## Measuring Output Gap: Is It Worth Your Time? — WP/20/24 (Sections I–IV)

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### Abstract and key result
- Paper: WP/20/24 by Jiaqian Chen and Lucyna Górnicka (February 2020).
- Main finding:
  - A structural VAR with an appropriate identification strategy provides improved estimates of the output gap with better real time properties and lower sensitivity to temporary shocks than usual filtering techniques; it also produces smaller out-of-sample forecast errors for inflation.
- Caution:
  - Results suggest caution in basing policy decisions solely on output gap estimates.

### Why the output gap matters
- Definition and uses:
  - Output gap = deviation of actual output from “potential” output; potential output is latent and may be defined as a trend/smooth component, sustainable aggregate supply consistent with stable inflation, or full utilization of factors of production.
  - Policy relevance:
    - Fiscal stance assessment via structural budget balance adjusted for business cycle.
    - Modern monetary policy relies on the Phillips Curve linking output gap, inflation, and inflation expectations.
- Core problem:
  - Potential output is unobserved; estimates are required and subject to methodological choices and errors.

### Approaches compared
- Structural identification (SVAR) approach:
  - If only supply shocks affect potential output, identify and aggregate such shocks (e.g., Blanchard-Quah style).
  - Challenges: identification difficult, especially for small open economies where global factors matter.
- Filtering (a-theoretical) approaches:
  - Fit GDP to a slow-moving trend; often augmented with inflation and unemployment.
  - Risks:
    - Filters do not distinguish supply vs demand shocks.
    - End-of-sample problem: filters assume average deviation of actual from potential is zero over the sample, causing revisions when new datapoints arrive and excluding protracted recessions.
    - Empirical evidence documents real-time errors and large biases.
- Methods compared in paper:
  - HP filter (λ=1,600 for quarterly data in baseline).
  - Multivariate Kalman (MVK).
  - MVK with financial variables (MVKfin / MVF).
  - Two-variable Blanchard-Quah (BQ) SVAR.
  - New small open-economy SVAR with short-, long-run-, and sign restrictions (building on Forbes et al. (2018)).

### Small open-economy SVAR: purpose, variables, identification
- Purpose:
  - Allow both domestic and global shocks to have permanent effects on output; distinguish domestic vs global demand and supply shocks.
- Variables included (six):
  - U.K. real GDP growth
  - U.K. CPI inflation at constant tax
  - U.K. shadow interest rate
  - Changes in the Sterling exchange rate index
  - U.K. import price inflation
  - Changes in foreign export prices
- Identification highlights (following Forbes et al. (2018)):
  - Six structural shocks identified via zero short-run and long-run restrictions and sign restrictions.
  - Only domestic supply shocks and persistent global shocks are assumed to affect the level of output in the long run; persistent global shocks capture foreign shocks with lasting effects on U.K. output and demand shocks with permanent impact.
  - Domestic developments assumed not to affect world export prices on impact or in the long run.
  - Sign restrictions include:
    - Domestic supply shock: negative correlation between GDP and CPI in the first 2 periods.
    - Domestic demand shock: positive on GDP, CPI, and exchange rate (positive demand → appreciation).
    - Monetary policy shock: lower interest rate associated with rise in GDP and CPI, and depreciation.
    - Exogenous exchange rate appreciation implies fall in CPI.
- Estimation:
  - Bayesian methods with Minnesota-style priors; Gibbs sampling used, saving final 1000 repetitions.
- Sample and lags:
  - SVAR estimated over 1982: Q1 to 2019: Q1.
  - Two lags of each variable; robustness checks with 4, 6 or 8 lags give similar estimated output gap.

### Identification restrictions (summary of short-run and long-run restrictions)
- Long-run: world (ex-UK) prices respond (+) to persistent and transitory global shocks; zero responses for domestic shocks.
- Short-run (impact and one following quarter):
  - UK GDP growth: UK supply (+), UK demand (+), UK monetary policy (-), Exogenous exchange rate (0), Persistent global (0), Transitory global (0).
  - UK CPI: UK supply (-), UK demand (+), UK monetary policy (-), Exogenous exchange rate (-), Persistent global (0), Transitory global (0).
  - UK interest rate: UK supply (+), UK demand (+), UK monetary policy (-), Exogenous exchange rate (0), Persistent global (0), Transitory global (0).
  - UK nominal ERI: UK supply (+), UK demand (+), UK monetary policy (+), Exogenous exchange rate (0), Persistent global (0), Transitory global (0).
  - UK import prices: Persistent global (+), Transitory global (+); zero restrictions on domestic shocks.
  - World (ex-UK) prices: 0 responses to domestic shocks; + to persistent and transitory global shocks.
- Notation: '+' ('-') indicates impulse response restricted to be positive (negative) in the quarter the shock hits and the following quarter; '0' denotes response restricted to be zero (on impact or in long run).

### Impulse response properties and consistency checks
- Calibration checks:
  - Loosening monetary policy by 100 basis points causes output to fall by about 0.5 percent (consistent with Burgess et al. (2013)).
  - Sterling appreciation of 1 percent causes import prices to fall by 0.5 percent (in line with Forbes et al. (2018)).
- Shock interpretations:
  - Positive domestic supply shock → permanent output increase and price level fall.
  - Positive domestic demand shock → only temporary improvement in output.
  - Persistent and transitory global shocks’ domestic responses align broadly with supply/demand interpretations.
- Overall: small open-economy SVAR yields impulse responses broadly consistent with economic intuition and literature benchmarks.

### Historical decomposition and output gap estimates (open-economy SVAR)
- Historical drivers:
  - Domestic and global supply shocks key drivers of sharp decline in growth during the GFC.
  - Series of negative domestic supply shocks after the Brexit referendum.
- Output gap defined as accumulation of past domestic supply and persistent global shocks.
- Key timing and levels (as reported):
  - Two periods of considerable overheating: late 1980s and prior to the GFC.
  - 2010: Q2–2012: Q4:
    - Open economy SVAR: output gap became positive in 2010: Q2, peaked in 2011: Q2, turned negative in 2013: Q2.
    - MVK: also suggests positive in 2010: Q2 (though smaller).
  - 2014: Q3–2016: Q3:
    - Open economy SVAR indicates a negative output gap between 2014: Q3–2016: Q3 while other methods suggest closed or positive gaps.
  - After the Brexit referendum (2016: Q3):
    - Other methods: output gap turned negative.
    - Open economy SVAR: points to a moderately positive output gap until only very recently.

### Data, real-time estimation samples, and mechanics
- MVKfin sample: 1991: Q1–2019: Q1 (shorter due to credit series availability).
- Small open economy SVAR sample: 1982: Q1–2019: Q1.
- Real-time data:
  - Bank of England’s “GDP Real Time Database” (monthly vintages from January 1990 until August 2016 as of writing).
  - Real-time estimation procedure:
    - For each quarter between 2005: Q1 and 2016: Q2, run regressions using data from 1982: Q1 up until that quarter, replacing GDP (and import price deflator where relevant) with the real-time vintage available in the middle month of that quarter.
    - First iteration based on 1982: Q1 to 2005: Q1.
    - HP filter: real GDP series “extended” by 8 quarters using the last available year-on-year growth rate to mitigate end-of-sample problem.

### Real-time performance findings
- Filtering methods:
  - Fail to signal, in real time, an overheating of the U.K. economy prior to the GFC.
  - Exhibit large ex-post revisions driven by the end-of-sample problem.
- BQ SVAR:
  - Signals slightly positive output gap starting in 2006: Q1.
  - Real-time estimates show a positive and quite large output gap between 2014 and 2016, inconsistent with a small pre-GFC positive gap.
- Small open-economy SVAR:
  - Real-time output gap estimates follow full-sample estimates more closely than the other methods.
  - Signals opening of a positive output gap already in 2004: Q1.
  - Real-time output gap reaches 1 percent around 2008: Q1, consistent with full-sample estimate.
- Additional regressions:
  - Ex-post revisions to filters are due to end-of-sample problem rather than data revisions before and during the GFC.

### Responses-to-shocks test: data, methodology, and shocks used
- Aim:
  - Check how potential output estimates respond to different types of shocks; test whether potential growth estimates are procyclical.
- External shock series:
  - Global permanent shocks:
    - U.S. TFP news shocks (Beaudry and Portier (2006), updated by Valerie Ramey).
  - Global temporary shocks:
    - U.S. monetary policy shocks identified using high frequency surprises (Gertler and Karadi (2015)).
    - U.S. fiscal shocks: U.S. military spending news shocks (Ramey (2016)).
  - Domestic shocks:
    - Domestic fiscal shocks following Blanchard and Perotti (2002).
    - U.K. monetary policy shocks: VAR with GDP growth, unemployment, inflation and interest rate (four lags) and Cholesky decomposition.
    - Productivity shocks: residuals from regression of output per worker on its lags.
- Regression specification for each shock s:
  - ∆ȳ̅_t = α_s + ∑_{k=0}^K φ^k_s ε_{t−k}^s + ζ_t^s
  - K = 6 lags used; one shock at a time; Newey-West standard errors.

### Impulse response findings (IRFs) to global and U.S. shocks
- Method:
  - IRFs of potential output constructed by summing coefficients up to horizons; significance shown with 66 percent confidence bands.
- U.S. productivity shock:
  - All five models yield a positive and statistically significant response of potential output to a positive global productivity shock.
- U.S. fiscal shock (transitory):
  - Two SVAR models: potential output responds in line with intuition.
  - MVK and MVKfin: statistically significant increase in potential growth after a U.S. fiscal shock.
  - HP filter: response of potential output is negative (marginally statistically significant) after 6 quarters.
  - Small open-economy SVAR: potential output initially increases but significance ceases after 3 quarters.
- Comparative assessment across six shocks (US prod., US fiscal, US monetary, UK prod., UK fiscal, UK monetary):
  - Qualitative coding summary:
    - Dark green: IRFs match intuition short and medium term.
    - Light green: match only in medium term.
    - Dark red / bright red: counterintuitive or borderline.
  - Findings:
    - SVAR-based potential output estimates tend to perform relatively well.
    - BQ SVAR: marginally significant to U.S. fiscal shock and insignificant to U.K. productivity shock over medium run.
    - Filter-based estimates show more counter-intuitive results (e.g., HP no significant response to U.K. productivity shock; MVKfin responds positively to U.K. monetary policy shock).
    - Small open-economy SVAR has the most intuitive impulse responses among models considered.

### Phillips curve / inflation forecasting tests
- Phillips curve specification (Blanchard (2016) framework):
  - π_t = θ ŷ_t + λ π_t^e + (1−λ) π_{t−1}^* + μ π_{m,t} + ε_t
  - Expectations: π_t^e = α + β π_{t−1}^* + η_t
  - Definitions:
    - π_t: headline consumer price inflation (quarterly inflation, annualized).
    - ŷ_t: output gap.
    - π_t^e: long-term inflation expectations.
    - π_{t−1}^*: average of last four quarterly inflation rates.
    - π_{m,t}: import price inflation relative to headline inflation.
- Tests:
  1. In-sample estimation for 1993: Q1–2016: Q2 using five real-time output gap measures (U.K. adopted inflation targeting in October 1992).
  2. Out-of-sample one-quarter-ahead iterative forecasting for each quarter 2006: Q1–2016: Q1 estimating Phillips curves using data from 1982: Q1 up to each quarter.
- In-sample regression highlights (1993: Q1–2016: Q2):
  - Lags of π^e:
    - pi e (-1): 0.37***, 0.36***, 0.34***, 0.34***, 0.20** (std. errors: (0.10), (0.10), (0.10), (0.09), (0.10))
    - pi e (-2): 0.16, 0.16, 0.15, 0.15, 0.08 (std. errors (0.10) each)
    - pi e (-3): 0.191*, 0.204**, 0.181*, 0.201**, 0.171* (std. errors (0.10) each)
    - pi e (-4): 0.104, 0.125, 0.0676, 0.122, 0.131 (std. errors ~ (0.09)–(0.10))
  - imported pi e / pi e: 0.0008***, 0.0008***, 0.0007**, 0.0008***, 0.0008*** (std. errors (0.0003))
  - Output gap coefficients (by method):
    - HP filter: 0.05* (std. error (0.03))
    - MVK: 0.10** (std. error (0.04))
    - MVKfin: 0.01 (std. error (0.03))
    - BQ SVAR: 0.07** (std. error (0.03))
    - Small open economy VAR: 0.40*** (std. error (0.09))
  - Observations: 105, 105, 97, 105, 105 (by column)
  - R-squared: 0.56, 0.57, 0.41, 0.57, 0.62
  - Significance: *** p<0.01, ** p<0.05, * p<0.1
- Out-of-sample forecasting performance (average squared errors, 2006: Q1–2016: Q2):
  - HP filter: 0.27
  - MVK: 0.26
  - MVKfin: 0.28
  - BQ SVAR: 0.24
  - Small open economy SVAR: 0.24
- Interpretation:
  - SVAR-based estimates produce the smallest forecast error, very close to MVK.
  - MVK embeds the Phillips curve in estimation, so low error expected by design.
  - Small open-economy SVAR appears to have the smallest errors in forecasting inflation, though differences may not be statistically significant.

### Robustness checks and sensitivity
- Variations tested:
  - Mitigate end-of-sample problem by extending sample by one year using Consensus forecasts for GDP and (for MVK/MVKfin) inflation.
  - MVK sensitivity to parameter priors (β), steady-state unemployment and potential growth, and detrended unemployment specifications.
  - BQ SVAR using alternative unemployment measures accounting for underemployment.
  - Small open-economy SVAR estimated without sign restrictions.
  - Alternative sample start: 1993: Q1.
- Findings:
  - Alternative specifications do not yield results considerably different from baseline.
  - BQ SVAR robust to unemployment measure; open-economy SVAR not much affected when imposing no sign restrictions.
  - All five methods give similar output gap estimates when starting sample in 1993: Q1.

### MVK and MVKfin model specifications and estimated parameters
- MVK state-space key equations (notations preserved):
  - ŷ_t = y_t − y̅_t  (C.1)
  - y̅_t = y̅_{t−1} + g_t + ε̅_t  (C.2)
  - g_t = θ g_ss + (1−θ) g_{t−1} + ε^g_t  (C.3)
  - ŷ_t = φ ŷ_{t−1} + ε^ŷ_t  (C.4)
  - π_t = λ π_{t+1} + (1−λ) π_{t−1} + β ŷ_t + ε^π_t  (C.5)
  - û_t = τ1 ŷ_t + τ2 û_{t−1} + ε^u_t  (C.6)
  - û_t = u_t − u̅_t  (C.7)
  - u̅_t = τ4(u̅_ss + (1−τ4) u̅_{t−1}) + g^u_t + ε̅^u_t  (C.8)
  - g^u_t = (1−τ3) g^u_{t−1} + ε^{g^u}_t  (C.9)
- Estimation:
  - Parameter values and variances are maximum likelihood estimates using Bayesian estimation.
  - Priors set at posteriors for the U.K. in Blagrave et al. (2015).
  - Full sample: 1982: Q1–2019: Q1.
- Priors and posteriors (preserved values):
  - λ: prior 0.25, posterior 0.23
  - β: prior 0.15, posterior 0.1
  - φ: prior 0.7, posterior 0.72
  - θ: prior 0.2, posterior 0.1
  - τ1: prior 0.4, posterior 0.33
  - τ2: prior 0.4, posterior 0.48
  - τ3: prior 0.1, posterior 0.09
  - τ4: prior 0.1, posterior 0.09
  - gss: 1.6
  - u̅ss: 4.5

- MVKfin / MVF specifics:
  - Decompose observed GDP: y_t = y̅_t + ŷ_t
  - ∆^2 y̅_t = ε̅_t
  - Variance ratio constraint λ ≡ Var(ŷ_t) / Var(ε̅_t) set at 1600 (trend captures frequencies above 8 years).
  - Observable covariates x_t included: real credit growth, real house price inflation, stock price inflation.
  - Estimation by maximum likelihood.
- Financial-variable parameter estimates (Estimate / Std. error / p-value):
  - Real HPI growth: 0.02 / 0.007 / 0.001
  - Real credit growth: 0.008 / 0.008 / 0.309
  - Stock price growth: 0.004 / 0.002 / 0.038

### Blanchard-Quah (BQ) SVAR specifics
- Specification:
  - Two-variable SVAR with quarterly real GDP growth and detrended unemployment rate over 1982: Q1–2019: Q1.
  - Lag length: 2 lags selected by AIC.
  - Identification: zero long-run impact of demand shocks on GDP growth per Blanchard and Quah (1989).
- Detrending rationale:
  - 5-quarter moving-average unemployment declined from over 8 percent in 1980s/early 1990s to around or somewhat below 5 percent before and after the GFC.
- Impulse responses and robustness:
  - Cumulative impulse responses reported to one standard deviation supply and demand shocks; 66 percent confidence bands presented.
  - Tests include one standard deviation positive U.S. monetary policy shock, U.K. productivity shock, U.K. fiscal shock, and U.K. monetary policy shock with 66 percent confidence bands.
- Key sample and lag facts:
  - Sample period for BQ SVAR: 1982: Q1–2019: Q1
  - Lag length in SVAR: 2 lags

### Main conclusions and policy-relevant implications
- Empirical findings:
  - Filtering-based models produce potential output estimates that respond to transitory shocks in a procyclical way (confirming Coibion et al. (2018)).
  - SVAR-based models that distinguish shock types yield potential output estimates with better impulse response properties.
  - The small open-economy SVAR performs best among considered models and has the best real-time properties.
- Policy guidance:
  - Output gap estimates are useful — answer to "Are output gap estimates worth economists’ time?" is yes — but:
    - Care is needed in identifying all relevant persistent shocks.
    - Model performance should be verified with established tests; good models should show minimal ex-post revisions to real-time estimates.
    - No perfect measure exists; policymakers should examine a range of output gap estimates and use broader indicators and judgment rather than rely on a single estimate.

*Source: WP/20/24 — Measuring Output Gap: Is It Worth Your Time? — Sections I–IV*

### Section 1

### Measuring Output Gap: Is It Worth Your Time? — Section I

### Abstract and Key Result
- Paper: WP/20/24 by Jiaqian Chen and Lucyna Górnicka (February 2020).
- Main finding: A structural VAR with an appropriate identification strategy provides improved estimates of the output gap with better real time properties and lower sensitivity to temporary shocks than usual filtering techniques; it also produces smaller out-of-sample forecast errors for inflation.
- Caution: Results suggest caution in basing policy decisions solely on output gap estimates.

### Importance of the Output Gap
- Definition: The output gap is a deviation of actual output from “potential” output; potential output is latent and can be defined in statistical or economic terms (trend/smooth component, sustainable aggregate supply consistent with stable inflation, or full utilization of factors of production).
- Policy relevance:
  - Fiscal policy stance assessment uses structural budget balance adjusted for the business cycle.
  - Modern monetary policy relies on the Phillips Curve linking the output gap, inflation, and inflation expectations.
- Problem: Potential output and thus the output gap are not directly observed; estimates are required.

### Approaches to Estimating Potential Output / Output Gap
- Structural identification approach:
  - Rationale: If only supply shocks affect potential output, identify and aggregate such shocks (e.g., SVARs in the spirit of Blanchard and Quah (1989) and Galí (1999)).
  - Challenge: Proper identification of supply shocks is difficult, especially for small open economies where global factors must be considered and the SVAR dimension and identification restrictions may need expansion.
- Filtering (a-theoretical) approaches:
  - General idea: Identify potential output by fitting GDP to a slow-moving trend; variables beyond GDP can be included when informative.
  - Common practice: Used by public institutions (Federal Reserve Bank, IMF, OECD) and often augmented with inflation and unemployment due to Phillips Curve and Okun’s Law links.
  - Risks:
    - Filtering methods do not distinguish supply vs demand shocks; both can move trend-based potential output estimates.
    - End-of-sample problem: Filters assume average deviation of actual from potential is zero over the sample, causing revisions when new datapoints arrive and excluding the possibility of protracted recessions.
    - Empirical evidence: Nelson and Nikolov (2003) and Office for Budget Responsibility (2019) document real-time errors and fiscal risks from output gap mismeasurement; Kangur et al. (2019) show IMF real-time estimates have large negative biases and are not useful to predict inflation.

### Contribution of the Paper
- Methods compared:
  - Two-variable Blanchard-Quah (BQ) SVAR.
  - Range of filtering techniques: HP filter, Multivariate Kalman (MVK), Multivariate Kalman with financial variables (MVKfin).
  - New proposal: A small open-economy SVAR with a mix of short-, long-term-, and sign restrictions (building on Forbes et al. (2018) to distinguish domestic and global demand and supply shocks).
- Intended advantages of the open-economy SVAR:
  - Distinguish domestic vs global demand and supply shocks.
  - Identify permanent shocks (long-term effects on output) as driving potential output, allowing permanent shocks to be domestic or global.
- Limitations acknowledged:
  - Blanchard (2018) point: Not all supply shocks necessarily have permanent effects on output.
  - Even expanded SVARs cannot represent an unlimited number of shocks or dynamics; model dimensionality remains limited.
- Policy guidance: Policymakers should examine a range of output gap estimates and apply judgment rather than rely on a single estimate.

### Technical Overview of Methods (as described)
- Hodrick-Prescott (HP) filter:
  - Objective: Fit a smooth trend τt to actual output yt by minimizing ∑(yt−τt)2 + λ∑[(τt+1−τt)−(τt−τt−1)]2.
  - Smoothing parameter: Hodrick and Prescott (1997) suggest λ=1,600 for quarterly data; Ravn and Uhlig (2002) recommend λ=6.25 for annual and λ=129,600 for monthly frequencies.
  - Production function approach: Decompose output into components (labor, capital, TFP), filter individually, and recombine.
  - Drawbacks: Filters do not distinguish shock types; HP filter has an I(2) component that can create spurious cycles.
- Multivariate Kalman (MVK) filter:
  - Augments GDP filtering with variables based on economic relationships (Phillips Curve, Okun’s law).
  - Typical specification (symbols preserved from source): ŷt = yt − y̅t; πt = λπt+1 + (1−λ)πt−1 + βŷt + επt; ût = τ1ût−1 + τ2ŷt + εut; ût = ut − u̅t.
- MVK with financial variables (MVKfin):
  - Borio et al. (2017) augment univariate Kalman filter with financial variables to capture financial imbalances:
    - ŷt = αŷt−1 + γ1rt + γ2Δcrt + γ3Δhpt, where rt is real interest rate, Δcrt is real credit growth, Δhpt is real house price growth.
- Blanchard-Quah SVAR (BQ SVAR):
  - Two-variable SVAR with GDP growth (Δyt) and unemployment (ut); structural shocks εts (supply) and εtd (demand) assumed uncorrelated, with zero long-term effect of demand shocks on output.
  - Potential output reconstructed as cumulative sum of supply shocks: ȳt = y0 + ∑εs,t−i.
  - Limitation: Highly restrictive (only two shock types), likely insufficient for small open economies.

### Empirical Application: U.K. (summary of Section I results and figures)
- Methods applied to U.K. data: HP filter (λ=1,600), BQ SVAR, MVK, MVKfin; specification and estimation details for MVK, MVKfin, and BQ SVAR in Appendix.
- Comparative observations:
  - Filtering methods produce similar dynamics (sharp decline during GFC, closing of a negative output gap around 2013–2014; MVK closes around 2010–2011).
  - Timing of pre-GFC positive gap differs:
    - HP: positive output gap starting in 2005.
    - MVK: positive starting mid-2006.
    - MVKfin: positive and uninterrupted since late 1990s.
  - BQ SVAR shows similar dynamics with pre-crisis positive gap around mid-2006 (smaller in absolute terms), return to potential by 2014, positive gaps 2014–2017, and a recent negative gap.
- Performance claims for the open-economy SVAR (from Section I, to be tested in later sections):
  - Less sensitivity to transitory shocks (e.g., monetary policy shocks).
  - Smaller ex-post real-time revisions when new data is added.
  - Stronger predictive power for inflation.

*Source: WP/20/24 — Measuring Output Gap: Is It Worth Your Time? — Section I*

### Section 2

### Section 2

### Data and estimation samples
- The MVKfin model was estimated on a shorter sample, 1991: Q1–2019: Q1, due to unavailability of data on credit to private sector for earlier years.
- The small open economy SVAR is estimated over 1982: Q1 to 2019: Q1.
- The SVAR includes two lags of each variable; robustness checks note the estimated output gap remains similar if more lags (4, 6 or 8) are used.

### Small open economy SVAR: purpose, variables, and identification
- Purpose: investigate whether a sufficiently rich and properly identified SVAR can overcome limitations of the benchmark BQ SVAR and yield output gap estimates with better properties, allowing both domestic and global shocks to have permanent effects on output.
- Six variables included:
  - U.K. real GDP growth
  - U.K. CPI inflation at constant tax
  - U.K. shadow interest rate
  - Changes in the Sterling exchange rate index
  - U.K. import price inflation
  - Changes in foreign export prices
- Identification strategy (following Forbes et al. (2018)):
  - Six structural shocks identified via a combination of zero short-run and long-run restrictions, and sign restrictions (Table 1).
  - Only domestic supply shocks and persistent global shocks are assumed to affect the level of output in the long run. Persistent global shocks incorporate any foreign shocks with a lasting effect on U.K. output, as well as any (foreign or domestic) demand shocks with a permanent impact on U.K. output.
  - Global shocks are distinguished from domestic shocks by assuming domestic developments do not affect world export prices neither on impact nor in the long run; global shocks may impact both world export prices and the U.K. economy.
  - Sign restrictions imposed include:
    - Domestic supply shock: negative correlation between GDP and CPI in the first 2 periods.
    - Domestic demand shock: positive correlation with i) GDP, ii) CPI, and iii) exchange rate (positive demand shock leads to appreciation).
    - Monetary policy shock: a lower interest rate is associated with a rise in GDP and CPI, and depreciation of the nominal exchange rate.
    - An exogenous exchange rate appreciation implies a fall in CPI.
- Estimation method: Bayesian methods with Minnesota-style priors (as in Binning (2013)); standard errors, percentiles and confidence intervals based on Gibbs sampling, saving final 1000 repetitions.

### Identification restrictions (summary of Table 1)
- Long-run restrictions (world (ex-UK) prices versus UK variables):
  - World (ex-UK) prices: persistent global shock (+) and transitory global shock (+); zero responses for domestic shocks.
- Short-run restrictions (impact and one following quarter):
  - UK GDP growth:
    - UK supply shock: +
    - UK demand shock: +
    - UK monetary policy shock: -
    - Exogenous exchange rate shock: 0
    - Persistent global shock: 0
    - Transitory global shock: 0
  - UK CPI:
    - UK supply shock: -
    - UK demand shock: +
    - UK monetary policy shock: -
    - Exogenous exchange rate shock: -
    - Persistent global shock: 0
    - Transitory global shock: 0
  - UK interest rate:
    - UK supply shock: +
    - UK demand shock: +
    - UK monetary policy shock: -
    - Exogenous exchange rate shock: 0
    - Persistent global shock: 0
    - Transitory global shock: 0
  - UK nominal ERI:
    - UK supply shock: +
    - UK demand shock: +
    - UK monetary policy shock: +
    - Exogenous exchange rate shock: 0
    - Persistent global shock: 0
    - Transitory global shock: 0
  - UK import prices:
    - UK supply shock: 0
    - UK demand shock: 0
    - UK monetary policy shock: 0
    - Exogenous exchange rate shock: 0
    - Persistent global shock: +
    - Transitory global shock: +
  - World (ex-UK) prices:
    - Short-run: 0 responses to domestic shocks; + to persistent and transitory global shocks.
- Note: '+' ('-') indicates impulse response restricted to be positive (negative) in the quarter the shock hits and the following quarter; '0' denotes response restricted to be zero (either on impact or in the long run).

### Impulse response properties and consistency with literature
- A loosening of monetary policy by 100 basis points causes output to fall by about 0.5 percent (consistent with Burgess et al. (2013)).
- An exchange rate shock that leads to sterling appreciation of 1 percent causes import prices to fall by 0.5 percent (in line with Forbes et al. (2018)).
- Positive domestic supply shock:
  - Causes output to increase permanently.
  - Causes the price level to fall.
- Positive domestic demand shock:
  - Leads only to a temporary improvement in output.
- Persistent and transitory global shocks:
  - Domestic variables’ responses to persistent and temporary global shocks broadly carry over the supply/demand interpretation above.

### Historical decomposition and output gap estimates from the open economy SVAR
- Historical decomposition of GDP growth shows:
  - Domestic and global supply shocks were key drivers of the sharp decline in growth during the GFC.
  - The U.K. economy experienced a series of negative domestic supply shocks after the Brexit referendum.
- Output gap (accumulation of past domestic supply and persistent global shocks) suggests:
  - Two periods of considerable overheating—late 1980s and prior to the GFC—both followed by strong declines in growth.
  - Between 2010: Q2–2012: Q4 the open economy SVAR suggests:
    - Output gap became positive in 2010: Q2.
    - Peaked in 2011: Q2.
    - Turned negative in 2013: Q2.
  - The MVK multivariate Kalman filter also suggests output gap turned positive in 2010: Q2 (though by less).
  - The open economy SVAR indicates a negative output gap between 2014: Q3–2016: Q3, while other methods suggest closed or positive output gaps during the same period.
  - Following the Brexit referendum in 2016: Q3:
    - All other methods suggest the output gap turned negative.
    - The open economy SVAR points to a moderately positive output gap until only very recently.

### Macroeconomic conditions around 2010: Q2–2012: Q4 and 2014: Q4–2016: Q3
- 2010: Q2–2012: Q4:
  - Headline growth rebounded from an average of minus 4 percent year-on-year per quarter in 2009 to an average of 2 percent between 2010: Q2 and 2011: Q4.
  - Domestic inflation (core services inflation) and wage growth accelerated compared to the growth rates in the previous four quarters.
  - Unemployment remained around 8 percent throughout the period.
- 2014: Q4–2016: Q3:
  - Growth accelerated, reaching an average rate of 2.8 percent.
  - Unemployment declined and wage growth picked up further.
  - Core services inflation as well as GDP deflator decelerated, suggesting a series of positive supply shocks.

### Comparison of methods and consistency with price dynamics
- When assessing output gap changes rather than levels:
  - MVK filter, BQ SVAR and open economy SVAR all suggest output gap declined between 2011: Q2–2013: Q3—consistent with falling core services inflation and wage growth, but inconsistent with accelerating GDP deflator.
  - Over 2014: Q3–2015: Q3, the three indicators of price pressures (GDP deflator, core services inflation, wage growth) change in different directions, making inference about output gap dynamics difficult.
  - All three inflation indicators suggest an improving output gap between 2016: Q2 and 2018: Q2—consistent only with the open economy SVAR.
- Overall conclusion: none of the methods considered gives an output gap estimate consistent with all the post-GFC price dynamics; the open economy VAR appears to perform marginally better.

### Tests of alternative output gap estimates: overview of procedures
- Three tests used to compare five alternative methods:
  1. Real-time properties of output gap estimates (following Orphanides and van Norden (2002)).
  2. How real-time potential output estimates respond to supply versus demand shocks (following Coibion et al. (2018)).
  3. Which output gap performs better in forecasting inflation.
- Real-time data source: Bank of England’s “GDP Real Time Database” (monthly vintages from January 1990 until August 2016 as of writing).
  - Use real-time GDP series and—for the small open economy SVAR—the real-time import price deflator series.
- Real-time estimation procedure:
  - For each quarter between 2005: Q1 and 2016: Q2 run separate regressions using only data from 1982: Q1 up until that quarter, replacing GDP (and import price deflator) with the real-time vintage available in the middle month of that quarter.
  - First iteration based on data from 1982: Q1 to 2005: Q1.
  - For HP filter approach, to mitigate end-of-sample problem, the real GDP series was “extended” by 8 quarters using the last available year-on-year growth rate.

### Real-time performance findings
- Filtering-based methods (three filters) fail to signal, in real time, an overheating of the U.K. economy prior to the GFC.
- BQ SVAR:
  - Signals a slightly positive output gap starting in 2006: Q1.
  - Real-time estimates suggest a positive and quite large output gap between 2014 and 2016, difficult to reconcile with the relatively small pre-GFC positive gap.
- Small open economy SVAR (extended SVAR):
  - Real-time output gap estimates follow full sample estimates more closely than the other four methods.
  - Signals opening of a positive output gap already in 2004: Q1.
  - Real-time output gap reaches 1 percent around 2008: Q1, in line with full-sample estimate.
- Additional regressions verify that ex-post revisions to filtering methods are due to the end-of-sample problem rather than data revisions before and during the GFC, confirming findings of Orphanides and van Norden (2002) and Marcellino and Musso (2011).

### Responses-to-shocks test: data and methodology
- Aim: check how potential output estimates respond to different types of shocks; test whether potential growth estimates are procyclical.
- External shock series used:
  - Global permanent shocks:
    - For global technology shocks, Beaudry and Portier (2006) U.S. TFP news shocks (updated by Valerie Ramey).
  - Global temporary shocks:
    - U.S. monetary policy shocks identified using high frequency surprises around policy announcements as external instruments (Gertler and Karadi (2015)).
    - Global fiscal shocks: U.S. military spending news shocks of Ramey (2016).
  - Domestic shocks:
    - Domestic fiscal shocks following Blanchard and Perotti (2002) SVAR specification and identification strategy.
    - U.K. monetary policy shocks: VAR with GDP growth, unemployment, inflation and interest rate (four lags) and Cholesky decomposition on this ordering.
    - Productivity shocks: residuals from a regression of output per worker on its lags.
- Regression specification for each shock s:
  - ∆ȳ̅_t = α_s + ∑_{k=0}^K φ^k_s ε_{t−k}^s + ζ_t^s
  - Due to small number of observations, K = 6 lags are used; one shock at a time; Newey-West standard errors applied.

*Source: IMF staff calculations, authors' calculations.*

### Section 3

### wpiea2020024-print-pdf - Section 3

### Impulse responses of potential output to global and U.S. shocks
- Method: Impulse responses (IRFs) of potential output constructed by summing coefficients 휑
푡−푘
푠
 up to a given horizon (e.g., 휑
0
푠
 for current period, 휑
0
푠
+휑
1
푠
 for one period after a shock, etc.).
- U.S. productivity shock (Figure 6):
  - All five models yield a positive and statistically significant response of potential output to a positive global productivity shock.
  - Identification: one standard deviation positive U.S. productivity shock, obtained using the Beaudry Portier (2006) short-run restrictions. Red dashed lines show 66 percent confidence bands.
- U.S. fiscal shock (Figure 7) — a transitory global shock:
  - Two SVAR models: potential output responds in line with intuition.
  - Two multivariate filters (MVK, MVKfin): yield a statistically significant increase in potential growth after a U.S. fiscal shock.
  - HP filter: response of potential output is negative (and marginally statistically significant) after 6 quarters.
  - Small open economy SVAR: potential output initially increases, but response ceases to be statistically significant after 3 quarters.
  - Identification: one standard deviation positive U.S. fiscal shock identified as in Ramey (2016). Red dashed lines show 66 percent confidence bands.

### Summary assessment of IRF properties across models and shocks (Figure 8)
- Scope: Five models (HP filter, MVK, MVKfin, BQ SVAR, Small open economy SVAR) and six types of shocks (US prod., US fiscal, US monetary, UK prod., UK fiscal, UK monetary).
- Qualitative coding:
  - Dark green: IRFs in line with economic intuition both short and medium term.
  - Light green: IRFs in line with intuition only in the medium term.
  - Dark red: IRFs counterintuitive in the medium term.
  - Bright red: IRFs borderline in the medium term.
- Findings:
  - SVAR-based potential output estimates tend to perform relatively well.
  - BQ SVAR: responds almost significantly to a U.S. fiscal shock and insignificantly to a U.K. productivity shock over the medium run.
  - Filter-based estimates show more counter-intuitive results:
    - HP filter: does not respond significantly to a U.K. productivity shock.
    - MVKfin: responds positively to a U.K. monetary policy shock.
  - Overall, the small open economy SVAR has the most intuitive impulse responses to the six shocks considered.

### Inflation forecasting (Phillips curve tests)
- Phillips curve specification (based on Blanchard (2016)):
  - 휋
푡
=휃푦̂
푡
+휆휋
푡
푒
+
(
1−휆
)
휋
푡−1
∗
+μ휋
푚,푡
+휀
푡
- Expectations equation:
  - 휋
푡
푒
=훼+훽휋
푡−1
∗
+휂
푡
- Definitions:
  - 휋
푡
: headline consumer price inflation (quarterly inflation, annualized).
  - 푦̂
푡
: output gap.
  - 휋
푡
푒
: long-term inflation expectations.
  - 휋
푡−1
∗
: average of the last four quarterly inflation rates.
  - 휋
푚,푡
: import price inflation relative to headline inflation.
- Tests performed:
  1. In-sample estimation of the Phillips curve for 1993: Q1 to 2016: Q2 using five real-time output gap measures.
     - Rationale: The U.K. adapted inflation targeting framework in October 1992; estimation starts 1993.
  2. Out-of-sample one-quarter-ahead iterative forecasting exercise for each quarter between 2006: Q1 and 2016: Q1, estimating separate Phillips curves using data from 1982: Q1 up to the given quarter.
- In-sample regression results (Table 3):
  - Lags of 휋
푡
푒
:
    - pi e (-1): 0.37***, 0.36***, 0.34***, 0.34***, 0.20** (standard errors: (0.10), (0.10), (0.10), (0.09), (0.10))
    - pi e (-2): 0.16, 0.16, 0.15, 0.15, 0.08 (standard errors (0.10) each)
    - pi e (-3): 0.191*, 0.204**, 0.181*, 0.201**, 0.171* (standard errors (0.10) each)
    - pi e (-4): 0.104, 0.125, 0.0676, 0.122, 0.131 (standard errors (0.10), (0.10), (0.10), (0.09), (0.09))
  - imported pi e / pi e: 0.0008***, 0.0008***, 0.0007**, 0.0008***, 0.0008*** (standard errors (0.0003))
  - Output gap coefficients (column-specific):
    - HP filter: 0.05* (standard error (0.03))
    - MVK: 0.10** (standard error (0.04))
    - MVKfin: 0.01 (standard error (0.03))
    - BQ SVAR: 0.07** (standard error (0.03))
    - Small open economy VAR: 0.40*** (standard error (0.09))
  - Constant: 0.0006, 0.0006, 0.001, 0.0006, 0.001** (standard errors (0.0006), (0.0006), (0.0007), (0.0006), (0.0006))
  - Observations: 105, 105, 97, 105, 105
  - R-squared: 0.56, 0.57, 0.41, 0.57, 0.62
  - Significance notation: *** p<0.01, ** p<0.05, * p<0.1
- Out-of-sample forecasting performance (Table 4):
  - Average of squared errors of out-of-sample projected inflation (in percentage points) over 2006: Q1–2016: Q2:
    - HP filter: 0.27
    - MVK: 0.26
    - MVKfin: 0.28
    - BQ SVAR: 0.24
    - Small open economy SVAR: 0.24
  - Interpretation:
    - SVAR-based estimates produce the smallest forecast error, very close to MVK.
    - Caveat: MVK approach embeds the Phillips curve in estimation, so a small error term is expected by design.
    - On net, the small open economy SVAR appears to have the smallest error terms in forecasting inflation, although differences with other methods may not be statistically significant.

### Robustness checks
- Approaches and variations tested:
  - Mitigate end-of-sample problem for three filtering approaches by extending sample by one year with Consensus forecasts of GDP and—in the case of MVK and MVKfin—of inflation.
  - MVK sensitivity: alternative parameter priors (e.g., lower or higher β), different steady-state unemployment and potential output growth, specification with detrended unemployment.
  - BQ SVAR: alternative measure of unemployment that accounts for underemployment.
  - Small open economy SVAR: estimated without imposing any sign restrictions.
  - Alternative sample start: estimation sample starting in 1993: Q1.
- Findings:
  - Alternative specifications do not yield results considerably different from baseline.
  - BQ SVAR results robust to different unemployment measures.
  - Small open economy SVAR results not much affected when imposing no sign restrictions.
  - All five methods do not yield considerably different output gap estimates when estimation sample starts in 1993: Q1.
  - Note: Some results are available upon request.

### Multivariate Kalman Filter specification and estimated parameters
- Key state-space equations (notations preserved):
  - 푦̂
푡
=푦
푡
−푦̅
푡
  (C.1)
  - 푦̅
푡
=푦̅
푡−1
+푔
푡
+휀
푡
푦
̅
  (C.2)
  - 푔
푡
=휃푔
푠푠
+
(
1−휃
)
푔
푡−1
+휀
푡
푔
  (C.3)
  - 푦̂
푡
=휑푦̂
푡−1
+휀
푡
푦
̂
  (C.4)
  - 휋
푡
=휆휋
푡+1
+
(
1−휆
)
휋
푡−1
+훽푦̂
푡
+휀
푡
휋
  (C.5)
  - 푢̂
푡
=휏
1
푦̂
푡
+휏
2
푢̂
푡−1
+휀
푡
푢
  (C.6)
  - 푢̂
푡
=푢
푡
−푢̅
푡
  (C.7)
  - 푢̅
푡
=휏
4
(
푢̅
푠푠
+
(
1−휏
4
)
푢̅
푡−1
)
+푔
푡
푢
+휀
푡
푢
̅
  (C.8)
  - 푔
푡
푢
=
(
1−휏
3
)
푔
푡−1
푢
+휀
푡
푔
푢
  (C.9)
- Estimation approach:
  - Parameter values and variances of shock terms are maximum likelihood estimates obtained using Bayesian estimation.
  - Priors set at posteriors estimated for the U.K. in Blagrave et al. (2015).
  - Full sample: 1982: Q1–2019: Q1.
- Priors and posteriors (table entries preserved):
  - λ: prior 0.25, posterior 0.23
  - β: prior 0.15, posterior 0.1
  - φ: prior 0.7, posterior 0.72
  - θ: prior 0.2, posterior 0.1
  - τ1: prior 0.4, posterior 0.33
  - τ2: prior 0.4, posterior 0.48
  - τ3: prior 0.1, posterior 0.09
  - τ4: prior 0.1, posterior 0.09
  - gss: 1.6
  - u̅ss: 4.5

### Multivariate Filter with Financial Variables (MVF / MVKfin) specification
- Approach: Based on Berger et al. (2015). Potential output estimated by decomposing observed GDP into trend and cycle.
  - 푦
푡
=푦̅
푡
+푦̂
푡
  - ∆
2
푦̅
푡
=휀̅
푡
  - Variance ratio constraint: 휆 ≡ 푉푎푟(푦̂
푡
) / 푉푎푟(휀̅
푡
) set at 1600, implying potential output captures output movements at frequencies above 8 years.
- Interpretation: Similar to HP filter with a specific constraint on the variance ratio to control frequency content of trend vs. cycle.

### Conclusions (Section V)
- Main empirical findings:
  - Filtering-based models produce potential output estimates that respond to transitory shocks in a procyclical way (confirming Coibion et al (2018)).
  - Models that distinguish between different types of shocks (SVAR-based) yield potential output estimates with better impulse response properties.
  - The small open economy SVAR performs best among considered models and is characterized by the best real-time properties of output gap estimates.
- Policy-relevant implications:
  - Output gap estimates are useful ("Are output gap estimates worth economists’ time?" — answer: yes), but:
    - Proper care needed when identifying all relevant persistent shocks.
    - Verify a model’s performance with established tests; a good model should be characterized by minimal ex-post revisions to real-time estimates.
    - No perfect measure of the output gap exists; using a broader range of indicators is advisable for a more accurate picture of the cyclical position of the economy.

*Source: wpiea2020024-print-pdf - Section 3*

### Section 4

### wpiea2020024-print-pdf - Section 4

### MVK filter specification and estimation
- Model equation for the output gap (preserving notation):
  - 푦푡 − 푦̅푡 = 휌(푦푡−1 − 푦̅푡−1) + 훽푥푡 + 휀푡
- The variance ratio Var(푦푡 − 푦̅푡) / Var(∆2 푦̅푡) is constrained to match the one implied by equations 1–3 and implicitly the frequency characteristic of the HP filter.
- Observable covariates 푥푡 included in the model:
  - real credit growth
  - real house price inflation
  - stock price inflation
- Estimation method: maximum likelihood.
- Table of estimates for financial variables (Estimate, Std. error, p-value):
  - Real HPI growth: 0.02, 0.007, 0.001
  - Real credit growth: 0.008, 0.008, 0.309
  - Stock price growth: 0.004, 0.002, 0.038
- Source note shown in figures: Authors' calculations.

### Blanchard-Quah (BQ) SVAR: specification and identification
- SVAR structure (as presented):
  - [1 0; 12? 1?] [∆y_t; u_t] = [A0,1 A0,2] + [B1,11 B1,12; B1,21 B1,22][∆y_{t−1}; u_{t−1}] + [ε_t^s; ε_t^d]
  - (Text presents the system in matrix form following Blanchard and Quah (1989).)
- Sample: 1982: Q1–2019: Q1.
- Lag selection: 2 lags of each variable—selected based on the AIC criterion.
- Identification: impose a zero long-run impact of demand shocks (ε_t^d) on GDP growth.
- Variables included: quarterly real GDP growth and detrended unemployment rate.
- Detrending rationale: detrended unemployment rate used to remove a downward trend observed in data; 5-quarter moving-average unemployment rate declined from over 8 percent in 1980s and early 1990s to around or somewhat below 5 percent before and after the GFC, respectively.

### Impulse response analysis and robustness checks
- Impulse responses reported for:
  - BQ SVAR: cumulative impulse responses of real GDP (GDP_G) and the unemployment rate (U) to a one standard deviation supply shock (Shock1) and a one standard deviation demand shock (Shock2).
  - Additional model variants/filters: HP filter, MVK filter, MVKfin filter, Extended SVAR.
- Tests of output gap estimates via shock experiments:
  - Responses of real-time potential output estimates to a one standard deviation positive U.S. monetary policy shock (following Gertler and Karadi (2015)); 66 percent confidence bands shown.
  - Responses to a one standard deviation positive U.K. productivity shock; 66 percent confidence bands shown.
  - Responses to a one standard deviation positive U.K. fiscal shock; 66 percent confidence bands shown.
  - Responses to a one standard deviation positive U.K. monetary policy shock; 66 percent confidence bands shown.
- Graphical results include horizon labels (Quarter 0–6 and longer horizons up to 40 in some cumulative charts) and plotted 66 percent confidence bands around impulse responses for each filter/SVAR variant.

### Key quantitative facts and diagnostics (preserved values)
- Sample period for BQ SVAR: 1982: Q1–2019: Q1
- Lag length in SVAR: 2 lags
- Unemployment trend facts: 5-quarter moving-average unemployment rate declined from over 8 percent to around or somewhat below 5 percent (1980s/early 1990s → before and after the GFC)
- Financial-variable estimates (Estimate / Std. error / p-value):
  - Real HPI growth: 0.02 / 0.007 / 0.001
  - Real credit growth: 0.008 / 0.008 / 0.309
  - Stock price growth: 0.004 / 0.002 / 0.038

*Source: wpiea2020024-print-pdf - Section 4*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020024-print-pdf.pdf_