyers of investment bonds F_t and long-term government bonds B^{FI}_t; assets financed by deposits D_t and net worth N_t; balance sheet (A.24): ̄B^{FI}_t + ̄F_t = D_t/P_t + N_t = L_t N_t.
- FI profit: prof_t ≡ [ (R^L_t − R_{t−1}) L_{t−1} + R_{t−1} ] N_{t−1} / Π_t (equation (A.25)), with R^L_t ≡ (1 + κ Q_t/Q_{t−1}).
- Dividend choice and net worth accumulation: maximize V_t = max_{N_t,div_t} E_t ∑ (β ζ)^j Λ^o_{t+j} div_{t+j} subject to div_t + N_t[1+ f(N_t)] ≤ prof_t (A.26–A.27).
  - FI impatience embedded via β ζ < β.
  - Portfolio adjustment cost f(N_t) = ψ_n/2 ( (N_t − ̄N)/̄N )^2.
- Incentive compatibility constraint (ICC) and leverage:
  - ICC: E_t V_{t+1} ≥ Θ_t L_t N_t E_t Λ^o_{t+1} Π_{t+1} R^L_{t+1} (A.28).
  - With binding ICC leverage L_t given by (A.29): L_t = E_t(Λ^o_{t+1} Π_{t+1}) / [ E_t(Λ^o_{t+1} Π_{t+1}) + (e^φ_t − 1) E_t(Λ^o_{t+1} Π_{t+1}) R^L_{t+1} / R_t ] — financial friction depends on exogenous credit shock e^φ_t and net worth dynamics.
- Term premium and long-term bond yields:
  - Term premium TP_t = 1 + R_{10,t} − R_{10,EH,t} (A.33).
  - Investment and long-bond dynamics explicitly modeled via Q_t, R^L_t, R_{10,t}.

### B. Production, investment, trade, and market clearing
- Production: Y_t = e^a_t K_t^α H_t^{1−α} (A.36 / B.27).
- Investment and capital accumulation:
  - Capital producers transform I_t into P^k_t e^{μ_t}[1 − S(I_t/I_{t−1})] I_t with S(·) = ψ_i/2 (I_t / I_{t−1} − 1)^2 (A.40).
  - Asset pricing for investment (A.41) and capital accumulation γ K_{t+1} = (1−δ) K_t + I_t e^{μ_t}[1 − S(...)] (B.37).
- Trade and terms of trade:
  - Home price index P_t = [ φ P_{h,t}^{1−χ} + (1−φ) P_{f,t}^{1−χ} ]^{1/(1−χ)} with χ common elasticity (A.42).
  - Terms of trade TOT_t = P_{f,t}/P_{h,t}, TOT evolution TOT_t / TOT_{t−1} = Π_{f,t}/Π_{h,t} (A.43).
- Resource constraint (detrended): Y_t = C_t + I_t + G_t + p_h,t EXP_t − p_f,t IMP_t + nominal adjustment-cost terms (A.51 / B.113).

### C. Stochastic shocks and equilibrium closure
- Wage and price markup shocks follow ARMA(1,1) (B.52).
- Other exogenous variables follow AR(1) (B.53) with common components for κ = [a, φ] to capture cross-country correlations.
- Currency-union model: eight structural shocks home, eight structural shocks foreign, and three common shocks including monetary policy — total of 19 exogenous disturbances. EA model features 9 structural shocks.

### D. Data, measurement, estimation setup
- Countries covered: Austria, Belgium, Finland, France, Germany, Ireland, Italy, Luxembourg, Netherlands, Portugal, Spain. Euro-Area variables aggregated by nominal GDP weights.
- Data sources (Table E.1): Eurostat (Y_N, I_TOT, I_G, C, G, R_G, T), OECD (PGDP, WH, R_L), Wu and Xia (REONIA shadow rate).
- Observable construction (Table E.2):
  - Y^o_t = ln(Y_{N,t} / P_t) × 100
  - I^o_t = ln((I_{TOT,t} − I_{G,t}) / P_t) × 100
  - C^o_t = ln(C_t / P_t) × 100
  - W^o_t = ln(W_t / P_t) × 100
  - G^o_t = ln((G_t − R_{G,t}) / P_t) × 100
  - T^o_t = ln(T_t / P_t) × 100
  - Π^o_t = ln(P_t / P_{t−1}) × 100
  - R^o_t = R_t / 4
  - TP^o_t = R^L_t − R_t / 4
  - PB/Y,o = (T_t − G_t)/Y_t
- Measurement equations include first differences and levels of detrended variables, with i.i.d. measurement errors ε^{me}_t and an additional measurement error for primary balance to avoid stochastic singularity (equations (E.1) and (E.2)).
- Priors and posteriors: prior distributions in Table E.3 and constants in Table E.4; posterior estimates in Tables E.5–E.6.

### E. Selected posterior estimation results (posterior means and diagnostics)
- Selected posterior means from Table E.5 (structural parameters; country-specific panels for France, Italy, Germany, Euro Area):
  - Inv. Frisch elasticity η:
    - France: 0.53 [0.46;0.59]
    - Italy: 0.51 [0.47;0.54]
    - Germany: 0.46 [0.42;0.50]
    - Euro Area: 0.50 [0.42;0.58]
  - Habits in consumption h:
    - France: 0.73 [0.68;0.79]
    - Italy: 0.77 [0.73;0.82]
    - Germany: 0.79 [0.74;0.84]
    - Euro Area: 0.79 [0.74;0.85]
  - Fraction of optimizing households ω:
    - France: 0.74 [0.68;0.80]
    - Italy: 0.90 [0.87;0.94]
    - Germany: 0.85 [0.80;0.91]
    - Euro Area: 0.85 [0.79;0.90]
  - Investment adjustment cost ψ_i:
    - France: 6.58 [5.66;7.49]
    - Italy: 4.29 [3.63;4.93]
    - Germany: 5.38 [4.19;6.56]
    - Euro Area: 4.97 [3.81;6.12]
  - Net worth adjustment costs ψ_n:
    - France: 0.70 [0.60;0.81]
    - Italy: 0.97 [0.87;1.08]
    - Germany: 0.89 [0.80;0.98]
    - Euro Area: 0.75 [0.59;0.91]
  - Price stickiness θ_p:
    - France: 65.9 [49.6;81.4]
    - Italy: 52.1 [44.8;58.3]
    - Germany: 66.3 [51.3;80.7]
    - Euro Area: 64.9 [47.0;82.6]
  - Wage stickiness θ_w:
    - France: 69.0 [56.3;81.1]
    - Italy: 51.4 [45.6;57.0]
    - Germany: 60.4 [50.5;71.6]
    - Euro Area: 69.4 [51.6;86.3]
  - Inflation Taylor rule coefficient ρ_π:
    - France: 1.94 [1.84;2.05]
    - Italy: 1.94 [1.84;2.07]
    - Germany: 1.70 [1.56;1.81]
    - Euro Area: 1.88 [1.74;2.02]
  - Interest rate smoothing ρ_r:
    - France: 0.80 [0.76;0.84]
    - Italy: 0.85 [0.81;0.88]
    - Germany: 0.85 [0.80;0.89]
    - Euro Area: 0.82 [0.78;0.86]
  - Elasticity of risk premium to NFA ψ_1:
    - France: 0.005 [0.003;0.008]
    - Italy: 0.003 [0.002;0.004]
    - Germany: 0.004 [0.002;0.005]

- Selected posterior means from Table E.6 (constants, shocks, measurement errors):
  - Trend γ:
    - France: 0.46 [0.42;0.50]
    - Italy: 0.35 [0.31;0.40]
    - Germany: 0.27 [0.22;0.33]
    - Euro Area: 0.44 [0.39;0.49]
  - Inflation constant ̄Π:
    - France: 0.15 [0.09;0.21]
    - Italy: 0.18 [0.14;0.23]
    - Germany: 0.08 [0.04;0.11]
    - Euro Area: 0.26 [0.17;0.35]
  - Term premium constant ̄TP:
    - France: 0.29 [0.18;0.39]
    - Italy: 0.39 [0.32;0.46]
    - Germany: 0.22 [0.11;0.32]
    - Euro Area: 0.39 [0.28;0.50]
  - Measurement errors σ_me:
    - France: 0.39 [0.34;0.44]
    - Italy: 0.61 [0.52;0.69]
    - Germany: 0.59 [0.51;0.68]
    - Euro Area: 0.47 [0.41;0.53]
  - Log-likelihood (Table E.6):
    - France: -1198.73
    - Italy: -11583.87
    - Germany: -1251.46
    - Euro Area: -595.95

### F. Dynamic Monetary and Fiscal Condition Indices (DMFCI) — regression and results
- DMFCI construction: regressions use model shocks and macro variables; Tables F.1–F.4 report percentiles of regression coefficients for Euro Area, France, Germany, Italy.
- Selected empirical regularities (from DMFCI tables and Appendix G):
  - Median ̄R^2 for regressions:
    - Euro Area median ̄R^2 = 0.64 (Table F.1)
    - France median ̄R^2 = 0.77 (Table F.2)
    - Germany median ̄R^2 = 0.87 (Table F.3)
    - Italy median ̄R^2 = 0.81 (Table F.4)
  - Euro Area monetary component DMFCI^M correlates with European Commission MCI: ρ = 0.69 (Figure G.1 caption).
  - Cross-country correlations of fiscal indices (Table G.1):
    - Correlation DMFCI^F: EA vs France = 0.78; EA vs Germany = 0.72; EA vs Italy = 0.57.
    - France vs Germany = 0.40; France vs Italy = 0.42; Germany vs Italy = 0.32.
  - Robustness to alternative output gap measures (trend vs efficient level):
    - Correlation between DMFCI changes computed with two alternative potential output definitions (Table G.2):
      - Total DMFCI correlations: EA 0.92; France 0.91; Germany 0.95; Italy 0.85.
      - Monetary component DMFCI^M correlations: EA 0.98; France 1.00; Germany 1.00; Italy 0.98.
      - Fiscal component DMFCI^F correlations: EA 0.87; France 0.90; Germany 0.94; Italy 0.90.
      - Fiscal-rest-of-EA DMFCI^F* correlations: France 0.81; Germany 0.90; Italy 0.83.
  - Historical decomposition vs DMFCI (Figure G.2):
    - Monetary components of DMFCI co-move with historical contribution of monetary shocks to output.
    - Fiscal components of DMFCI exhibit almost no comovement with historical contribution of fiscal shocks to output.

### G. Practical implications and model mechanisms (from appendices)
- Financial intermediation and QE channel:
  - Central bank purchases of long-term government bonds change FI portfolio composition; increased central bank holdings reduce FI holdings of long-term government bonds, freeing FI net worth to purchase investment bonds and increase private investment (Appendix A.2 discussion).
  - Financial frictions (ICC, leverage L_t, portfolio adjustment costs f(N_t)) limit arbitrage and cause sluggish net worth adjustments, amplifying the real effects of central bank asset purchases.
- Keynesian fiscal transmission:
  - Presence of rule-of-thumb households (share 1−ω) increases Keynesian effect of fiscal policy; larger 1−ω → larger fiscal multiplier (Appendix A.1.2).
- Term premium and long-rate targeting:
  - Term premium TP_t explicitly modeled (A.33); long-rate policies affect R^L_t and thus FI returns and leverage incentives.

*Source: wpiea2020086-print-pdf - References (Appendices A–G), https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020086-print-pdf.pdf*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020086-print-pdf.pdf_