## 1. Optimized Response Rule Coefficients

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---

### Introduction and context
- Since the early 1990s, inflation in the U.K. has remained low and steady, and its correlation with measures of economic slack has weakened (Figure 1).
- The link between these variables is often referred to as “the short-run Phillips curve.”
- Flattening of the Phillips curve has been observed in other industrial countries and has been attributed, in part, to globalization forces (Bean, 2006).
- IMF (2006), Borio and Filardo (2006), and Mody and Ohnsorge (2006) provide empirical cross-country evidence for globalization-related decline in the sensitivity of prices to domestic demand pressures.
- IMF (2006) finds that globalization can account for more than half of the decline in the sensitivity of prices to domestic output, while improved monetary policy credibility and the low inflation environment account for the remainder.
- Core question addressed: whether and how the central bank’s response to shocks should change when the Phillips curve flattens due to globalization.
- Scope: focuses on globalization-related explanation for Phillips-curve flattening and how it affects monetary policy response; Section II describes the general equilibrium model, Section III derives best achievable combinations of inflation and output gap variability and discusses efficient parameterization of the monetary policy response rule for different slopes of the Phillips curve. Conclusions are in Section IV.

### Mechanisms driving Phillips curve flattening
- Three observations underpin the argument that globalization may lead to a flattening of the Phillips curve:
  - First, due to increased competition from abroad, businesses have less scope to raise prices when demand rises.
  - Second, increased trade and investment flows have made goods prices less sensitive to domestic demand pressures. Goods price inflation has borne very little relationship to estimates of demand pressures in the U.K. over the last decade, while service price inflation has been well predicted by domestic demand pressures.
  - Third, labor mobility (both actual and virtual) has also increased in recent years; this could lead to further flattening if it results in declining sensitivity of service sector wages and prices to domestic demand shifts.
- Complementary explanation: good monetary policy has helped anchor inflation expectations (Laxton and N’Diaye, 2002), contributing in the initial years after the introduction of inflation targeting in the United Kingdom.

### Policy implications and challenges (qualitative)
- Reduced sensitivity of inflation to demand shocks implies:
  - Easier management of temporary imbalances between demand and supply, since demand shocks transmit less into inflation.
  - Harder response to inflation shocks: the central bank must move interest rates and aggregate demand more to offset inflation shocks and bring inflation back to target.
- Identification of cause matters: if flattening is due to increased monetary policy credibility rather than globalization, policy implications differ.

### Model description and key equations
- Model type: New Keynesian open economy macroeconomic model with rational expectations; U.K. treated as a small open economy; foreign sector is a weighted average of the euro area and the United States.
- Key behavioral equations (variable definitions preserved exactly as in the source):
  - Output gap (IS curve): ygap equation (given as equation (1)).
  - Inflation (expectations-augmented Phillips curve): π equation (given as equation (2)).
  - Exchange rate relationship: z equation (given as equation (3)).
  - Monetary policy response function (inflation-forecast-based interest rate rule): rs equation (given as equation (4)).
  - Variable definitions:
    - ygap = output gap
    - rs = nominal interest rate
    - rr = real interest rate
    - z = real exchange rate
    - rrgap = gap between the real interest rate and its equilibrium value
    - zgap = gap between the real exchange rate index and its equilibrium value
    - ygap* = foreign output gap
    - π = quarterly annualized rate of CPI inflation
    - π4 = four-quarter moving average of π
    - πT = target rate of inflation
- Model described as relatively simple, capturing key channels of monetary policy transmission and offering clarity and tractability.

### Estimation, priors, and scenarios analyzed
- Parameters estimated from U.K. quarterly data over the period 1993 to 2005 using a Bayesian approach.
- Prior distributions based on prior distributions for Canada (Berg, Karam, and Laxton (2006)), augmented with UK-specific evidence where available.
- Bayesian estimation allows for measurement errors in the data; measurement errors are omitted in the stochastic simulations.
- Simulations assume rational expectations: all agents know the true model, see current period shocks correctly, and assume all future shocks will be zero.
- Two versions of the model to examine a structural change in the Phillips curve:
  - Baseline (estimated) Phillips-curve coefficient on the output gap: 0.3.
  - Alternative (flatter) Phillips-curve coefficient on the output gap: 0.1.
- Rationale for the flat Phillips-curve scenario:
  - Ball (2006) estimate: a 10 percentage points increase in trade openness reduces the output gap coefficient by 0.08.
  - In the U.K., the ratio of trade (imports plus exports excluding oil) to GDP increased by 12 percentage points in the last 15 years (1991 to 2006), translating into a coefficient decline of 0.1.
  - Additional cross-border labor flows can contribute to flattening; a greater decline of 0.2 is assumed in the alternative specification.
- Note: flattening could alternatively reflect increased monetary policy credibility (modeled as an increase in the coefficient on inflation expectations in the Phillips curve).

### Impulse-response dynamics (qualitative summary)
- Under a flatter Phillips curve (output gap coefficient reduced from 0.3 to 0.1):
  - Demand shocks have much smaller effects on inflation; therefore interest rates need to move very little.
  - Weaker policy response makes demand shocks more persistent, increasing output gap volatility.
  - Inflation shocks cause greater volatility in both inflation and output because unwinding such shocks requires more aggressive interest-rate and demand adjustments.
- If monetary policy credibility (higher coefficient on expected inflation in the Phillips curve) is the driver of flattening, dynamics differ: better-anchored expectations reduce the required output-gap movements to counter inflation shocks and improve the inflation-output volatility trade-off.

### Evaluating alternative monetary policy response rules
- Optimal monetary policy response rule: inflation-forecast-based interest rate rule defined by equation (4), chosen to minimize the quadratic loss function:
  - Loss function (equation (5)): L = sum_{t=0}^{∞} [λπ (πt − πT)^2 + λy (ygapt)^2 + λr (Δrst)^2]
- Preferences of the monetary authority represented by the ratio λy / λπ; varying this ratio traces different policy priorities between inflation and output-gap variability.
- Stochastic-simulation methodology:
  - Use estimated model parameters and estimated shock distributions.
  - For each policy parametrization, generate 5,000 artificially-generated outcomes by drawing from the estimated distribution of residuals and solving the model.
  - Average the standard deviations of inflation and the output gap across draws to evaluate policy performance.
- Parameter sweep used to construct efficient frontiers and optimized rules:
  - Monetary policy response coefficients on the output gap and deviations of inflation from target are varied in the interval 0.5 to 12 with a step of 0.5.
  - Interest-rate smoothing coefficient is set at 0.5 in all simulations.
- Main findings from the efficient policy frontier and optimized rules:
  - The achievable combinations of inflation and output variability are less favorable when the Phillips curve is flatter.
  - With a flatter Phillips curve, the central bank needs to generate greater volatility in domestic demand to achieve a given level of inflation variability (and vice versa).
  - Directional change in optimized rule coefficients when the Phillips curve flattens:
    - The optimal coefficient on the expected inflation deviation declines.
    - The optimal coefficient on the output gap increases.
  - Intuition: with a flat Phillips curve, policymakers must respond more to demand pressures to prevent costly-to-reverse inflation deviations, while responses to pure inflation shocks are attenuated because rapid unwinding would create larger output-gap variability.
- Numerical specifics preserved exactly:
  - 5,000 artificial outcomes per frontier point.
  - Coefficient variation interval: 0.5 to 12 (step 0.5).
  - Interest-rate smoothing coefficient: 0.5.
  - Example preference illustrated: λy / λπ = 3.

### Policy implications and recommendations (operational)
- Communication and expectations management:
  - The implications of a flatter Phillips curve for volatility and speed of adjustment must be clearly communicated so that inflation expectations remain anchored around the target.
  - Enhancing public understanding of structural changes (and possible policy changes) is important to keep expectations anchored.
- Optimal policy stance under a flatter Phillips curve:
  - A central bank that cares about both the inflation target and deviations of output from potential should:
    - Respond relatively less to deviations of inflation from target.
    - Respond relatively more to deviations of output from potential.
  - Effects of following such an optimized rule:
    - Reduces output volatility after an inflation shock.
    - Reduces volatility of both output and inflation after a demand shock (relative to leaving the rule unchanged).
  - Even with an attenuated response coefficient on inflation, interest rates may need to stay above the long-run equilibrium rate longer after a positive inflation shock because deviations of inflation from target are more persistent under a flatter Phillips curve.
  - Conversely, the cumulative interest-rate gap would be smaller after a demand shock.
- Caveats and conditional factors:
  - Results are contingent on the assumption that the joint distribution of shocks going forward is the same as the historical distribution used for estimation.
  - Structural changes that flatten the Phillips curve may also alter the distribution of shocks.
  - If the flattening is due to increased monetary policy credibility rather than globalization, monetary policy implications differ; empirical identification of the cause is important.
  - If agents’ expectations deviate from rational expectations, the monetary policy response to inflation shocks would be more aggressive than the optimal response under stable expectations.

### APPENDIX I — A. The Model (core equations and identities)
- Model context:
  - Open-economy model specified in gap and rate-of-change terms so that, under inflation targeting, all variables are stationary.
  - For simulation purposes, the equilibrium values for the real interest rate and the real exchange rate are assumed to be time invariant.
  - Core behavioral equations are split into domestic and foreign sectors; stochastic error processes and identities complete the model.
- Core domestic-sector equations:
  - Aggregate Demand (IS curve):
    - ygap_t = β1·ygap_{t-1} + β2·ygap_{t-2} + β3·ygap_{t-4} + β4·rrgap_{t-1} + β5·zgap_{t-1} + β14·ygap^*_t + ε_ygap_t
  - Inflation (New Keynesian Phillips-type specification):
    - π_t = δ1·π_{t-1} + δ2·π_{t-2} + (1−δ3)·π_{4,t−1} + δ4·Δz_t + ε_π_t
  - Real exchange rate:
    - z_t = φ1·z_{t-1} + φ2·(z_{t-4} − z^*_{t-4}) + φ3·(rr_t − rr^*_t) + ε_z_t
  - Monetary policy reaction function (Taylor-like rule):
    - rs_t = α1·rs_{t-1} + α2·rs_{t-4} + γ·(π_{4,t+1} − π_T) + (1−γ)·(π_{t+1} − π_T) + α1·(rr_eq − rr)_t + ε_rs_t
- Core foreign-sector equations (analogues):
  - Aggregate Demand:
    - ygap^*_t = β^*_1·ygap^*_{t-1} + β^*_2·ygap^*_{t-2} + β^*_3·ygap^*_{t-4} + β^*_4·rrgap^*_{t-1} + ε_ygap^*_t
  - Inflation:
    - π^*_t = δ^*_1·π^*_{t-1} + δ^*_2·π^*_{t-2} + (1−δ^*_3)·π^*_{4,t-1} + δ^*_4·ygap^*_t + ε_π^*_t
  - Monetary policy reaction function:
    - rs^*_t = α^*_1·rs^*_{t-1} + α^*_2·rs^*_{t-4} + γ^*·(π^*_{4,t+1} − π^*_T) + (1−γ^*)·(π^*_{t+1} − π^*_T) + α^*_1·(rr^*_{eq} − rr^*)_t + ε_rs^*_t
- Stochastic processes (autoregressive specifications) and identities as specified in the source:
  - π_{4,t} = .25·(π_{t} + π_{t-1} + π_{t-2} + π_{t-3})
  - π^*_{4,t} = .25·(π^*_{t} + π^*_{t-1} + π^*_{t-2} + π^*_{t-3})
  - rr_{t-1} = rs_{t-1} − π_{t-1}
  - rr^*_{t-1} = rs^*_{t-1} − π^*_{t-1}
  - rrgap_t = rr_t − rr_eq_t
  - rrgap^*_t = rr^*_t − rr^*_{eq,t}
  - zgap_t = z_t − z^_{eq,t}

### APPENDIX I — B. Data
- Sample and measurement:
  - Model estimated using quarterly data from 1993 to 2005.
  - For the U.K., the price index is RPIX until the last quarter of 2003, and headline CPI thereafter; the inflation target is adjusted respectively.
  - Foreign aggregate demand, interest rates and CPI inflation are proxied by an index comprising the euro area and the United States. (The euro area data is approximated by a weighted average of Germany, France, and Italy.)
  - Real exchange rate: trade-weighted real exchange rate index published by the Bank of England.
  - Nominal short-term interest rates: 90-day Treasury bills rates.
  - Equilibrium values for potential output, the real interest rate, and the real exchange rate constructed using a variant of the HP filter with end-sample constraints.
- NAIRU: estimated using the methodology of Boone et al. (2002).

### APPENDIX I — C. Estimation Results (posterior means)
- Estimation approach:
  - Bayesian, full-system estimation conducted in DYNARE; observable variables: output gaps (real GDP), nominal short-term interest rates, inflation rates, and logs of real exchange rates; estimation allows for measurement error.
  - Estimates presented for the U.K. and foreign counterpart (sample period 1993Q1 to 2005Q4).
- Table 1A — United Kingdom Model Parameter Estimation Results (Posterior Means shown):
  - Domestic parameters (posterior means shown):
    - β1: 0.677
    - β2: 0.094
    - β3: 0.108
    - β4: 0.034
    - β5: 0.173
    - δ1: 0.201
    - δ2: 0.286
    - δ3: 0.084
    - Φ: 0.261
    - γ: 0.557
    - α1: 1.966
    - α2: 0.557
  - Foreign parameters (posterior means shown):
    - β^*_1: 0.748
    - β^*_2: 0.103
    - β^*_3: 0.105
    - δ^*_1: 0.190
    - δ^*_2: 0.237
    - γ^*: 0.588
    - α^*_1: 2.001
    - α^*_2: 0.530
- Table 2A — Error Processes and Measurement Errors (Posterior Means shown):
  - Domestic:
    - ρ_ygap: 0.772
    - std. dev. ξ_ygap: 0.157
    - std. dev. measurement error ygap: 0.123
    - ρ_π: 0.499
    - std. dev. ξ_π: 0.286
    - std. dev. measurement error π: 2.206
    - ρ_rs: 0.767
    - std. dev. ξ_rs: 0.171
    - std. dev. measurement error rs: 0.123
    - std. dev. ε_z: 5.992
  - Foreign:
    - ρ_ygap^*: 0.765
    - std. dev. ξ_ygap^*: 0.176
    - std. dev. measurement error ygap^*: 0.149
    - ρ_π^*: 0.510
    - std. dev. ξ_π^*: 0.165
    - std. dev. measurement error π^*: 1.042
    - ρ_rs^*: 0.837
    - std. dev. ξ_rs^*: 0.163
    - std. dev. measurement error rs^*: 0.147

*Source: APPENDIX I of the provided IMF working paper chapter (estimation sample period 1993Q1 to 2005Q4).*

### 1. Optimized Response Rule Coefficients ............................................................................9

### 1. Optimized Response Rule Coefficients

### Introduction and context
- Since the early 1990s, inflation in the U.K. has remained low and steady, and its correlation with measures of economic slack has weakened (Figure 1).
- The link between these variables is often referred to as “the short-run Phillips curve.”
- Flattening of the Phillips curve has been observed in other industrial countries and has been attributed, in part, to globalization forces (Bean, 2006).
- IMF (2006), Borio and Filardo (2006), and Mody and Ohnsorge (2006) provide empirical cross-country evidence for globalization-related decline in the sensitivity of prices to domestic demand pressures.
- IMF (2006) finds that globalization can account for more than half of the decline in the sensitivity of prices to domestic output, while improved monetary policy credibility and the low inflation environment account for the remainder.

### Mechanisms driving Phillips curve flattening
- Three observations underpin the argument that globalization may lead to a flattening of the Phillips curve:
  - First, due to increased competition from abroad, businesses have less scope to raise prices when demand rises.
  - Second, increased trade and investment flows have made goods prices less sensitive to domestic demand pressures. Goods price inflation has borne very little relationship to estimates of demand pressures in the U.K. over the last decade, while service price inflation has been well predicted by domestic demand pressures.
  - Third, labor mobility (both actual and virtual) has also increased in recent years. This could lead to further flattening if it results in declining sensitivity of service sector wages and prices to domestic demand shifts; workers may not press for higher wages when the domestic labor market tightens for fear that their jobs may be taken by foreign labor.

- A complementary explanation is that good monetary policy has helped anchor inflation expectations (Laxton and N’Diaye, 2002), which likely contributed in the initial years after the introduction of inflation targeting in the United Kingdom.

### Policy implications and challenges
- A reduced sensitivity of inflation to demand shocks implies:
  - It becomes easier for central bankers regarding temporary imbalances between demand and supply, since demand shocks transmit less into inflation.
  - It becomes harder to respond to inflation shocks: the central bank must move interest rates and aggregate demand more to offset inflation shocks and bring inflation back to target.

- Core question addressed: whether and how the central bank’s response to shocks should change when the Phillips curve flattens due to globalization.

### Scope and structure of the analysis
- The paper focuses on the implications of the globalization-related explanation for the flattening of the Phillips curve, as this is likely to be more relevant at the current juncture and going forward.
- This paper explores how a globalization-induced flattening of the Phillips curve would affect the monetary policy response to economic disturbances.
- Section II describes the general equilibrium model used in the analysis.
- Section III derives the best achievable combinations of inflation and output gap variability and discusses the efficient parameterization of the monetary policy response rule for different slopes of the Phillips curve.

_ _wp0776 - 1. Optimized Response Rule Coefficients_

### conclusions are in Section IV.

### _wp0776 - conclusions are in Section IV.

### Model description and key equations
- The analysis uses a New Keynesian open economy macroeconomic model with rational expectations.
- Key behavioral equations for the U.K.:
  - Output gap (IS curve): ygap equation (given as equation (1) in the source).
  - Inflation (expectations-augmented Phillips curve): π equation (given as equation (2)).
  - Exchange rate relationship: z equation (given as equation (3)).
  - Monetary policy response function (inflation-forecast-based interest rate rule): rs equation (given as equation (4)).
- Variable definitions preserved exactly as in the source:
  - ygap = output gap
  - rs = nominal interest rate
  - rr = real interest rate
  - z = real exchange rate
  - rrgap = gap between the real interest rate and its equilibrium value
  - zgap = gap between the real exchange rate index and its equilibrium value
  - ygap* = foreign output gap
  - π = quarterly annualized rate of CPI inflation
  - π4 = four-quarter moving average of π
  - πT = target rate of inflation
- The U.K. is treated as a small open economy; the foreign sector is a weighted average of the euro area and the United States.
- The model is described as relatively simple, capturing key channels of monetary policy transmission and offering clarity and tractability.

### Estimation, priors, and scenarios analyzed
- Parameters estimated from U.K. quarterly data over the period 1993 to 2005 using a Bayesian approach.
- Prior distributions were based on prior distributions for Canada (Berg, Karam, and Laxton (2006)), augmented with UK-specific evidence where available.
- The Bayesian estimation allows for measurement errors in the data; these measurement errors are omitted in the stochastic simulations.
- Simulations assume rational expectations: all agents know the true model, see current period shocks correctly, and assume all future shocks will be zero.
- Two versions of the model are employed to examine a structural change in the Phillips curve:
  - Baseline (estimated) Phillips-curve coefficient on the output gap: 0.3.
  - Alternative (flatter) Phillips-curve coefficient on the output gap: 0.1 (implying smaller responsiveness of inflation to the output gap).
- Rationale for the flat Phillips-curve scenario:
  - Ball (2006) estimate: a 10 percentage points increase in trade openness reduces the output gap coefficient by 0.08.
  - In the U.K., the ratio of trade (imports plus exports excluding oil) to GDP increased by 12 percentage points in the last 15 years (1991 to 2006), translating into a coefficient decline of 0.1.
  - Additional cross-border labor flows can contribute to flattening; a greater decline of 0.2 is assumed in the alternative specification.
- Note on alternative structural interpretation: flattening could alternatively reflect increased monetary policy credibility (which would be modeled as an increase in the coefficient on inflation expectations in the Phillips curve).

### Impulse-response dynamics (qualitative summary)
- Under a flatter Phillips curve (output gap coefficient reduced from 0.3 to 0.1):
  - Demand shocks have much smaller effects on inflation; therefore interest rates need to move very little.
  - Weaker policy response makes demand shocks more persistent, increasing output gap volatility.
  - Inflation shocks cause greater volatility in both inflation and output because unwinding such shocks requires more aggressive interest-rate and demand adjustments.
- When monetary policy credibility (higher coefficient on expected inflation in the Phillips curve) is the driver of flattening, dynamics differ: better-anchored expectations reduce the required output-gap movements to counter inflation shocks and improve the inflation-output volatility trade-off.

### Evaluating alternative monetary policy response rules
- Optimal monetary policy response rule: the inflation-forecast-based interest rate rule defined by equation (4), chosen to minimize the quadratic loss function:
  - Loss function (equation (5)): L = sum_{t=0}^{∞} [λπ (πt - πT)^2 + λy (ygapt)^2 + λr (Δrst)^2] (preserved in form as in the source).
- Preferences of the monetary authority represented by the ratio λy / λπ; varying this ratio traces different policy priorities between inflation and output-gap variability.
- Stochastic-simulation methodology:
  - Use estimated model parameters and estimated shock distributions.
  - For each policy parametrization, generate 5,000 artificially-generated outcomes by drawing from the estimated distribution of residuals and solving the model.
  - Average the standard deviations of inflation and the output gap across draws to evaluate policy performance.
- Parameter sweep used to construct efficient frontiers and optimized rules:
  - Monetary policy response coefficients on the output gap and deviations of inflation from target are varied in the interval 0.5 to 12 with a step of 0.5.
  - Interest-rate smoothing coefficient is set at 0.5 in all simulations.
- Main findings from the efficient policy frontier and optimized rules:
  - The achievable combinations of inflation and output variability are less favorable when the Phillips curve is flatter.
  - With a flatter Phillips curve, the central bank needs to generate greater volatility in domestic demand to achieve a given level of inflation variability (and vice versa).
  - Directional change in optimized rule coefficients when the Phillips curve flattens:
    - The optimal coefficient on the expected inflation deviation declines.
    - The optimal coefficient on the output gap increases.
  - Intuition: with a flat Phillips curve, policymakers must respond more to demand pressures to prevent costly-to-reverse inflation deviations, while responses to pure inflation shocks are attenuated because rapid unwinding would create larger output-gap variability.
- Numerical/scenario specifics preserved exactly where provided:
  - 5,000 artificial outcomes per frontier point.
  - Coefficient variation interval: 0.5 to 12 (step 0.5).
  - Interest-rate smoothing coefficient: 0.5.
  - Example preference illustrated: λy / λπ = 3.

### Policy implications and recommendations
- Communication and expectations management:
  - The implications of a flatter Phillips curve for volatility and speed of adjustment must be clearly communicated so that inflation expectations remain anchored around the target.
  - Enhancing public understanding of structural changes (and possible policy changes) is important to keep expectations anchored.
- Optimal policy stance under a flatter Phillips curve:
  - A central bank that cares about both the inflation target and deviations of output from potential should:
    - Respond relatively less to deviations of inflation from target.
    - Respond relatively more to deviations of output from potential.
  - Effects of following such an optimized rule:
    - Reduces output volatility after an inflation shock.
    - Reduces volatility of both output and inflation after a demand shock (relative to leaving the rule unchanged).
  - Even with an attenuated response coefficient on inflation, interest rates may need to stay above the long-run equilibrium rate longer after a positive inflation shock because deviations of inflation from target are more persistent under a flatter Phillips curve.
  - Conversely, the cumulative interest-rate gap would be smaller after a demand shock.
- Caveats and conditional factors:
  - Results are contingent on the assumption that the joint distribution of shocks going forward is the same as the historical distribution used for estimation.
  - Structural changes that flatten the Phillips curve may also alter the distribution of shocks (for example, heightened competition may change firms’ reactions to cost shocks).
  - If the flattening is due to increased monetary policy credibility rather than globalization, the monetary policy implications differ; empirical identification of the cause of flattening is important when setting policy.
  - If agents’ expectations are more influenced by actual inflation deviations than the model implies (violation of rational expectations), then the monetary policy response to inflation shocks would be more aggressive than the optimal response under stable expectations.

*Source: _wp0776 - conclusions are in Section IV.*

### APPENDIX I

### APPENDIX I

### A. The Model
- Model context:
  - Open-economy model specified in gap and rate-of-change terms so that, under inflation targeting, all variables are stationary.
  - For simulation purposes, the equilibrium values for the real interest rate and the real exchange rate are assumed to be time invariant.
  - Core behavioral equations are split into domestic and foreign sectors; stochastic error processes and identities complete the model.

- Core domestic-sector equations (as specified):
  - Aggregate Demand (IS curve):
    - ygap_t = β1·ygap_{t-1} + β2·ygap_{t-2} + β3·ygap_{t-4} + β4·rrgap_{t-1} + β5·zgap_{t-1} + β14·ygap^*_t + ε_ygap_t
    - (Source notation: ygap denotes the output gap, rrgap is the gap between the real interest rate and its equilibrium value, zgap is the gap between the real exchange rate index and its equilibrium value, ygap* is the foreign output gap, ε_ygap is the stochastic error process.)
  - Inflation (New Keynesian Phillips-type specification):
    - π_t = δ1·π_{t-1} + δ2·π_{t-2} + (1−δ3)·π_{4,t−1} + δ4·Δz_t + ε_π_t
    - (π is the quarterly annualized rate of CPI inflation; π_4 is a four-quarter moving average; Δz is first difference in the real exchange rate index; ε_π is the stochastic error.)
  - Real exchange rate:
    - z_t = φ1·z_{t-1} + φ2·(z_{t-4} − z^*_{t-4}) + φ3·(rr_t − rr^*_t) + ε_z_t
    - (z is log of the real exchange rate index; rr is domestic real interest rate; rr* is foreign real interest rate; ε_z is stochastic error.)
  - Monetary policy reaction function (Taylor-like rule):
    - rs_t = α1·rs_{t-1} + α2·rs_{t-4} + γ·(π_{4,t+1} − π_T) + (1−γ)·(π_{t+1} − π_T) + α1·(rr_eq − rr)_t + ε_rs_t
    - (rs is annualized short-term policy rate; rr_eq is equilibrium real interest rate; π_T is target inflation; ε_rs is stochastic error.)

- Core foreign-sector equations (analogues of domestic):
  - Aggregate Demand:
    - ygap^*_t = β^*_1·ygap^*_{t-1} + β^*_2·ygap^*_{t-2} + β^*_3·ygap^*_{t-4} + β^*_4·rrgap^*_{t-1} + ε_ygap^*_t
  - Inflation:
    - π^*_t = δ^*_1·π^*_{t-1} + δ^*_2·π^*_{t-2} + (1−δ^*_3)·π^*_{4,t-1} + δ^*_4·ygap^*_t + ε_π^*_t
  - Monetary policy reaction function:
    - rs^*_t = α^*_1·rs^*_{t-1} + α^*_2·rs^*_{t-4} + γ^*·(π^*_{4,t+1} − π^*_T) + (1−γ^*)·(π^*_{t+1} − π^*_T) + α^*_1·(rr^*_{eq} − rr^*)_t + ε_rs^*_t

- Stochastic processes (autoregressive specifications):
  - ygap_t = ρ_ygap·ygap_{t-1} + ξ_ygap_t
  - ygap^*_t = ρ_ygap^*·ygap^*_{t-1} + ξ_ygap^*_t
  - π_t = ρ_π·π_{t-1} + ξ_π_t
  - π^*_t = ρ_π^*·π^*_{t-1} + ξ_π^*_t
  - rs_t = ρ_rs·rs_{t-1} + ξ_rs_t
  - rs^*_t = ρ_rs^*·rs^*_{t-1} + ξ_rs^*_t

- Identities:
  - π_{4,t} = .25·(π_{t} + π_{t-1} + π_{t-2} + π_{t-3})
  - π^*_{4,t} = .25·(π^*_{t} + π^*_{t-1} + π^*_{t-2} + π^*_{t-3})
  - rr_{t-1} = rs_{t-1} − π_{t-1}
  - rr^*_{t-1} = rs^*_{t-1} − π^*_{t-1}
  - rrgap_t = rr_t − rr_eq_t
  - rrgap^*_t = rr^*_t − rr^*_{eq,t}
  - zgap_t = z_t − z^_{eq,t}

### B. Data
- Sample and measurement:
  - Model estimated using quarterly data from 1993 to 2005.
  - For the U.K., the price index is RPIX until the last quarter of 2003, and headline CPI thereafter; the inflation target is adjusted respectively.
  - Foreign aggregate demand, interest rates and CPI inflation are proxied by an index comprising the euro area and the United States. (The euro area data is approximated by a weighted average of Germany, France, and Italy.)
  - Real exchange rate: trade-weighted real exchange rate index published by the Bank of England.
  - Nominal short-term interest rates: 90-day Treasury bills rates.
  - Equilibrium values for potential output, the real interest rate, and the real exchange rate constructed using a variant of the HP filter with end-sample constraints to prevent convergence to observed data and to allow forcing toward user-specified end values.

- NAIRU:
  - The NAIRU (in Figure 1) estimated using the methodology of Boone et al. (2002).

### C. Estimation Results
- Estimation approach:
  - Bayesian, full-system estimation conducted in DYNARE.
  - Observable variables: output gaps (real GDP), nominal short-term interest rates, inflation rates, and logs of real exchange rates.
  - Estimation allows for measurement error in observable variables.
  - Estimates presented for the U.K. and foreign counterpart in Tables 1A and 2A (sample period 1993Q1 to 2005Q4).

- Table 1A — United Kingdom Model Parameter Estimation Results (Posterior Means shown):
  - Domestic parameters (prior mean; distribution; posterior mean):
    - β1: prior mean 0.85; distribution gamma; posterior mean 0.677
    - β2: prior mean 0.10; distribution beta; posterior mean 0.094
    - β3: prior mean 0.10; distribution gamma; posterior mean 0.108
    - β4: prior mean 0.05; distribution beta; posterior mean 0.034
    - β5: prior mean 0.15; distribution beta; posterior mean 0.173
    - δ1: prior mean 0.20; distribution gamma; posterior mean 0.201
    - δ2: prior mean 0.30; distribution gamma; posterior mean 0.286
    - δ3: prior mean 0.10; distribution gamma; posterior mean 0.084
    - Φ:  prior mean 0.30; distribution beta; posterior mean 0.261
    - γ:  prior mean 0.55; distribution beta; posterior mean 0.557
    - α1: prior mean 2.00; distribution gamma; posterior mean 1.966
    - α2: prior mean 0.55; distribution beta; posterior mean 0.557
  - Foreign parameters (prior mean; distribution; posterior mean):
    - β^*_1: prior mean 0.85; distribution gamma; posterior mean 0.748
    - β^*_2: prior mean 0.10; distribution beta; posterior mean 0.103
    - β^*_3: prior mean 0.10; distribution gamma; posterior mean 0.105
    - δ^*_1: prior mean 0.20; distribution beta; posterior mean 0.190
    - δ^*_2: prior mean 0.30; distribution gamma; posterior mean 0.237
    - γ^*: prior mean 0.56; distribution beta; posterior mean 0.588
    - α^*_1: prior mean 2.00; distribution gamma; posterior mean 2.001
    - α^*_2: prior mean 0.50; distribution beta; posterior mean 0.530

- Table 2A — Error Processes and Measurement Errors (Posterior Means shown):
  - Domestic:
    - ρ_ygap: prior mean 0.75; distribution beta; posterior mean 0.772
    - std. dev. ξ_ygap: prior mean 0.25; distribution inverse gamma; posterior mean 0.157
    - std. dev. measurement error ygap: prior mean 0.20; distribution inverse gamma; posterior mean 0.123
    - ρ_π: prior mean 0.50; distribution beta; posterior mean 0.499
    - std. dev. ξ_π: prior mean 0.25; distribution inverse gamma; posterior mean 0.286
    - std. dev. measurement error π: prior mean 0.20; distribution inverse gamma; posterior mean 2.206
    - ρ_rs: prior mean 0.750; distribution beta; posterior mean 0.767
    - std. dev. ξ_rs: prior mean 0.25; distribution inverse gamma; posterior mean 0.171
    - std. dev. measurement error rs: prior mean 0.20; distribution inverse gamma; posterior mean 0.123
    - std. dev. ε_z: prior mean 4.00; distribution inverse gamma; posterior mean 5.992
  - Foreign:
    - ρ_ygap^*: prior mean 0.75; distribution beta; posterior mean 0.765
    - std. dev. ξ_ygap^*: prior mean 0.25; distribution inverse gamma; posterior mean 0.176
    - std. dev. measurement error ygap^*: prior mean 0.20; distribution inverse gamma; posterior mean 0.149
    - ρ_π^*: prior mean 0.50; distribution beta; posterior mean 0.510
    - std. dev. ξ_π^*: prior mean 0.25; distribution inverse gamma; posterior mean 0.165
    - std. dev. measurement error π^*: prior mean 0.20; distribution inverse gamma; posterior mean 1.042
    - ρ_rs^*: prior mean 0.750; distribution beta; posterior mean 0.837
    - std. dev. ξ_rs^*: prior mean 0.25; distribution inverse gamma; posterior mean 0.163
    - std. dev. measurement error rs^*: prior mean 0.20; distribution inverse gamma; posterior mean 0.147

*Source: APPENDIX I of the provided IMF working paper chapter (estimation sample period 1993Q1 to 2005Q4).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp0776.pdf_