## 1. Golden Rule: Procyclical Fiscal Policy During the Cycle

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### I. Purpose and approach
- Objective: examine a forward-looking implementation of the golden rule that is independent of cycle dating by aiming for current balance or better over a rolling time horizon (two- to five-year horizons examined, focus on the three-year horizon).
- Rationale for golden rule (introduced 1998): over the economic cycle, the government should borrow only to invest and not to fund current spending; requires the current budget to be in balance or surplus on average over the economic cycle.
- Drawbacks of the “over the cycle” formulation:
  - Automatic stabilizers may be constrained if procyclical tightening is required toward the end of a cycle.
  - Ex-post testing requires precise dating of the economic cycle; cycle dating is controversial.
- Method: compare fiscal performance under alternative implementations using a simple open economy stochastic model calibrated to U.K. data.

### II. Model and simulation design
- Model type and estimation:
  - Small open-economy “New Keynesian” model with rational expectations, specified in gap and rate-of-change terms; estimated by Bayesian technique for 1992:Q4–2006:Q4.
  - Parameters estimated; selected posterior means reported (e.g., β1: 0.743; θ1: 0.182; α2: 1.849).
- Simulation structure:
  - Seven stochastic shocks each period; drawn from normal distributions with mean 0 and model-estimated standard deviations.
  - Simulation: 1,000 draws lasting 100 quarters each (100 quarters = 25 years); starting from steady state.
  - Three-step period sequence each period: (Step 1) update forecasts and monetary policy, (Step 2) identify economic cycles (illustrative simulations assume ex-post perfect observability), (Step 3) adjust current and future fiscal balance path to meet the tested fiscal rule; discretionary fiscal policy updated only once a year.
- Cycle definition used:
  - Crossing points where output gap = 0 and require four consecutive quarters of same sign after a sign change (shortest cycle = eight quarters = two years).
  - Historical U.K. cycles identified: 1978:Q1–1987:Q2, 1987:Q3–1997:Q3, 1997:Q4–2003:Q2.
  - Model-generated cycle length (average): Mean 27.1 quarters; Median 25.0 quarters.
- Fiscal calibration and rule mechanics:
  - Automatic stabilizers Ast: Treasury 2003 elasticities (output gap contemporaneous elasticity 0.5, lagged 0.2; total elasticity 0.7).
  - Equilibrium overall fiscal balance calibrated at -2 percent of GDP (zero current balance + 2 percent of GDP net investment) consistent with stabilizing public sector net debt at 40 percent of GDP.
  - Forward-looking rule operational target: aim for current balance or better at a rolling horizon; three-year specification aims to achieve fiscal balance three-year ahead (t+12 FBgap) and eliminate projected debt gap over next four years (t+28 E(CumFBΔ)); adjustments introduced gradually (example: 1 percent of GDP projected deficit at t+12 implies one-twelfth adjustment each successive quarter).
  - Non-negativity constraint: if t+12 expected fiscal balance is in surplus, discretionary policy lets only automatic stabilizers operate for at least four quarters.

### III. Key simulation findings and quantitative outcomes
- Difficulty meeting “over the cycle” targets:
  - Even with ex-post perfect cycle observability, stochastic shocks and uncertainty about cycle end make meeting “over the cycle” targets difficult.
  - Under the “over the cycle” golden rule, the 95 percent confidence interval for the cumulative balance at the end of the cycle is ±1 percent of GDP.
  - There is a 45 percent probability that the cumulative balance falls into deficit at the end of the cycle; probability of missing by more than ½ percentage point of GDP is about 15 percent.
- Procyclicality risk:
  - “Over the cycle” formulation generates substantial risk of procyclical fiscal tightening near cycle ends:
    - Fiscal policy becomes procyclical by more than ½ percentage points of GDP in over half of the 2,767 simulated cycles during the final stage.
    - About 12 percent of simulated cycles experienced fiscal policy that was procyclical during the entire last fourth of each cycle.
  - Under the forward-looking three-year rule, the risk of such procyclicality is virtually zero.
- Macroeconomic volatility (standard deviations, average over the cycle):
  - Output standard deviation:
    - No Fiscal policy 1.08
    - Over the cycle 0.86
    - Forward-looking 0.80
    - Automatic Stabilizer 0.81
  - Prices standard deviation:
    - No Fiscal policy 0.80
    - Over the cycle 0.74
    - Forward-looking 0.71
    - Automatic Stabilizer 0.72
  - Forward-looking three-year rule enhances economic stability by about 5 to 7 percent relative to the “over the cycle” formulation.
- Debt volatility and average debt path:
  - Starting public sector net debt assumed at 40 percent of GDP.
  - Under “over the cycle” rule: 95 percent of the time net debt remains within ±5 percentage points of the debt ceiling (between 35 and 45 percent of GDP).
  - Under forward-looking three-year rule: net debt would be within ±7 percentage points of GDP from the debt ceiling 95 percent of the time.
  - Despite larger fluctuation, on average over the cycle net debt declines by 0.3 percentage points of GDP under the forward-looking three-year alternative (asymmetric implementation: surpluses are not spent when economy projected above trend).
- Mitigation with a debt band:
  - Introducing a debt band width D and forcing a fiscal adjustment (D2 = 1, i.e., 1 percent of GDP) when cumulative balance goes outside the band reduces debt volatility.
  - Example: debt band width ±3 percentage points of GDP reduces forward-looking formulation volatility to ±5 percentage points, but reintroduces a small risk of procyclicality (still considerably smaller than under the “over the cycle” rule).
- Robustness across horizons (Tables summary, average over the cycle):
  - Output standard deviation: 2-year 0.82; 3-year 0.80; 5-year 0.81.
  - Prices standard deviation: 2-year 0.71; 3-year 0.71; 5-year 0.71.
  - Fiscal balance standard deviation: 2-year 0.55; 3-year 0.57; 5-year 0.56.
  - Cumulative balance standard deviation: 2-year 3.15; 3-year 3.31; 5-year 3.36.
  - Change in cumulative balance over the cycle (Average): 2-year 0.43; 3-year 0.26; 5-year 0.11.

### IV. Policy insights and recommendations
- Operational recommendation:
  - Shift golden rule implementation from current balance “over the cycle” to current balance “at a two–five-year ahead rolling horizon” to:
    - Reduce risk of procyclical fiscal adjustments near cycle ends.
    - Lower volatility of output and prices (forward-looking three-year rule reduces volatility relative to over-the-cycle by about 5–7 percent).
- Trade-offs and containment:
  - Forward-looking implementation slightly increases debt-to-GDP variability (from ±5 to ±7 percentage points of GDP at the 95 percent interval under a three-year rule), but average net debt can decline (example: -0.3 percentage points of GDP under three-year rule).
  - Debt volatility under forward-looking rule can be contained by adding a debt band (for example, ±3 percentage points) that triggers an automatic fiscal measure (1 percent of GDP) when breached; this containment reintroduces a small procyclicality risk but substantially reduces debt volatility.
- Implementation caveats:
  - Model limitations: simple specification; abstracts from asset price movements and other supply-side features; equilibrium real exchange rate and potential output growth not solved endogenously.
  - Success of forward-looking rule depends on unbiased macroeconomic projections; absent reliable ex-ante forecasts, fiscal slippages could accumulate rapidly.
  - Without an ex-post test, credibility of ex-ante forecasts is especially important.

*Source: _wp07199 - 1. Golden Rule: Procyclical Fiscal Policy During the Cycle (PDF chapter/section).*

### 1. Golden Rule: Procyclical Fiscal Policy During the Cycle...................................................13

### 1. Golden Rule: Procyclical Fiscal Policy During the Cycle

### Main sections
- 1. Golden Rule: Procyclical Fiscal Policy During the Cycle...................................................13
- 2. Comparison of Economic Performance Over the Cycle......................................................14
- 3. Forward-Looking Rule with a Debt Band: Procyclical Fiscal Policy During the Cycle .....15
- 4. Alternative Rule: Different Horizons...................................................................................16

### Figures
- 1. Fiscal Developments ..............................................................................................................4
- 2. The Golden Rule and the Economic Cycle............................................................................6
- 3. Sequence of Decisions in Each Period...................................................................................8
- 4. Economic Cycles ................................................................................................................10
- 5. Distribution of Cumulative Balance at the End of the Cycle...............................................12

### Appendixes
- I. Data Description and Sources...............................................................................................18
- II. The Model ...........................................................................................................................19

### Appendix Tables
- 1. Model Parameter Estimation Results...................................................................................22
- 2. Estimation Results for the Error Processes and Measurements Errors................................23

*Source: _wp07199 - 1. Golden Rule: Procyclical Fiscal Policy During the Cycle (PDF chapter/section).*

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

### _wp07199 - References..............................................................................................................

### I. INTRODUCTION — Purpose and approach
- The golden rule (introduced 1998): over the economic cycle, the government should borrow only to invest and not to fund current spending; requires the current budget to be in balance or surplus on average over the economic cycle.
- Rationale: protect investment spending given significant infrastructure needs; allow automatic stabilizers to operate without jeopardizing long-term fiscal sustainability.
- Drawbacks of the “over the cycle” formulation:
  - Automatic stabilizers may be constrained if procyclical tightening is required toward the end of a cycle to meet the rule.
  - Ex-post testing requires precise dating of the economic cycle, which is controversial because alternative methodologies yield different cycle dating.
- Objective of the paper: examine an alternative, forward-looking implementation of the golden rule that is independent of cycle dating by aiming for current balance or better over a rolling time horizon (two- to five-year horizons examined, focus on the three-year horizon).
- Method: compare fiscal performance under alternative implementations using a simple open economy stochastic model calibrated to U.K. data.

### II. BACKGROUND — Motivation and practical issues
- Historical context:
  - Large fiscal slippage FY1989/90–FY1993/94 contributed to the current fiscal framework.
  - New framework pillars: (i) fiscal rules including the golden rule and a “sustainable investment rule” (public sector net debt over the economic cycle targeted below 40 percent of GDP), and (ii) a multiyear spending framework.
- Cycle identification problems:
  - Different definitions and estimation approaches (HP filter, production function, turning point/NBER approaches, U.K. Treasury on-trend-point approach) can yield significantly different cycle datings.
  - Judgment plays an important role; perceived conflict of interest because the Treasury defines cycles and conducts self-assessment of rule observance.
  - Example: Treasury redefined cycle start from 1999 to 1997 after national accounts revisions in 2005; NAO audit found the decision reasonable but redating shifted a sizable current surplus into the present cycle.

### III. THE MODEL AND THE SIMULATION — Structure and implementation
- Model type: small open-economy “New Keynesian” model with rational expectations, specified in gap and rate-of-change terms; estimated by Bayesian technique for 1992:Q4–2006:Q4.
- U.K. model core behavioral equations (gap terms):
  - (1) Aggregate demand (IS function)
  - (2) Inflation equation
  - (3) Real exchange rate equation
  - (4) Monetary policy reaction function
  - (5) Fiscal policy reaction function: FBgap determined as function of output gap and discretionary term aiming at policy objective; debt defined as cumulative fiscal balance (CumFB).
- Shocks and simulation design:
  - Seven stochastic shocks each period (output, prices, interest rates in U.K. and ROW, and real exchange rate); drawn from normal distributions with mean 0 and the model-estimated standard deviations.
  - Simulation: 1,000 draws lasting 100 quarters each (100 quarters = 25 years); starting from steady state.
  - Three-step period simulation sequence: (Step 1) update forecasts and monetary policy, (Step 2) identify economic cycles (assumed ex-post perfectly observable in illustrative simulations), (Step 3) adjust current and future fiscal balance path to meet the tested fiscal rule; discretionary fiscal policy updated only once a year in the simulations.
- Cycle definition used:
  - Crossing points where output gap = 0 and require four consecutive quarters of same sign after a sign change to define crossing points (implies shortest cycle = eight quarters = two years).
  - Historical U.K. cycles identified: 1978:Q1–1987:Q2, 1987:Q3–1997:Q3, 1997:Q4–2003:Q2.
  - Model-generated cycle length (average): Mean 27.1 quarters; Median 25.0 quarters.
- Fiscal balance decomposition:
  - Automatic stabilizers Ast computed using Treasury 2003 elasticities (output gap contemporaneous elasticity 0.5, lagged 0.2; total elasticity 0.7).
  - Equilibrium overall fiscal balance calibrated at -2 percent of GDP (zero current balance + 2 percent of GDP net investment) consistent with stabilizing public sector net debt at 40 percent of GDP.
  - Golden rule (“over the cycle”) discretionary policy specification (equation (6)) adjusts gradually over remaining cycle to meet current balance objective and maintain non-negativity of cumulative balance.
- Forward-looking alternative (“forward-looking rule”):
  - Operational target: aim for current balance or better at a forward-looking rolling horizon; examined horizons between two- and five-year ahead, with focus on three-year ahead.
  - Three-year forward-looking specification (equation (7)): aims to achieve fiscal balance three-year ahead (t+12 FBgap) and eliminate projected debt gap over next four years (t+28 E(CumFBΔ)); adjustments introduced gradually (example: 1 percent of GDP projected deficit at t+12 implies one-twelfth adjustment each successive quarter).
  - Non-negativity constraint: if t+12 expected fiscal balance is in surplus, discretionary policy lets only automatic stabilizers operate for at least four quarters.

### IV. SIMULATION RESULTS — Key findings and quantitative outcomes
- Uncertainty and cycle-dating:
  - Even when cycles are perfectly observable ex-post, uncertainty about cycle end and stochastic shocks produce significant difficulty in meeting “over the cycle” targets.
  - Under the “over the cycle” golden rule, the 95 percent confidence interval for the cumulative balance at the end of the cycle is ±1 percent of GDP, implying the golden rule would be breached nearly half of the time.
  - Specifically, there is a 45 percent probability that the cumulative balance falls into deficit at the end of the cycle; probability of missing by more than ½ percentage point of GDP is about 15 percent.
- Procyclicality risk:
  - “Over the cycle” formulation generates substantial risk of procyclical fiscal tightening near cycle ends: in simulated cycles, fiscal policy becomes procyclical by more than ½ percentage points of GDP in over half of the 2,767 simulated cycles during the final stage.
  - About 12 percent of simulated cycles experienced fiscal policy that was procyclical during the entire last fourth of each cycle.
  - Under the forward-looking three-year rule, the risk of such procyclicality is virtually zero.
- Volatility of output and prices:
  - Automatic stabilizers significantly reduce volatility compared with no fiscal policy.
  - Comparison (standard deviation, average over the cycle; in percent where specified):
    - Output standard deviation: No Fiscal policy 1.08; Over the cycle 0.86; Forward-looking 0.80; Automatic Stabilizer 0.81.
    - Prices standard deviation: No Fiscal policy 0.80; Over the cycle 0.74; Forward-looking 0.71; Automatic Stabilizer 0.72.
  - Forward-looking three-year rule enhances economic stability by about 5 to 7 percent relative to the “over the cycle” formulation.
- Debt volatility (cumulative balance / debt-to-GDP ratio):
  - Starting public sector net debt assumed at 40 percent of GDP.
  - Under “over the cycle” rule: 95 percent of the time net debt remains within ±5 percentage points of the debt ceiling (i.e., between 35 and 45 percent of GDP).
  - Under forward-looking three-year rule: net debt would be within ±7 percentage points of GDP from the debt ceiling 95 percent of the time.
  - Despite larger fluctuation, on average over the cycle net debt declines by 0.3 percentage points of GDP under the forward-looking three-year alternative (asymmetric implementation: surpluses are not spent when economy projected above trend).
- Mitigation via debt band:
  - Introducing a debt band (width D) around equilibrium debt-to-GDP ratio and forcing a fiscal adjustment (D2 = 1, i.e., 1 percent of GDP) when cumulative balance goes outside the band reduces debt volatility.
  - Simulation: debt band width ±3 percentage points of GDP reduces forward-looking formulation volatility to ±5 percentage points, but reintroduces a small risk of procyclicality (still considerably smaller than under the “over the cycle” rule).
- Robustness across horizons:
  - Results robust for two- to five-year rolling horizons: standard deviations of output and prices remain broadly unchanged between 2-year, 3-year, and 5-year rules.
  - Table 4 (average over the cycle):
    - Output standard deviation: 2-year 0.82; 3-year 0.80; 5-year 0.81.
    - Prices standard deviation: 2-year 0.71; 3-year 0.71; 5-year 0.71.
    - Fiscal balance standard deviation: 2-year 0.55; 3-year 0.57; 5-year 0.56.
    - Cumulative balance standard deviation: 2-year 3.15; 3-year 3.31; 5-year 3.36.
    - Change in cumulative balance over the cycle (Average): 2-year 0.43; 3-year 0.26; 5-year 0.11.

### V. POLICY INSIGHTS AND RECOMMENDATIONS
- Operational change recommended: shift golden rule implementation from current balance “over the cycle” to current balance “at a two–five-year ahead rolling horizon” to:
  - Reduce risk of procyclical fiscal adjustments near cycle ends.
  - Lower volatility of output and prices (forward-looking three-year rule reduces volatility relative to over-the-cycle by about 5–7 percent).
- Trade-offs:
  - Forward-looking implementation slightly increases debt-to-GDP variability (from ±5 to ±7 percentage points of GDP at the 95 percent interval under a three-year rule), but average net debt can decline (example: -0.3 percentage points of GDP under three-year rule).
  - Debt volatility under forward-looking rule can be contained by adding a debt band (e.g., ±3 percentage points) that triggers an automatic fiscal measure (1 percent of GDP) when breached, at modest cost of reintroducing a small procyclicality risk.
- Important implementation caveats:
  - Model limitations: simple specification; abstracts from asset price movements and other supply-side features; equilibrium real exchange rate and potential output growth not solved endogenously.
  - Success of forward-looking rule depends on unbiased macroeconomic projections; absent reliable ex-ante forecasts, fiscal slippages could accumulate rapidly.
  - Without an ex-post test, credibility of ex-ante forecasts is especially important.

*Source: IMF working paper content (as provided).*

### References

### _wp07199 - References

### Key cited works
- Artis, M., 2002, Dating the Business Cycle in Britain, CGBCR Discussion Paper Series, available at http://www.ses.man.ac.U.K./cgbcr/discussi.htm.
- Berg, A., P. Karam, and D. Laxton, 2006, Practical Model-Based Monetary Policy Analysis—How-To Guide, IMF Working Paper No. 06/81 (Washington, D.C.: International Monetary Fund).
- Emmerson, C., C. Frayne, and S. Love, 2004, Updating the U.K.’s Code for Fiscal Stability, IFS Working Paper No. 04/29.
- U.K. Treasury, 1997, Fiscal Policy: Lessons from the Last Economic Cycle, available at http://www.hm-treasury.gov.U.K./mediastore/otherfiles/lessons.pdf.
- U.K. Treasury, 1999, Fiscal Policy: Public Finances and the Cycle, available at http://www.hm-treasury.gov.uk/media/415/CB/cycles.pdf.
- U.K. Treasury, 2003a, End of Year Fiscal Report, available at http://www.hm-treasury.gov.uk/media/324/70/end_of_year_352.pdf.
- U.K. Treasury, 2003b, Evidence on the UK Economic Cycle, available at http://www.hm-treasury.gov.uk/media/2E6/A5/economic_cycles190705.pdf.
- Honjo, K., and B. Hunt, 2006, Stabilizing Inflation in Iceland, IMF Working Paper 06/262 (Washington, D.C.: International Monetary Fund).
- Kell, M., 2001, Assessment of Fiscal Rules in the United Kingdom, IMF Working Paper No. 01/91 (Washington, D.C.: International Monetary Fund).
- Koeva, P., 2003, The Implementation of the Golden Rule, (Washington, D.C.: International Monetary Fund).
- Van den Nord, P., 2000, The Size and Role of Automatic Fiscal Stabilizers in the 1990s and Beyond, OECD Working Paper No. 230.

### Appendix I — Data description and sources
- Output gap (percentage points): Staff estimates using quarterly real GDP series from 1992:Q4 to 2006:Q2 based on a production function approach. Source: GDP (ONS).
- Consumer prices: Synthetic combination between RPIX and CPI using quarterly RPIX index from 1992:Q1 to 2003:Q4 and the rate of change from CPI index from 2004:Q1 onwards. Source: ONS.
- Exchange rate: Quarterly effective exchange rate index 1992:Q4 to 2006:Q2. Source: Bank of England (BoE).
- Real exchange rate: Calculated using domestic and foreign inflation and nominal effective exchange rate from the BoE. Equilibrium values are exogenous and are derived using a variant of the Hodrick Prescott (1997) filter that allows for additional constraints to be added to the minimization problem to prevent the resulting equilibrium value from converging to the actual observed data at the end of the sample period.
- Nominal short-term interest rates: 90-day treasury bills or equivalent in percentage points.
- Fiscal balance: Annualized quarterly public sector overall fiscal balance in percent of GDP. Source: ONS.

### The model — structure and behavioral equations
- Model specification:
  - Open-economy model for the U.K. and the rest of the world, specified in gap and rate-of-change terms; under inflation targeting all variables are stationary.
  - Equilibrium values for the real interest rate, the real exchange rate, and potential growth rate are assumed time invariant for simulation.
  - Model equations (U.K. economy):
    - (1) Aggregate Demand (IS function): ygap_t = β1⋅ygap_{t-1} + β2⋅ygap_{t+1} + β3⋅rrgap_{t-1} + β4⋅zgap_{t-1} + β5⋅ygap*_t + β6⋅FBgap_{t-1} + β14⋅? + ε_t  [as presented in source: (1) ygapt t t t t t t t FBgapygapzgaprrgapygapygapygapεββββββ+⋅−⋅+⋅+⋅+⋅+⋅= −−−+−16 514131211]
    - (2) Inflation: π_t = δ1⋅π_{t+1} + δ2⋅ygap_{t-?} + δ3⋅Δz_t + δ4⋅π_{4,t} + ε_t  [as presented: (2) ,)1( 312 4 11 4 41 π εδδπδπδπ ttttt zygap+Δ⋅+⋅+⋅−+⋅= −−+]
    - (3) Real exchange rate: z_t determined by φ and leads/lags and ε_t [as in source: (3) Real exchange rate * 11 (1)() / 4/ 4, z ttttt t zzzrrrrφφε +− =⋅ +−⋅ + − +]
    - (4) Monetary Policy Reaction Function: rs_t related to lagged rs, inflation gap, ygap, real exchange rate gap and equilibrium rates with shocks ε_t [as in source: (4) Monetary Policy Reaction Function 44 1112  43 (1)  (_()), Trs tttt  ttt rsrsrr eqygapααπαππαε −+ =⋅ +− ⋅++⋅  − +⋅   +]
    - (5) Fiscal Policy Reaction Function: FBgap_t = θ1⋅ygap_{t-1} + θ2⋅Dgap_{t+1} + ε_t [as in source: (5) Fiscal Policy Reaction Function 112  1 . FBgap tttt FBgapygapDgapθθ ε −+ =⋅    −⋅+]
  - Variables and definitions:
    - ygap = output gap
    - rr = domestic real interest rate
    - rr_eq = equilibrium real interest rate
    - rs = annualized short-term policy rate
    - z = log of the real exchange rate index (an increase implies a depreciation)
    - π = quarterly annualized rate of CPI inflation
    - π4 = four-quarter moving average of quarterly annualized CPI inflation
    - πT = target rate of inflation
    - FB = fiscal balance
    - D = government debt
    - Dgap = deviation from the government’s debt target D*
- Fiscal rule design:
  - Allows for automatic stabilizers while targeting stable public debt.
  - Fiscal policy responds to period t-1’s output gap by reducing (adding) demand stimulus when there is positive (negative) output gap.
  - Forward-looking component aims at achieving government’s public debt target.
  - For this exercise, debt is defined as the cumulative fiscal balance and the debt target is set equal to zero (implying equilibrium fiscal balance is zero and no debt accumulation over time).

### The rest of the world — model equations
- Equations for foreign sector (as presented):
  - (6) ygap*_t = β*1⋅ygap*_{t-1} + β*2⋅ygap*_{t+1} + β*3⋅rrgap*_{t-1} + ε*_t  [source: (6) ** 1 * 3 * 1 * 2 * 1 * 1 *ygap ttttt rrgapygapygapygapεβββ+⋅+⋅+⋅= −+−]
  - (7) Inflation equation for foreign sector analogous to (2). [source: (7) ,)1( ** 1 * 2 *4 1 * 1 *4 4 * 1 *π εδπδπδπ ttttt ygap+⋅+⋅−+⋅= −−+]
  - (8) Monetary policy reaction function for the rest of the world analogous to (4). [source: (8) * ****4**4*    *  ** 1112 43 (1)  (_()), Trs tttt   ttt rsrsrr eqygapααπαππαε −+ =⋅ +− ⋅+ +⋅ − +⋅   +]

### Stochastic processes and identities
- Stochastic processes (as presented):
  - (9) ygap_{t+1} = ρ_ygap⋅ygap_t + ξ_ygap,t+1 ε
  - (10) ygap*_{t+1} = ρ_ygap*⋅ygap*_t + ξ_ygap*,t+1 ε
  - (11) π_{t+1} = ρ_π⋅π_t + ξ_π,t+1 ε
  - (12) π*_{t+1} = ρ_π*⋅π*_t + ξ_π*,t+1 ε
  - (13) rs_{t+1} = ρ_rs⋅rs_t + ξ_rs,t+1 ε
  - (14) rs*_{t+1} = ρ_rs*⋅rs*_t + ξ_rs*,t+1 ε
- Identities (as presented):
  - (15) π_t = (π_{t-1} + π_{t-2} + π_{t-3} + π_{t-4})/4
  - (16) π*_t = (π*_{t-1} + π*_{t-2} + π*_{t-3} + π*_{t-4})/4
  - (17) rs_{t+1} − π_{t+1} = rr_t
  - (18) rs*_{t+1} − π*_{t+1} = rr*_t
  - (19) rrgap_t = rr_t − rr_eq_t
  - (20) rrgap*_t = rr*_t − rr_eq*_t
  - (21) zgap_t = z_t − z_eq_t

### Estimation and simulation approach
- Parameters estimated using a Bayesian technique allowing for measurement error in observable variables.
- Priors and posterior estimates are presented for the United Kingdom and the rest of the world (Tables 1 and 2).
- All simulations conducted using custom-built TROLL programs.

### Table 1 — United Kingdom and Rest of the World: Model parameter posterior means (sample period 1992Q1 to 2006Q4)
- United Kingdom posterior means (selected):
  - β1: 0.743
  - β2: 0.093
  - β3: 0.104
  - β4: 0.037
  - β5: 0.200
  - β6: 0.165
  - δ1: 0.192
  - δ2: 0.272
  - δ3: 0.087
  - θ1: 0.182
  - θ2: 0.104
  - φ: 0.335
  - α1: 0.537
  - α2: 1.849
  - α3: 0.520
- Rest of the World posterior means (selected):
  - β*1: 0.762
  - β*2: 0.102
  - β*3: 0.111
  - δ*1: 0.185
  - δ*2: 0.230
  - α*1: 0.558
  - α*2: 1.980
  - α*3: 0.230

### Table 2 — Estimation results for error processes and measurement errors (sample period 1992Q1 to 2006Q4)
- United Kingdom posterior means (selected):
  - ρ_ygap: 0.810
  - std. dev. ξ_ygap: 0.175
  - std. dev. mes. er. ygap: 0.121
  - ρ_π: 0.491
  - std. dev. ξ_π: 0.304
  - std. dev. mes. er. π: 2.137
  - ρ_rs: 0.765
  - std. dev. ξ_rs: 0.157
  - std. dev. mes. er. rs: 0.117
  - std. dev. ε_z: 5.608
  - std. dev. ε_FBgap: 1.248
  - std. dev. ε_D: 0.051
- Rest of the World posterior means (selected):
  - ρ_ygap*: 0.751
  - std. dev. ξ_ygap*: 0.178
  - std. dev. mes. er. ygap*: 0.149
  - ρ_π*: 0.530
  - std. dev. ξ_π*: 0.150
  - std. dev. mes. er. π*: 1.046
  - ρ_rs*: 0.823
  - std. dev. ξ_rs*: 0.154
  - std. dev. mes. er. rs*: 0.152

*Source: _wp07199 - References (IMF staff estimates and model documentation).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp07199.pdf_