## _wp1052 - References

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---

### Introduction
- Purpose: Assess whether output gap estimates are useful indicators for inflation forecasts when estimated with information available in real time.
- Frameworks referenced: Phillips curve and Taylor-rule logic (Taylor, 1993).
- Key methodological principle: Use real-time GDP vintages available to policymakers at each point in time rather than ex post revised series.

### Real-Time GDP datasets
- United Kingdom dataset:
  - Matrix dimensions: 205 x 126.
  - Vintages: 126 vintages starting in 1976Q1 and ending in 2007Q2.
  - Longest vintage contains 205 observations (1956Q1 until 2007Q1).
  - Source: Bank of England (Appendix 1).
- United States dataset:
  - Matrix dimensions: 245 x 171.
  - Vintages: 171 vintages starting in 1965Q4 and ending in 2008Q2.
  - Longest vintage contains 245 observations (1947Q1 until 2008Q1).
  - Source: Federal Reserve Bank of Philadelphia (Appendix 2).
- Germany dataset:
  - Matrix dimensions: 183 x 140.
  - Vintages: 140 vintages, first vintage perspective 1973Q1, last vintage perspective 2007Q4.
  - Longest vintage contains 183 observations (1962Q1 until 2007Q3) — treated as ex post GDP for researchers today.
  - Constructed from 140 issues of Bundesbank’s Saisonbereinigte Wirtschaftszahlen; Clausen and Meier (2005) provided vintages 1973–1998 and this paper adds 36 vintages to 2007Q4 (Appendix 3).

### Revisions and summary statistics on real GDP revisions
- Revisions measured as percentage-point differences between ex post growth rates and real-time reported growth rates.
- Summary statistics:
  - Germany: Mean 0.26; St. Dev. 0.83; Corr 0.92
  - UK: Mean 0.83; St. Dev. 1.20; Corr 0.79
  - US: Mean 0.41; St. Dev. 0.96; Corr 0.91
- Notable revision episodes:
  - Germany: 1975Q1 and 1991Q1 — output significantly underestimated in real time and revised upwards.
  - UK: Substantial upward revisions, especially until 1991 (real-time growth systematically underestimated).
  - US: Largest revisions in 1966Q1 (upwards 2½ percentage points) and 1975Q2 (revision of 4 percentage points from an initial -6 percent to -2 percent).
- Overall bias: Mean revision to original data is upwards in all three countries (initial underreporting of growth).

### Estimating output gaps (methods and definitions)
- Output gap definition: og = y - y* (log of observed output minus log of unobserved trend potential output).
- Output gap series defined:
  - a_t: real-time output gap (recursive potential estimated for each vintage using data available at that time).
  - b_t: quasi-real-time output gap (recursive potential estimated using ex post data but subtracting ex post output — Orphanides and van Norden approach).
  - c_t: ex post output gap (potential estimated once using revised ex post data; subtract revised output).
- Detrending methods used:
  - Linear detrending:
    - Ex post trend on whole sample (ex post output gap).
    - Real-time recursively estimated trend.
    - Real-time rolling window trend: repeated trend regressions using rolling windows with ten years of data.
  - Hodrick-Prescott (HP) filter:
    - End-of-sample problem mitigated by adding forecasts for next 12 quarters generated from an AR(8) model for change in real GDP.
- Expectation: Real-time output gaps may differ from ex post due to data revisions and smaller information sets at the time of estimation.

### Simulating inflation forecasting (procedure)
- Model 1 (bivariate Phillips curve):
  - Equation (1): π_{t+h} = α + β og_{t} + ε_{t+h}
  - π is annual CPI inflation; og is the output gap series; h is forecast horizon.
- Model 2 (benchmark AR(1)):
  - Equation (2): π_{t+h} = α + β π_{t-1+h} + ε_{t+h}
- Estimation:
  - GMM used for the Phillips curve to address endogeneity; lagged values of the output gap used as instruments.
  - Forecast horizons: 1, 4, 8, and 12 quarters ahead.
  - Recursive estimation: initial training periods and then re-estimate adding one quarter at a time while always starting from the same initial observation.
  - Starting training periods:
    - Germany: 1962Q1 to 1972Q4.
    - UK: 1956Q1 to 1975Q4.
    - US: 1947Q1 to 1965Q3.
- Evaluation metric: Root Mean Squared Error (RMSE) of h-period ahead forecasts; output gap estimate considered useful if bivariate model RMSE < AR(1) RMSE.

### Results — Inflation forecast error ratios (ratio = RMSE(bivariate) / RMSE(AR(1)))
- Interpretation: Ratios above 1 mean output gaps did not help reduce forecast errors.

- United Kingdom (UK):
  - Ex Post GDP Data and Potential Output Estimated with HP-Filter:
    - 1 Quarter: 0.95
    - 4 Quarters: 0.92
    - 8 Quarters: 0.91
    - 12 Quarters: 0.92
  - Ex Post GDP Data and Potential Output Estimated with Linear Trend:
    - 1 Quarter: 0.95
    - 4 Quarters: 0.91
    - 8 Quarters: 0.86
    - 12 Quarters: 0.83
  - Real-Time GDP Data and Potential Output Estimated with HP-Filter:
    - 1 Quarter: 1.03
    - 4 Quarters: 1.01
    - 8 Quarters: 0.86
    - 12 Quarters: 0.92
  - Real-Time GDP Data and Potential Output Estimated with Linear Trend (Roll. Wind.):
    - 1 Quarter: 1.03
    - 4 Quarters: 1.04
    - 8 Quarters: 0.88
    - 12 Quarters: 0.93

- United States (US):
  - Ex Post GDP Data and Potential Output Estimated with HP-Filter:
    - 1 Quarter: 0.89
    - 4 Quarters: 0.87
    - 8 Quarters: 0.90
    - 12 Quarters: 0.92
  - Ex Post GDP Data and Potential Output Estimated with Linear Trend:
    - 1 Quarter: 0.92
    - 4 Quarters: 0.89
    - 8 Quarters: 0.89
    - 12 Quarters: 0.92
  - Real-Time GDP Data and Potential Output Estimated with HP-Filter:
    - 1 Quarter: 0.84
    - 4 Quarters: 0.86
    - 8 Quarters: 1.00
    - 12 Quarters: 1.28
  - Real-Time GDP Data and Potential Output Estimated with Linear Trend (Roll. Wind.):
    - 1 Quarter: 0.89
    - 4 Quarters: 1.02
    - 8 Quarters: 1.34
    - 12 Quarters: 1.91

- Germany:
  - Ex Post GDP Data and Potential Output Estimated with HP-Filter:
    - 1 Quarter: 0.98
    - 4 Quarters: 0.89
    - 8 Quarters: 0.81
    - 12 Quarters: 0.78
  - Ex Post GDP Data and Potential Output Estimated with Linear Trend:
    - 1 Quarter: 1.01
    - 4 Quarters: 1.04
    - 8 Quarters: 1.10
    - 12 Quarters: 1.17
  - Real-Time GDP Data and Potential Output Estimated with HP-Filter:
    - 1 Quarter: 1.02
    - 4 Quarters: 1.14
    - 8 Quarters: 1.55
    - 12 Quarters: 2.24
  - Real-Time GDP Data and Potential Output Estimated with Linear Trend (Roll. Wind.):
    - 1 Quarter: 1.01
    - 4 Quarters: 1.11
    - 8 Quarters: 1.60
    - 12 Quarters: 2.41

### Key findings and interpretation
- Ex post output gaps:
  - For all three countries, ex post output gaps (both detrending methods, with one exception) tend to improve inflation forecast accuracy relative to AR(1).
  - Exception: Germany — only the HP-filter ex post output gap helps forecast inflation better than the naive benchmark; linear-trend ex post series for Germany do not.
- Real-time output gaps:
  - Generally perform worse than ex post output gaps and often fail to beat the AR(1) benchmark.
  - Germany: Neither HP-filter nor linear-trend real-time output gaps reduce forecast errors relative to AR(1) across horizons.
  - United States: Real-time output gaps useful for 1-quarter ahead forecasts; HP-filter real-time gap also useful for 4-quarter ahead forecasts; performance declines at longer horizons.
  - United Kingdom: Real-time output gaps did not help for 1- and 4-quarter horizons but improved forecasts at 8- and 12-quarter horizons.
- Overall implication: Ex post output gap measures may overstate the practical usefulness of Phillips-curve frameworks for inflation forecasting because real-time estimation uncertainty and data revisions reduce predictive gains in practice.
- Caveat: Output gaps can still be useful for guiding forecast judgments when combined with other indicators of spare capacity.

### Conclusions with regard to the finding that using ex post data helps reduce forecast errors
- Main findings on forecasting performance:
  - The simple Phillips curves as specified in the paper and based on ex post output gaps generally improve the accuracy of inflation forecasts compared to an AR(1) forecast.
  - Real-time output gaps often do not help forecasting inflation.
  - The usefulness of ex post output gap estimates indicates that when looking at ‘historic’ revised GDP, output gaps can measure the extent of slack in the economy and do have an explanatory value for the dynamics of inflation.
  - When simulating a real-time perspective close to reality, output gaps (as calculated by the two methodologies used in this paper) might not always be as helpful in forecasting inflation in practice due to:
    - the uncertainty and difficulty in estimating potential output in real-time, and
    - revisions to the output series itself.
  - The finding that the output gap generated by linear detrending using ex post data does not help forecast inflation in Germany is not necessarily explained by the structural break in the German data after unification. To avoid a jump due to reunification, within the vintages from 1995Q2 onwards, the authors link the data for West Germany with that of the reunified country by using West Germany’s growth rates from 1993Q4 backwards.
  - In a related exercise, Groen, Kapetanios, and Price (2009) find that the inflation forecasts in the Bank of England’s Inflation Report dominate those generated by autoregressive processes.

- Interpretation and implications for policy and forecasting practice:
  - Ex post output gaps are informative about historic slack and inflation dynamics; this supports the theoretical concept of Phillips curves in a historical/revised-data context.
  - The poorer real-time performance highlights the challenge for policymakers and forecasters who must rely on unrevised real-time GDP vintages: real-time estimates of potential output and output gaps are noisy and may weaken the practical forecasting value of Phillips-curve relationships.
  - Output gaps can still be a useful tool in guiding forecast judgments, especially when used in conjunction with other indicators of spare capacity.
  - The findings call more into question how to measure slack in the economy in real-time, rather than invalidating the theoretical Phillips curve.

### Data and measurement notes (series and definitions preserved)
- United Kingdom real-time GDP dataset:
  - Matrix dimensions: 205 x 126, with 126 vintages starting in 1976Q1 and ending in 2007Q2.
  - The last and longest vintage contains 205 observations (1956Q1 until 2007Q1).
  - Source: Bank of England’s GDP(E) real-time database.
- United States real-time GDP dataset:
  - Matrix dimensions: 245 x 171, with 171 vintages starting in 1965Q4 and ending in 2008Q2.
  - The last and longest vintage contains 245 observations (1947Q1 until 2008Q1).
  - Source: Federal Reserve Bank of Philadelphia, ROUTPUT series.
- Germany real-time GNP/GDP dataset:
  - Matrix dimensions: 183 x 140, with 140 vintages starting in 1973Q1 and ending in 2007Q4.
  - The last and longest vintage contains 183 observations (1962Q1 until 2007Q3).
  - All 140 series were manually entered, each series originating from a different issue of the Bundesbank’s Saisonbereinigte Wirtschaftszahlen.
  - The dataset used in this paper draws on Clausen and Meier (2005) for 1973Q1 to 1998Q4 and adds 36 vintages until 2007Q4.
- Series notation:
  - a_t: real-time output gap series.
  - b_t: quasi-real-time output gap series.
  - c_t: ex post output gap series.

### Recommended extensions and avenues for further research
- Examine sub-samples of the time periods chosen to assess whether the magnitude of revisions has changed over time and what effect this might have on forecast errors.
- Apply other methodologies of estimating the output gap to assess what these other methods imply for the differences between real-time and ex post estimates.
- Explore additional indicators of spare capacity to complement output gaps in real-time forecasting.

*Source: IMF working paper content unit _wp1052 - References*

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

### _wp1052 - References

### Introduction
- Purpose: Assess whether output gap estimates are useful indicators for inflation forecasts when estimated with information available in real time.
- Frameworks referenced: Phillips curve and Taylor-rule logic (Taylor, 1993).
- Key methodological principle: Use real-time GDP vintages available to policymakers at each point in time rather than ex post revised series.

### Real-Time GDP datasets
- United Kingdom dataset:
  - Matrix dimensions: 205 x 126.
  - Vintages: 126 vintages starting in 1976Q1 and ending in 2007Q2.
  - Longest vintage contains 205 observations (1956Q1 until 2007Q1).
  - Source: Bank of England (Appendix 1).
- United States dataset:
  - Matrix dimensions: 245 x 171.
  - Vintages: 171 vintages starting in 1965Q4 and ending in 2008Q2.
  - Longest vintage contains 245 observations (1947Q1 until 2008Q1).
  - Source: Federal Reserve Bank of Philadelphia (Appendix 2).
- Germany dataset:
  - Matrix dimensions: 183 x 140.
  - Vintages: 140 vintages, first vintage perspective 1973Q1, last vintage perspective 2007Q4.
  - Longest vintage contains 183 observations (1962Q1 until 2007Q3) — treated as ex post GDP for researchers today.
  - Constructed from 140 issues of Bundesbank’s Saisonbereinigte Wirtschaftszahlen; Clausen and Meier (2005) provided vintages 1973–1998 and this paper adds 36 vintages to 2007Q4 (Appendix 3).

### Revisions and summary statistics on real GDP revisions
- Revisions are measured as percentage-point differences between ex post growth rates and real-time reported growth rates.
- Summary statistics:
  - Germany: Mean 0.26; St. Dev. 0.83; Corr 0.92
  - UK: Mean 0.83; St. Dev. 1.20; Corr 0.79
  - US: Mean 0.41; St. Dev. 0.96; Corr 0.91
- Notable revision episodes:
  - Germany: 1975Q1 and 1991Q1 — output significantly underestimated in real time and revised upwards.
  - UK: Substantial upward revisions, especially until 1991 (real-time growth systematically underestimated).
  - US: Largest revisions in 1966Q1 (upwards 2½ percentage points) and 1975Q2 (revision of 4 percentage points from an initial -6 percent to -2 percent).
- Overall bias: Mean revision to original data is upwards in all three countries (initial underreporting of growth).

### Estimating output gaps (methods and definitions)
- Output gap definition: og = y - y* (log of observed output minus log of unobserved trend potential output).
- Output gap series defined (Table 1 reference):
  - at: real-time output gap (recursive potential estimated for each vintage using data available at that time).
  - bt: quasi-real-time output gap (recursive potential estimated using ex post data but subtracting ex post output — Orphanides and van Norden approach).
  - ct: ex post output gap (potential estimated once using revised ex post data; subtract revised output).
- Detrending methods used:
  - Linear detrending:
    - Ex post trend on whole sample (ex post output gap).
    - Real-time recursively estimated trend (illustrative).
    - Real-time rolling window trend: repeated trend regressions using rolling windows with ten years of data (accounts for breaks).
  - Hodrick-Prescott (HP) filter:
    - End-of-sample problem mitigated by adding forecasts for next 12 quarters generated from an AR(8) model for change in real GDP.
- Expectation: Real-time output gaps may differ from ex post due to data revisions and smaller information sets at the time of estimation.

### Simulating inflation forecasting (procedure)
- Model 1 (bivariate Phillips curve):
  - Equation (1): π_{t+h} = α + β og_{t} + ε_{t+h}
  - π is annual CPI inflation; og is the output gap series; h is forecast horizon.
- Model 2 (benchmark AR(1)):
  - Equation (2): π_{t+h} = α + β π_{t-1+h} + ε_{t+h}
- Estimation:
  - GMM used for the Phillips curve to address endogeneity; lagged values of the output gap used as instruments.
  - Forecast horizons: 1, 4, 8, and 12 quarters ahead.
  - Recursive estimation: initial training periods and then re-estimate adding one quarter at a time while always starting from the same initial observation.
  - Starting training periods:
    - Germany: 1962Q1 to 1972Q4.
    - UK: 1956Q1 to 1975Q4.
    - US: 1947Q1 to 1965Q3.
- Evaluation metric: Root Mean Squared Error (RMSE) of h-period ahead forecasts; output gap estimate considered useful if bivariate model RMSE < AR(1) RMSE.

### Results — Inflation forecast error ratios (Table 2: ratio = RMSE(bivariate) / RMSE(AR(1)))
- Interpretation: Ratios above 1 (bold in source) mean output gaps did not help reduce forecast errors.
- United Kingdom (UK):
  - Using Ex Post GDP Data and Potential Output Estimated with HP-Filter:
    - 1 Quarter: 0.95
    - 4 Quarters: 0.92
    - 8 Quarters: 0.91
    - 12 Quarters: 0.92
  - Using Ex Post GDP Data and Potential Output Estimated with Linear Trend:
    - 1 Quarter: 0.95
    - 4 Quarters: 0.91
    - 8 Quarters: 0.86
    - 12 Quarters: 0.83
  - Using Real-Time GDP Data and Potential Output Estimated with HP-Filter:
    - 1 Quarter: 1.03
    - 4 Quarters: 1.01
    - 8 Quarters: 0.86
    - 12 Quarters: 0.92
  - Using Real-Time GDP Data and Potential Output Estimated with Linear Trend (Roll. Wind.):
    - 1 Quarter: 1.03
    - 4 Quarters: 1.04
    - 8 Quarters: 0.88
    - 12 Quarters: 0.93
- United States (US):
  - Using Ex Post GDP Data and Potential Output Estimated with HP-Filter:
    - 1 Quarter: 0.89
    - 4 Quarters: 0.87
    - 8 Quarters: 0.90
    - 12 Quarters: 0.92
  - Using Ex Post GDP Data and Potential Output Estimated with Linear Trend:
    - 1 Quarter: 0.92
    - 4 Quarters: 0.89
    - 8 Quarters: 0.89
    - 12 Quarters: 0.92
  - Using Real-Time GDP Data and Potential Output Estimated with HP-Filter:
    - 1 Quarter: 0.84
    - 4 Quarters: 0.86
    - 8 Quarters: 1.00
    - 12 Quarters: 1.28
  - Using Real-Time GDP Data and Potential Output Estimated with Linear Trend (Roll. Wind.):
    - 1 Quarter: 0.89
    - 4 Quarters: 1.02
    - 8 Quarters: 1.34
    - 12 Quarters: 1.91
- Germany:
  - Using Ex Post GDP Data and Potential Output Estimated with HP-Filter:
    - 1 Quarter: 0.98
    - 4 Quarters: 0.89
    - 8 Quarters: 0.81
    - 12 Quarters: 0.78
  - Using Ex Post GDP Data and Potential Output Estimated with Linear Trend:
    - 1 Quarter: 1.01
    - 4 Quarters: 1.04
    - 8 Quarters: 1.10
    - 12 Quarters: 1.17
  - Using Real-Time GDP Data and Potential Output Estimated with HP-Filter:
    - 1 Quarter: 1.02
    - 4 Quarters: 1.14
    - 8 Quarters: 1.55
    - 12 Quarters: 2.24
  - Using Real-Time GDP Data and Potential Output Estimated with Linear Trend (Roll. Wind.):
    - 1 Quarter: 1.01
    - 4 Quarters: 1.11
    - 8 Quarters: 1.60
    - 12 Quarters: 2.41

### Key findings and interpretation
- Ex post output gaps:
  - For all three countries, ex post output gaps (both detrending methods, with one exception) tend to improve inflation forecast accuracy relative to AR(1).
  - Exception: Germany — only the HP-filter ex post output gap helps forecast inflation better than the naive benchmark; linear-trend ex post series for Germany do not.
- Real-time output gaps:
  - Generally perform worse than ex post output gaps and often fail to beat the AR(1) benchmark.
  - Germany: Neither HP-filter nor linear-trend real-time output gaps reduce forecast errors relative to AR(1) across horizons.
  - United States: Real-time output gaps useful for 1-quarter ahead forecasts; HP-filter real-time gap also useful for 4-quarter ahead forecasts; performance declines at longer horizons.
  - United Kingdom: Real-time output gaps did not help for 1- and 4-quarter horizons but improved forecasts at 8- and 12-quarter horizons.
- Overall implication: Ex post output gap measures may overstate the practical usefulness of Phillips-curve frameworks for inflation forecasting because real-time estimation uncertainty and data revisions reduce predictive gains in practice.
- Caveat: Output gaps can still be useful for guiding forecast judgments when combined with other indicators of spare capacity.

*Source: IMF working paper content unit _wp1052 - References*

### conclusions with regard to the finding that using ex post data helps reduce forecast errors

### conclusions with regard to the finding that using ex post data helps reduce forecast errors

### Main findings on forecasting performance
- The simple Phillips curves as specified in the paper and based on ex post output gaps generally improve the accuracy of inflation forecasts compared to an AR(1) forecast.
- Real-time output gaps often do not help forecasting inflation.
- The usefulness of ex post output gap estimates indicates that when looking at ‘historic’ revised GDP, output gaps can measure the extent of slack in the economy and do have an explanatory value for the dynamics of inflation.
- When simulating a real-time perspective close to reality, output gaps (as calculated by the two methodologies used in this paper) might not always be as helpful in forecasting inflation in practice due to:
  - the uncertainty and difficulty in estimating potential output in real-time, and
  - revisions to the output series itself.
- The finding that the output gap generated by linear detrending using ex post data does not help forecast inflation in Germany is not necessarily explained by the structural break in the German data after unification. To avoid a jump due to reunification, within the vintages from 1995Q2 onwards, the authors link the data for West Germany with that of the reunified country by using West Germany’s growth rates from 1993Q4 backwards.
- In a related exercise, Groen, Kapetanios, and Price (2009) find that the inflation forecasts in the Bank of England’s Inflation Report dominate those generated by autoregressive processes.

### Interpretation and implications for policy and forecasting practice
- Ex post output gaps are informative about historic slack and inflation dynamics; this supports the theoretical concept of Phillips curves in a historical/revised-data context.
- The poorer real-time performance highlights the challenge for policymakers and forecasters who must rely on unrevised real-time GDP vintages: real-time estimates of potential output and output gaps are noisy and may weaken the practical forecasting value of Phillips-curve relationships.
- Output gaps can still be a useful tool in guiding forecast judgments, especially when used in conjunction with other indicators of spare capacity.
- The findings call more into question how to measure slack in the economy in real-time, rather than invalidating the theoretical Phillips curve.

### Data and measurement notes (datasets and series definitions preserved from the paper)
- United Kingdom real-time GDP dataset:
  - Matrix dimensions: 205 x 126, with 126 vintages starting in 1976Q1 and ending in 2007Q2.
  - The last and longest vintage contains 205 observations (1956Q1 until 2007Q1).
  - The series is considered the ex post GDP data in this paper.
  - Source: Bank of England’s GDP(E) real-time database.
- United States real-time GDP dataset:
  - Matrix dimensions: 245 x 171, with 171 vintages starting in 1965Q4 and ending in 2008Q2.
  - The last and longest vintage contains 245 observations (1947Q1 until 2008Q1).
  - The series is considered the ex post GDP data in this paper.
  - Source: Federal Reserve Bank of Philadelphia, ROUTPUT series.
- Germany real-time GNP/GDP dataset:
  - Matrix dimensions: 183 x 140, with 140 vintages starting in 1973Q1 and ending in 2007Q4.
  - The last and longest vintage contains 183 observations (1962Q1 until 2007Q3).
  - All 140 series were manually entered, with each series originating from a different issue of the Bundesbank’s Saisonbereinigte Wirtschaftszahlen.
  - The dataset used in this paper draws on Clausen and Meier (2005) for 1973Q1 to 1998Q4 and adds 36 vintages until 2007Q4.
- Series notation used in the paper:
  - a_t refers to the real-time output gap series.
  - c_t refers to the ex post output gap series.
  - b_t constitutes the quasi-real-time output gap series.
  - These series are estimated by the authors using the respective real-time GDP datasets.

### Recommended extensions and avenues for further research
- Examine sub-samples of the time periods chosen to assess whether the magnitude of revisions has changed over time and what effect this might have on forecast errors.
- Apply other methodologies of estimating the output gap to assess what these other methods imply for the differences between real-time and ex post estimates.
- Explore additional indicators of spare capacity to complement output gaps in real-time forecasting.

*Source: Conclusions section of the provided IMF working paper content.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp1052.pdf_