## 7. Euro Area Macro-Financial Model: Government Bond Yields and Model

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---

### Introduction and key findings
- The financial crisis severely impaired traditional monetary transmission channels (interest rate, bank lending, broad credit, expectations).
- Evidence on interest rate pass-through:
  - Policy rate changes transmitted to market rates but with weaker efficiency, longer lags, and more noise.
  - Bank lending and broad credit channels also disrupted; policy reaction needed to stabilize the economy became stronger.
- Non-standard ECB measures had measurable effects:
  - Some evidence that money market spreads and the government bond yield curve have been lower and flatter than predicted by standard macro variables since September 2008.
  - These measures included lengthening maturities of refinancing operations and a sizeable increase in the ECB’s balance sheet.

### ECB policy response to the crisis (operations and chronology)
- Core ECB actions and features:
  - Unlimited provision of liquidity through “fixed rate tenders with full allotment”.
  - Extension of the collateral list, increasing private sector assets to 56 percent of the nominal value of securities on the list.
  - Extension of the maturity of long-term refinancing operations, initially to six months, and then to twelve months (late June 2009).
  - Liquidity provision in foreign currencies through swap lines with the Federal Reserve.
  - Outright purchases of covered bonds (for an amount of 60 billion euros).
- Counterparty participation and volumes:
  - Counterparties participating in refinancing operations increased from 1,700 before the crisis to 2,200.
  - Number of active counterparties before the crisis was about 450; increased to 750 during the crisis; more than 1,100 participated in the one-year operation.
- Market outcomes:
  - Term money market spreads dropped sharply after unlimited longer-term funds at fixed rates began end-October 2008; liquidity premia virtually eliminated though spreads remained elevated.
  - In January 2009 the ECB restored the interest rate corridor from ±50 basis points to ±100 basis points around the policy rate.
  - Overnight interbank market volumes stabilized at around 40–60 billion euros in 2009.

### Transmission channels: conceptual structure
- Interest rate channel:
  - short nominal rates → expectations & sticky prices → long nominal and real rates → user cost of capital → investment/output.
- Bank lending channel:
  - central bank affects credit supply to bank-dependent borrowers via interest rates and quantity of base money.
- Broad credit channel:
  - monetary policy affects borrowers’ net worth and the external finance premium through financial frictions.
- Expectations channel:
  - anchoring inflation expectations is vital; central bank credibility and predictability strengthen this channel.

### Methodology overview
- Bi-variate VARs:
  - Used to assess pass-through of policy rate (EONIA approximating policy rate) to market rates.
  - Market rates studied: one-, three-, six-month, and one-year Euribor; corporate bond yields (3- to 5-year, ratings A, AA, AAA, BBB); new loans to non-financial corporations (various size/maturity buckets); new loans to households; consumer credit maturities.
  - Sample periods: January 1999–April 2009 for interbank rates and corporate yields; January 2003–April 2009 for loans data.
  - Lag length set at 3 months. VARs estimated in levels and first differences using OLS; BVAR robustness checks reported as qualitatively similar.
- Theory-based general equilibrium (New Keynesian-style) model:
  - Core equations: inflation (monthly, seasonally adjusted at annual rate), output gap, monetary policy reaction.
  - Extensions: time-varying inflation target to capture underlying inflation perceptions; loan demand and supply block to test bank-lending channel; corporate bond spread equation to capture external finance premium and broader credit channel.
  - Estimation: Bayesian methods with monthly data; sample January 1995–February 2009 (bank-lending version) and January 1999–February 2009 (credit version).
  - Variables: EONIA, average loan rates (1–5 years), AAA/AA/A/BBB corporate yields (3–5 year), corporate-government spread, HICP inflation, M3, bank loans, activity index (industrial production 30% + retail sales 70%).

### Empirical results — VAR analysis
- Pass-through characteristics (full-sample VARs):
  - Impact on 3-month Euribor is close to one-for-one; maximum impact within a month.
  - Initial impact on corporate bond yields and new loans to non-financial corporations is quick; full adjustment more protracted.
  - Impact magnitudes on corporate bonds: 0.6 to 0.7 percentage point for AA- and AAA-rated bonds versus 1.2 for BBB-rated bonds.
  - Pass-through to loans to households for house purchases is somewhat lower and slower; other consumer loans adjust faster.
- Crisis effects on pass-through:
  - Time for full adjustment of market rates increased to over 12 months from 3–6 months before the crisis.
  - Transmission to lower-grade corporate bonds particularly impaired: initial response of BBB yields switched from positive pre-crisis to negative post-crisis.
  - Variance of VAR residuals for market rates increased since beginning of 2008 in most cases, signaling less reliable pass-through.
- Selected VAR residual standard deviation comparisons (structural models):
  - Loan rates model (Jan. 02 - Jul. 07): 0.22; (Aug. 07 - Dec. 08): 0.40.
  - Term spread model (Jan. 02 - Jul. 07): 0.11; (Aug. 07 - Dec. 08): 0.20.
  - Short-term interest rates model (Jan. 02 - Jul. 07): 0.07; (Aug. 07 - Dec. 08): 0.12.
- Interpretation:
  - Tightening lending standards and quantity effects (loan originations cut) further impair monetary policy effectiveness beyond interest-rate pass-through.

### Empirical results — Theory-based model
- Speed and strength of transmission:
  - Policy feedback pass-through time increased to about 2½ years from about 1½ year before the crisis (response of policy rates to demand and supply shocks).
  - Time for full transmission of monetary policy shocks to inflation increased from about 2 years before the crisis to about 3 years during the crisis.
- Structural parameter changes:
  - Policy reaction to deviation of expected inflation from target increased from around 1.2 before the crisis to 1.4 during the crisis.
  - Policy response to output gap remained at around 0.5 across samples.
- Model residuals and volatility:
  - Variability of residuals jumped after mid-2008; transmission became slower and more unpredictable (more noise).
- Contribution of channels to variance (impulse/variance decomposition):
  - Interest rate channel: over 30 percent of inflation variation; close to 50 percent of output variation.
  - Expectations: around 40 percent of inflation variation; about 30 percent of output variation.
  - Bank-lending channel: explains about 15 percent of output variation.
  - Broader credit channel: explains about 10 percent of output variation.
- Inflation expectations:
  - Model-derived inflation expectations declined significantly in Q4 2008 and recovered since beginning of 2009 as policies eased.
  - Market-based measures correlated with model-derived expectations with a correlation coefficient of 75 percent (noted caveats given market conditions).

### Monetary policy, liquidity trap risks, and non-standard measures
- Liquidity-trap discussion:
  - Policy rates at or near zero constrain conventional easing; expectations about future inflation/policy matter crucially.
- Practical non-standard policy tools:
  - Shaping public expectations about future policy rate settings (explicit commitment to low rates, providing term funds at policy rate).
  - Increasing money supply (quantitative easing) via asset purchases or easing collateral—can lower nominal yields and encourage portfolio rebalancing.
  - Credit easing—direct support to dysfunctional credit markets (e.g., covered bonds, asset-backed securities).
- Empirical assessment of non-standard ECB measures on government bond yields (macro-finance VAR-based model):
  - State variables: output gap (HP-detrended weighted activity index), year-on-year HICP inflation, monthly EONIA, one-year Euribor.
  - Data: monthly observations January 1999–January 2009; VAR(3) stacked to VAR(1).
  - Pricing kernel with market prices of risk linear in state variables.
  - Estimation in two stages: VAR coefficients estimated; then stochastic pricing kernel loadings estimated by nonlinear least squares fitting bond yield data (1999M1–2009M1).
- Model fit and residuals:
  - Predicted yields track actual yields closely; long-run “risk-free” yields slightly above 3 percent for long-term yields during 1999M1–2009M1.
  - Short-term (two-year) model residuals show no obvious trend; long-term government bond residuals have been more or less consistently negative since 2004–05.
  - Residuals turned sharply negative in October 2008 (coincident with ECB non-standard measures).
- Interpretation of yield-curve movements:
  - Since October 2008, actual yield curve lower and flatter than predicted by macro variables.
  - Possible channels:
    - Increase in monetary base and relative supply of money to bonds → portfolio rebalancing → lower nominal bond yields.
    - Flattening may reflect markets interpreting ECB actions as implicit commitments to keep policy rates low longer than the simple VAR-implied state.
  - Caveats: short investigation period; indirect analysis of ECB measures; alternative explanations (e.g., capital flows, “flight to safety”); flattening most pronounced at long end while expectations channel would predict larger short-end effects.

### Conclusions and policy implications
- Transmission has continued to operate but with:
  - Lower efficiency, longer lags, and more noise since the crisis onset.
  - Stronger policy reaction required to stabilize output and inflation.
- Inflation expectations fell sharply in Q4 2008 but recovered as policies eased in early 2009.
- Non-standard ECB measures likely contributed to:
  - Reducing term spreads in money markets (preliminary evidence).
  - Some beneficial effects on government bond term spreads and lowering/flattening the yield curve (evidence preliminary and indirect).
- Policy considerations:
  - Non-standard measures that extend maturities, increase monetary base, and directly support dysfunctional credit markets can assist transmission when conventional policy rates are constrained, but they carry costs (e.g., central bank bearing credit risk, potential reversal challenges).
  - Anchoring inflation expectations and signaling persistence of low policy rates are central to mitigating liquidity-trap/deflation risks.

### Appendix I. Small Theory-Based Model for the Euro Area — key equations and data
- Inflation Equation (Phillips Curve):
  - πt = 1200(p_sa_t - p_sa_{t-1}); π12_t = 100(p_t - p_{t-12}).
  - Variables: πt (monthly inflation, seasonally adjusted at annual rate), πt+12^e (expected year-on-year inflation), π12_t (year-on-year inflation), yt (output gap), επ_t (supply shock).
- Aggregate Demand:
  - Includes forward-looking expected output gap, lagged MA output gap terms, (r_t - eq_r_t), l̂_t r_t (loan real rate deviation), sp_t (corporate-government spread), demand shock.
- Monetary Policy Reaction Function:
  - it (nominal policy rate) responds with interest rate smoothing, natural interest rate, forward-looking responses to expected inflation deviation from target and the output gap.
- Extensions:
  - Time-varying inflation target: πs_t = ss_π + ρ(πs_{t-1} - ss_π) + επ_t.
  - Loan demand (A5): l_t depends on lagged loan interest rates, output growth, loan demand shock.
  - Loan supply (A6): inverse loan supply relates lagged loan interest rates, money growth μ_t, loan supply shock.
  - Corporate spread (A7): sp_t = σ1 sp_{t-1} + σ2 y_{t-1} + σ3 i_{t-1} + sp_t ε.
- Equilibrium trends and real rates:
  - g_t = (1 - τ_g) g_{t-1} + τ_g g^* + g_t ε; y_t^* evolves with g_t; r_t^{n} follows AR(1); r_t = i_t - π_{t+12}^e.
- Data and estimation:
  - Bayesian estimation.
  - Sample periods: bank-lending channel version January 1995 to February 2009; broader credit channel version January 1999 to February 2009.
  - Monthly, seasonally adjusted series: activity index (industrial production 30% + retail sales 70%), EONIA, loan rates (1–5 years), corporate yields (3–5 years, AAA/AA/A/BBB), sp_t, HICP inflation, M3, bank loans.

### Euro Area: Estimates of model parameters (selected)
- λ1: 0.77; St. dev.: 0.05; t-stat: 16.65; Prior distrib.: beta; Prior st. dev.: 0.05
- λ2: 0.12; St. dev.: 0.06; t-stat: 1.92; Prior distrib.: gamm; Prior st. dev.: 0.10
- β1: 0.74; St. dev.: 0.10; t-stat: 7.57; Prior distrib.: gamm; Prior st. dev.: 0.10
- β2: 0.14; St. dev.: 0.05; t-stat: 2.73; Prior distrib.: beta; Prior st. dev.: 0.05
- β3: 0.20; St. dev.: 0.05; t-stat: 4.22; Prior distrib.: gamm; Prior st. dev.: 0.05
- θl: 0.08; St. dev.: 0.04; t-stat: 1.76; Prior distrib.: gamm; Prior st. dev.: 0.05
- θsp: 1/0.05; St. dev.: 0.02; t-stat: 2.08; Prior distrib.: gamm; Prior st. dev.: 0.05
- γ1: 0.83; St. dev.: 0.15; t-stat: 5.40; Prior distrib.: beta; Prior st. dev.: 0.15
- γ2: 1.38; St. dev.: 0.27; t-stat: 5.14; Prior distrib.: gamm; Prior st. dev.: 0.35
- γ3: 0.47; St. dev.: 0.15; t-stat: 3.20; Prior distrib.: gamm; Prior st. dev.: 0.15
- τ: 0.04; St. dev.: 0.02; t-stat: 2.10; Prior distrib.: beta; Prior st. dev.: 0.05
- πss: 1.98; St. dev.: 0.16; t-stat: 11.99; Prior distrib.: norm; Prior st. dev.: 0.50
- g*: 1.80; St. dev.: 0.50; t-stat: 3.64; Prior distrib.: norm; Prior st. dev.: 0.50
- ρ: 0.05; St. dev.: 0.02; t-stat: 2.95; Prior distrib.: beta; Prior st. dev.: 0.07
- r*: 1.49; St. dev.: 0.21; t-stat: 7.23; Prior distrib.: norm; Prior st. dev.: 0.50
- λi: 0.34; St. dev.: 0.12; t-stat: 2.91; Prior distrib.: gamm; Prior st. dev.: 0.10
- λy: 0.73; St. dev.: 0.30; t-stat: 2.45; Prior distrib.: gamm; Prior st. dev.: 0.10
- τi: 0.15; St. dev.: 0.04; t-stat: 3.89; Prior distrib.: gamm; Prior st. dev.: 0.10
- λm: 0.33; St. dev.: 0.19; t-stat: 1.69; Prior distrib.: gamm; Prior st. dev.: 0.10
- σ1: 0.48; St. dev.: 0.07; t-stat: 6.45; Prior distrib.: gamm; Prior st. dev.: 0.10
- σ2: 1.12; St. dev.: 0.12; t-stat: 9.68; Prior distrib.: gamm; Prior st. dev.: 0.10

_Italic: IMF staff estimates and analysis as presented in the chapter._

### References..............................................................................................................

### _wp09185 - References..............................................................................................................

### References
- References................................................................................................................................33

### Tables
- 1. VAR Parameter Estimates ...................................................................................................20
- 2. Risk Factor Loadings ...........................................................................................................20

### Figures
- 1. Euro Area: Recent Developments of the ECB’s Liquidity Operations ...............................22
- 2. Euro Area: Cost of Borrowing by Businesses and Households...........................................23
- 3. Euro Area: Pass-through of The ECB Policy Rate Changes to Market Rates.....................24
- 4. Euro Area: The Impact of Crisis on Policy Rate Pass-through ...........................................25
- 5. Euro Area: VAR Residuals of Market Rates .......................................................................26
- 6. Euro Area: Effectiveness of Monetary Policy .....................................................................26

*Source: _wp09185 - References (PDF).*

### 7. Euro Area Macro-Financial Model: Government Bond Yields and Model ........................27

### 7. Euro Area Macro-Financial Model: Government Bond Yields and Model

### Introduction and key findings
- The financial crisis severely impaired traditional monetary transmission channels (interest rate, bank lending, broad credit, expectations).
- Evidence on interest rate pass-through:
  - Policy rate changes transmitted to market rates but with weaker efficiency, longer lags, and more noise.
  - Bank lending and broad credit channels also disrupted; policy reaction needed to stabilize the economy became stronger.
- Non-standard ECB measures had measurable effects:
  - Some evidence that money market spreads and the government bond yield curve have been lower and flatter than predicted by standard macro variables since September 2008.
  - These measures included lengthening maturities of refinancing operations and a sizeable increase in the ECB’s balance sheet.

*Source: IMF staff analysis in the chapter.*

### ECB policy response to the crisis (operations and chronology)
- Core ECB actions and features:
  - Unlimited provision of liquidity through “fixed rate tenders with full allotment”.
  - Extension of the collateral list, increasing private sector assets to 56 percent of the nominal value of securities on the list.
  - Extension of the maturity of long-term refinancing operations, initially to six months, and then to twelve months (late June 2009).
  - Liquidity provision in foreign currencies through swap lines with the Federal Reserve.
  - Outright purchases of covered bonds (for an amount of 60 billion euros).
- Counterparty participation and volumes:
  - Counterparties participating in refinancing operations increased from 1,700 before the crisis to 2,200.
  - Number of active counterparties before the crisis was about 450; increased to 750 during the crisis; more than 1,100 participated in the one-year operation.
- Market outcomes:
  - Term money market spreads dropped sharply after unlimited longer-term funds at fixed rates began end-October 2008; liquidity premia virtually eliminated though spreads remained elevated.
  - In January 2009 the ECB restored the interest rate corridor from ±50 basis points to ±100 basis points around the policy rate.
  - Overnight interbank market volumes stabilized at around 40–60 billion euros in 2009.

### Transmission channels: conceptual structure
- Interest rate channel: short nominal rates → expectations & sticky prices → long nominal and real rates → user cost of capital → investment/output.
- Bank lending channel: central bank affects credit supply to bank-dependent borrowers via interest rates and quantity of base money.
- Broad credit channel: monetary policy affects borrowers’ net worth and the external finance premium through financial frictions.
- Expectations channel: anchoring inflation expectations is vital; central bank credibility and predictability strengthen this channel.

### Methodology overview
- Bi-variate VARs:
  - Used to assess pass-through of policy rate (EONIA approximating policy rate) to market rates.
  - Market rates studied: one-, three-, six-month, and one-year Euribor; corporate bond yields (3- to 5-year, ratings A, AA, AAA, BBB); new loans to non-financial corporations (various size/maturity buckets); new loans to households; consumer credit maturities.
  - Sample periods: January 1999–April 2009 for interbank rates and corporate yields; January 2003–April 2009 for loans data.
  - Lag length set at 3 months. VARs estimated in levels and first differences using OLS; BVAR robustness checks reported as qualitatively similar.
- Theory-based general equilibrium (New Keynesian-style) model:
  - Core equations: inflation (monthly, seasonally adjusted at annual rate), output gap, monetary policy reaction.
  - Extensions: time-varying inflation target to capture underlying inflation perceptions; loan demand and supply block to test bank-lending channel; corporate bond spread equation to capture external finance premium and broader credit channel.
  - Estimation: Bayesian methods with monthly data; sample January 1995–February 2009 (bank-lending version) and January 1999–February 2009 (credit version).
  - Variables: EONIA, average loan rates (1–5 years), AAA/AA/A/BBB corporate yields (3–5 year), corporate-government spread, HICP inflation, M3, bank loans, activity index (industrial production 30% + retail sales 70%).

### Empirical results — VAR analysis
- Pass-through characteristics (full-sample VARs):
  - Impact on 3-month Euribor is close to one-for-one; maximum impact within a month.
  - Initial impact on corporate bond yields and new loans to non-financial corporations is quick; full adjustment more protracted.
  - Impact magnitudes on corporate bonds: 0.6 to 0.7 percentage point for AA- and AAA-rated bonds versus 1.2 for BBB-rated bonds.
  - Pass-through to loans to households for house purchases is somewhat lower and slower; other consumer loans adjust faster.
- Crisis effects on pass-through:
  - Time for full adjustment of market rates increased to over 12 months from 3–6 months before the crisis.
  - Transmission to lower-grade corporate bonds particularly impaired: initial response of BBB yields switched from positive pre-crisis to negative post-crisis.
  - Variance of VAR residuals for market rates increased since beginning of 2008 in most cases, signaling less reliable pass-through.
- Selected VAR residual standard deviation comparisons (structural models):
  - Loan rates model (Jan. 02 - Jul. 07): 0.22; (Aug. 07 - Dec. 08): 0.40.
  - Term spread model (Jan. 02 - Jul. 07): 0.11; (Aug. 07 - Dec. 08): 0.20.
  - Short-term interest rates model (Jan. 02 - Jul. 07): 0.07; (Aug. 07 - Dec. 08): 0.12.
- Interpretation:
  - Tightening lending standards and quantity effects (loan originations cut) further impair monetary policy effectiveness beyond interest-rate pass-through.

### Empirical results — Theory-based model
- Speed and strength of transmission:
  - Policy feedback pass-through time increased to about 2½ years from about 1½ year before the crisis (response of policy rates to demand and supply shocks).
  - Time for full transmission of monetary policy shocks to inflation increased from about 2 years before the crisis to about 3 years during the crisis.
- Structural parameter changes:
  - Policy reaction to deviation of expected inflation from target increased from around 1.2 before the crisis to 1.4 during the crisis.
  - Policy response to output gap remained at around 0.5 across samples.
- Model residuals and volatility:
  - Variability of residuals jumped after mid-2008; transmission became slower and more unpredictable (more noise).
- Contribution of channels to variance (impulse/variance decomposition):
  - Interest rate channel: over 30 percent of inflation variation; close to 50 percent of output variation.
  - Expectations: around 40 percent of inflation variation; about 30 percent of output variation.
  - Bank-lending channel: explains about 15 percent of output variation.
  - Broader credit channel: explains about 10 percent of output variation.
- Inflation expectations:
  - Model-derived inflation expectations declined significantly in Q4 2008 and recovered since beginning of 2009 as policies eased.
  - Market-based measures correlated with model-derived expectations with a correlation coefficient of 75 percent (noted caveats given market conditions).

### Monetary policy, liquidity trap risks, and non-standard measures
- Liquidity-trap discussion:
  - Policy rates at or near zero constrain conventional easing; expectations about future inflation/policy matter crucially.
  - Practical non-standard policy tools:
    - Shaping public expectations about future policy rate settings (explicit commitment to low rates, providing term funds at policy rate).
    - Increasing money supply (quantitative easing) via asset purchases or easing collateral—can lower nominal yields and encourage portfolio rebalancing.
    - Credit easing—direct support to dysfunctional credit markets (e.g., covered bonds, asset-backed securities).
- Empirical assessment of non-standard ECB measures on government bond yields (macro-finance VAR-based model following Bernanke, Reinhart, and Sack 2004 / Rudebusch, Swanson, and Wu 2006):
  - State variables: output gap (HP-detrended weighted activity index), year-on-year HICP inflation, monthly EONIA, one-year Euribor.
  - Data: monthly observations January 1999–January 2009; VAR(3) stacked to VAR(1).
  - Pricing kernel with market prices of risk linear in state variables.
  - Estimation in two stages: VAR coefficients estimated; then stochastic pricing kernel loadings estimated by nonlinear least squares fitting bond yield data (1999M1–2009M1).
- Model fit and residuals:
  - Predicted yields track actual yields closely; long-run “risk-free” yields slightly above 3 percent for long-term yields during 1999M1–2009M1.
  - Short-term (two-year) model residuals show no obvious trend; long-term government bond residuals have been more or less consistently negative since 2004–05.
  - Residuals turned sharply negative in October 2008 (coincident with ECB non-standard measures).
- Interpretation of yield-curve movements:
  - Since October 2008, actual yield curve lower and flatter than predicted by macro variables.
  - Possible channels:
    - Increase in monetary base and relative supply of money to bonds → portfolio rebalancing → lower nominal bond yields.
    - Flattening may reflect markets interpreting ECB actions as implicit commitments to keep policy rates low longer than the simple VAR-implied state.
  - Caveats: short investigation period; indirect analysis of ECB measures; alternative explanations (e.g., capital flows, “flight to safety”); flattening most pronounced at long end while expectations channel would predict larger short-end effects.

### Conclusions and policy implications
- Transmission has continued to operate but with:
  - Lower efficiency, longer lags, and more noise since the crisis onset.
  - Stronger policy reaction required to stabilize output and inflation.
- Inflation expectations fell sharply in Q4 2008 but recovered as policies eased in early 2009.
- Non-standard ECB measures likely contributed to:
  - Reducing term spreads in money markets (preliminary evidence).
  - Some beneficial effects on government bond term spreads and lowering/flattening the yield curve (evidence preliminary and indirect).
- Overall message for policy:
  - Non-standard measures that extend maturities, increase monetary base, and directly support dysfunctional credit markets can assist transmission when conventional policy rates are constrained, but they carry costs (e.g., central bank bearing credit risk, potential reversal challenges).
  - Anchoring inflation expectations and signaling persistence of low policy rates are central to mitigating liquidity-trap/deflation risks.

_Italic: IMF staff estimates and analysis as presented in the chapter._

### Appendix I. Small Theory-Based Model for the Euro Area

### Appendix I. Small Theory-Based Model for the Euro Area

### Inflation Equation (Phillips Curve)
- Core equation (A1) describes monthly inflation dynamics including expected inflation, lagged year-on-year inflation, output gap terms, and a supply shock:
  - Variables:
    - πt: monthly inflation (seasonally adjusted at annual rate)
    - πt+12^e: expected year-on-year inflation
    - π12_t: year-on-year inflation
    - yt: measure of the output gap
    - επ_t: supply shock
  - Interpretation:
    - Prices are set as a markup over marginal cost (captured by the output gap).
    - Degree of price flexibility and inflation inertia captured by the weight on the expected inflation term.
- Empirical implementation details:
  - πt = 1200(p_sa_t - p_sa_{t-1}), where p_sa_t is the logarithm of the seasonally adjusted HICP.
  - π12_t = 100(p_t - p_{t-12}), where p_t is the logarithm of the HICP index.
  - Economic activity approximated by weighted average of industrial production (30 percent share) and retail sales indexes.
  - Initial values for the output gap y_t for Bayesian estimation use the log-difference of the actual index from its Hodrick-Prescott filtered value.

### Aggregate Demand
- Core aggregate demand equation (A2) includes:
  - e_t y_{t+1}: expected output gap next period (forward-looking term).
  - Lagged moving average output gap terms (inertia).
  - (r_t - eq_r_t): deviation of actual from equilibrium real interest rates (interest rate channel).
  - l̂_t r_t: deviation of the real rate of loans from equilibrium (bank lending channel).
  - sp_t: corporate-government bond yield spread (broader credit channel).
  - y_t ε: demand shock.
- Interpretation:
  - Forward-looking term captures intertemporal smoothing by agents.
  - Bank lending and broader credit channels captured separately to analyze transmission beyond policy rates.
- Monthly leads and lags chosen to reproduce dynamic responses consistent with quarterly models and plausible price-setting contract lengths and capacity-utilization lags.

### Monetary Policy Reaction Function
- Policy rule (A3) specification:
  - it: nominal policy interest rate
  - πt^e: expected inflation used in policy rule
  - π: inflation target
  - it ε: monetary policy shock
  - Components:
    - Interest rate smoothing term (degree of smoothing captured by first term).
    - Natural interest rate term (eq_r or r* component).
    - Forward-looking Taylor-type responses to expected inflation deviation from target and to the output gap.

### Extensions: Time-Varying Inflation Target and Credit Channels
- Inflation target dynamics (A4):
  - πs_t = ss_π + ρ(πs_{t-1} - ss_π) + επ_t
  - ss_π: steady-state inflation
  - επ_t: shock to underlying inflation
  - Rationale: time-varying inflation target provides insights on central bank credibility.
- Bank lending channel: added loan demand and loan supply block to core equations (A1)–(A4).
  - Loan Demand (A5):
    - l_t: growth of loans provided by banking system
    - l_{t-1} i_t: interest rate on loans (lagged)
    - g_t: growth of output
    - ld_t ε: loan demand shock
  - Loan Supply (A6) — inverse loan supply (determines loan interest rates):
    - i_{t-1}^l: interest rate on loans (lagged)
    - μ_t: money growth
    - ls_t ε: loan supply shock
  - Money demand closure: money growth μ_t determined by output growth and policy interest rates.
- Broader credit channel: corporate bond yield spread equation (A7):
  - sp_t = σ1 sp_{t-1} + σ2 y_{t-1} + σ3 i_{t-1} + sp_t ε
  - sp_t ε: shock to corporate spread
  - The corporate bond spread captures the external finance premium (risk and liquidity premia).

### Equilibrium Trends and Real Rates
- Equilibrium growth (A8):
  - g_t = (1 - τ_g) g_{t-1} + τ_g g^* + g_t ε
  - g_t: quarterly real GDP growth at annual rate
  - g^*: steady-state real GDP growth
  - g_t ε: shock to steady-state growth
- Equilibrium output (A9):
  - y_t^* = y_{t-1}^* + g_t + y_t^* ε
  - y_t^*: log level of equilibrium real GDP
  - y_t^* ε: shock to equilibrium output
- Equilibrium real interest rate (A10):
  - r_t^{n} = ρ r_{t-1}^{n} + (1 - ρ) r^* + r_t^{n} ε
  - r^*: steady-state real interest rate
  - r_t^{n} ε: shock to equilibrium real interest rates
- Real interest rate identity (A11):
  - r_t = i_t - π_{t+12}^e

### Data and Estimation
- Estimation approach:
  - Bayesian methods.
- Sample periods:
  - Version with bank-lending channel: January 1995 to February 2009.
  - Version with broader credit channel: January 1999 to February 2009.
- Monthly seasonally adjusted data series used:
  - Economic activity: weighted average of industrial production (30 percent share) and retail sales.
  - Short-rates: approximated by EONIA interest rate.
  - Loan rates: average of interest rates on new loans to euro area residents of 1- to 5-year maturity.
  - Corporate bond yields: AAA-, AA-, A-, BBB-rated corporate bond yields of 3- to 5-year maturity.
  - sp_t: spread of corporate bond yield and government bond yield.
  - HICP inflation.
  - Money supply: approximated by M3 aggregate.
  - Bank loans to euro area residents.

### Euro Area: Estimates of Model Parameters
- λ1: 0.77; St. dev.: 0.05; t-stat: 16.65; Prior distrib.: beta; Prior st. dev.: 0.05
- λ2: 0.12; St. dev.: 0.06; t-stat: 1.92; Prior distrib.: gamm; Prior st. dev.: 0.10
- β1: 0.74; St. dev.: 0.10; t-stat: 7.57; Prior distrib.: gamm; Prior st. dev.: 0.10
- β2: 0.14; St. dev.: 0.05; t-stat: 2.73; Prior distrib.: beta; Prior st. dev.: 0.05
- β3: 0.20; St. dev.: 0.05; t-stat: 4.22; Prior distrib.: gamm; Prior st. dev.: 0.05
- θl: 0.08; St. dev.: 0.04; t-stat: 1.76; Prior distrib.: gamm; Prior st. dev.: 0.05
- θsp: 1/0.05; St. dev.: 0.02; t-stat: 2.08; Prior distrib.: gamm; Prior st. dev.: 0.05
- γ1: 0.83; St. dev.: 0.15; t-stat: 5.40; Prior distrib.: beta; Prior st. dev.: 0.15
- γ2: 1.38; St. dev.: 0.27; t-stat: 5.14; Prior distrib.: gamm; Prior st. dev.: 0.35
- γ3: 0.47; St. dev.: 0.15; t-stat: 3.20; Prior distrib.: gamm; Prior st. dev.: 0.15
- τ: 0.04; St. dev.: 0.02; t-stat: 2.10; Prior distrib.: beta; Prior st. dev.: 0.05
- πss: 1.98; St. dev.: 0.16; t-stat: 11.99; Prior distrib.: norm; Prior st. dev.: 0.50
- g*: 1.80; St. dev.: 0.50; t-stat: 3.64; Prior distrib.: norm; Prior st. dev.: 0.50
- ρ: 0.05; St. dev.: 0.02; t-stat: 2.95; Prior distrib.: beta; Prior st. dev.: 0.07
- r*: 1.49; St. dev.: 0.21; t-stat: 7.23; Prior distrib.: norm; Prior st. dev.: 0.50
- λi: 0.34; St. dev.: 0.12; t-stat: 2.91; Prior distrib.: gamm; Prior st. dev.: 0.10
- λy: 0.73; St. dev.: 0.30; t-stat: 2.45; Prior distrib.: gamm; Prior st. dev.: 0.10
- τi: 0.15; St. dev.: 0.04; t-stat: 3.89; Prior distrib.: gamm; Prior st. dev.: 0.10
- λm: 0.33; St. dev.: 0.19; t-stat: 1.69; Prior distrib.: gamm; Prior st. dev.: 0.10
- σ1: 0.48; St. dev.: 0.07; t-stat: 6.45; Prior distrib.: gamm; Prior st. dev.: 0.10
- σ2: 1.12; St. dev.: 0.12; t-stat: 9.68; Prior distrib.: gamm; Prior st. dev.: 0.10

*Source: IMF staff estimates. 1/ The sample starts in January 1999.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp09185.pdf_