## wpiea2019183-print-pdf

## Source details

**Canonical URL:** [wpiea2019183-print-pdf](https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019183-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2019/wpiea2019183-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2019/wpiea2019183-print-pdf.pdf.json)

---

### Motivation and Contributions
- Main contributions:
  - The paper claims and elaborates that, properly defined, the output gap should have a negative mean over the long run.
  - Links the New-Keynesian model notion of the output gap (distance from the fully flexible price allocation) to the Friedman-Phelps concept of the output gap (the gap relevant to buildup of inflationary or deflationary pressures).
  - Shows that downward nominal wage rigidity implies a negative mean output gap in a New-Keynesian model extended with labor search and matching frictions.
  - Demonstrates that standard output gap estimation methods (filters) impose symmetry by constraining the estimated series average to zero, producing systematic bias when the true output gap has a non-zero mean.
  - Uses the Abbritti and Fahr (2013) model to simulate artificial data where the true output gap is observable and shows that standard filters underestimate potential output relative to the model-implied level.
- Relevance for policy:
  - The size and evolution of the output gap is central to rules-based interest rate setting (e.g., Taylor Rule) and structural budget balance calculations, which rely on an estimate of the output gap.
  - Biased estimates of potential output caused by symmetric filters can mislead monetary policy committees and fiscal authorities relying on structural budget concepts.
- Empirical findings cited:
  - International Wage Flexibility Project (IWFP) finds strong evidence of downward nominal wage rigidity with heterogeneity across countries (Dickens et al., 2007).
  - In the US, the share of workers with wage changes “frozen” at zero rose from 12% in 2006 to 16% in 2011 (Daly et al., 2013).

### Output Gaps, Wage Rigidity and Filters
- Conceptual linkages:
  - Friedman-Phelps “natural rate” conception: the natural rate of unemployment is isomorphic to a zero output gap where inflation is stable; if output/employment rise above this level, inflation accelerates; if below, inflation declines.
  - In New Keynesian models, price rigidities cause deviations of output from its natural level; fully flexible-price allocation corresponds to a closed output gap.
  - With labor search and matching frictions (Pissarides, 2000), the model’s output gap corresponds closely to the Friedman-Phelps concept: demand shocks coupled with sluggish price/wage adjustment produce inflation dynamics consistent with Friedman-Phelps.
- Intuition for negative mean output gap from downward nominal wage rigidity:
  - Empirical wage-change histograms show little weight below zero and a sharp spike at zero, indicating strong downward nominal wage rigidity.
  - When nominal wages cannot be cut, negative shocks induce larger quantity (employment) adjustments than price (wage) adjustments.
  - Symmetric positive and negative shocks yield cumulative negative deviations larger than cumulative positive deviations, producing an asymmetric cycle with a negative mean output gap.
  - Downward nominal wage rigidity binds more strongly in deep recessions than in shallow downturns; the negative mean is driven mainly by periods of elevated volatility.
- Quantitative model-based results:
  - The mean output gap in the Abbritti and Fahr (2013) model is negative, amounting to about -1% of potential output.
  - The negative mean is driven mainly by periods of elevated volatility; the average is closer to zero when turbulent periods are excluded.
  - Filters’ bias examples:
    - The Hodrick-Prescott (HP) filter has a mean-zero cyclical component by construction.
    - The IMF’s multivariate filter behaves similarly (approximately zero-mean).
    - Applying filters to simulated model data yields underestimated potential output (i.e., overestimated output gap).
    - The bias of the HP filter is about 1% of potential when averaged over the full generated sample.
    - The HP filter bias increases to about 3% when focusing on the most turbulent quarter of the sample.
- Mechanism of estimation bias:
  - Filters impose symmetry (zero mean) on estimated output gaps; when the true data-generating process has asymmetric adjustment (negative-mean gap), filters must “shift down” potential output so that estimated negative and positive deviations cancel, producing biased (downward) potential estimates.
  - Bias is most prominent in deep recessions when downward nominal wage rigidity is most binding.
- Related interpretive implications:
  - Results are consistent with critiques that official potential output estimates move with cyclical shocks that have little long-run impact (Coibion, Gorodnichenko and Ulate, 2019).
  - Fitting an HP filter to asymmetric time series generates potential output estimates that track demand shocks and are therefore biased, particularly after deep recessions (e.g., widespread downward revisions to potential output after the Great Recession).

### Model summary (selected features relevant to the output gap)
- Model: New-Keynesian DSGE with search and matching frictions and asymmetric wage adjustment costs (Abbritti and Fahr, 2013).
- Labor market: search and matching frictions; matching function with constant returns to scale; contemporaneous hiring.
- Households: representative household with pooled consumption; period utility log(Ct); unemployed earn bt; wages wt for employed.
- Demand shocks:
  - Risk-free nominal bonds yield gross nominal interest rate Rt exp(εb,t), with εb,t = ρb εb,t−1 + ηb,t and ηb,t ∼ N(0, σ2,b) — a serially correlated risk premium shock introducing demand fluctuations.
- Production: wholesale firms use capital and labor; productivity exp(Zt) with Zt = ρz Zt−1 + ηz,t and ηz,t ∼ N(0, σ2,z).
- Wage adjustment costs (asymmetric):
  - cw,t(πw,t) = φw −1 /2 (πw,t − γ)2 + 1/ψ2 {exp[−ψ(πw,t − γ)] + ψ(πw,t − γ) − 1}
  - πw,t is wage inflation; γ is long-run deterministic output growth rate in steady state.
  - φw governs convexity; ψ governs asymmetry (ψ > 0: reducing wage inflation below steady state is more costly than raising it).
- Wage determination: Nash bargaining with effective bargaining power ωt = η / [η + (1−η) τt,t+1], where τt,t+1 depends on cw,t and its derivatives and is state-dependent.
- Price setting: retail firms face Dixit-Stiglitz preferences; optimal price setting with convex adjustment costs yields a Phillips Curve: Γ′t Πt = ϵ(φt + Γt) − (ϵ − 1) + Et[βt,t+1 Yt+1/Yt Γ′t+1 Πt+1].
- Monetary policy: standard Taylor rule with interest rate smoothing (monetary policy shocks abstracted from in this paper).
- Output gap definition:
  - gapt = Yrigidityt − Yflext, where Yflext denotes output under price and wage flexibility and Yrigidityt output with price and wage rigidity.
  - This definition corresponds to the Friedman-Phelps conception: gapt = 0 implies stable inflation and unemployment equal to NAIRU.

### Calibration (section 2.3)
- Calibration strategy and targets:
  - Only parameters for exogenous shocks are adjusted relative to the baseline calibration of Abbritti and Fahr (2013).
  - The original shock calibration implies a standard deviation (SD) of gap_t of 0.55 percent of potential output.
  - This original SD is considered unreasonably small relative to empirical amplitudes of estimated output gaps for advanced economies (typically 3 to 4 times larger).
  - Calibration objective: match (i) the SD of IMF output gap estimates and (ii) the SD of IMF-estimated potential around its long-run trend, while leaving the mean of the output gap endogenously determined by the model.
- Data and empirical moments used:
  - Time series: IMF’s 2017 World Economic Outlook, 1981–2017.
  - Country focus: G7 countries (non-weighted average).
  - Target moments for the output gap from WEO (non-weighted G7 average):
    - SD of gap_t = 1.77% of potential output
    - First-order autocorrelation (AC) of gap_t = 0.61
- Demand shock calibration:
  - Parameters chosen: volatility parameter σ^2_b and AR(1) coefficient ρ_b.
  - σ^2_b and ρ_b are selected so that the model’s SD and first-order AC of gap_t match the WEO moments above.
  - Rationale: the model’s output gap is almost exclusively driven by demand shocks, so these parameters have a clear mapping into the SD and AC of the model’s output gap.
- Productivity (supply) shock calibration:
  - AR(1) coefficient of productivity shocks: ρ_z = 0.95.
  - The parameter σ^2_z governs the standard deviation of potential output around its long-run trend.
  - Empirical counterpart for σ^2_z:
    - Detrend IMF potential output series using multiple breakpoint test (eViews global information criteria), allowing up to 5 breaks in a regression of potential on a time trend and a constant, and allowing changes in trend growth at detected break points.
    - The standard deviation of the resulting residuals is 0.3% percent of potential (averaged over the G7 countries).
- Implications for model dynamics:
  - Productivity shocks induce comparatively weak but highly persistent movements of output around its long-run trend.
  - Demand shocks give rise to stronger but considerably less persistent fluctuations.
  - The calibrated magnitudes imply that output fluctuations in the model align well with empirical data.
- Simulation setup and sample statistics:
  - Simulated sample: 400,000 quarters of data.
  - Perturbation method: second order accurate perturbation to account for model asymmetries (following Abbritti and Fahr (2013)).
  - Over the full 400,000-period sample, the model-implied output gap averages -1.02 percent of potential output.
  - Sub-sample analysis (turbulent vs. calm windows defined by a 3-year moving average threshold):
    - When one third of the sample is counted as turbulent, the output gap averages -1.9% of potential in the turbulent third and -0.6% in the calm two-thirds.
  - Table of average output gap in sub-samples (true model-implied values, percent of potential output):
    - 10% / 90% split: -3.61 / -0.72
    - 25% / 75% split: -2.68 / -0.46
    - 33% / 67% split: -2.40 / -0.33
    - 50% / 50% split: -1.95 / -0.08
    - Full sample: -1.03
- Key interpretation:
  - Asymmetric wage adjustment costs, together with symmetric shocks, produce a negative mean output gap in the model because large negative demand shocks are amplified more than large positive shocks.
  - Calibration ensures demand shocks are the primary drivers of high-frequency output gap variability, while productivity shocks shape low-frequency, persistent movements in potential output.

### The Hodrick-Prescott Filter (section 5.1)
- Zero-mean property and mechanics:
  - The HP filter decomposes an output series into a trend component (interpreted as potential) and a cyclical component (interpreted as output gap).
  - The cyclical component from the HP filter has a mean of zero for any sample size (the appendix proves this).
  - For quarterly data the HP filtering in the illustrative exercise uses λ = 1600.
  - The HP construction forces estimated potential to “cut through” actual output so that cumulative deviations sum to zero; as a result estimated potential is centered on actual output.
- Quantitative bias in level of potential:
  - HP-estimated potential tends to lie below true potential when the true output gap is negative and overshoots when the true gap is positive; because negative gaps are more frequent and larger, this creates a negative average true gap relative to the HP estimate.
  - Sample results:
    - The HP-estimated gap has a mean of zero over the whole sample, while the true gap averages -1.02 percent of potential. The difference, -1.02 percent, is the filter’s average bias.
  - Sub-sample results:
    - Excluding the most turbulent third of the sample: true output gap averages -0.33 percent while the HP estimates have a mean of 0.09 percent; resulting bias is 0.39 percent on average.
    - Focusing on the most turbulent quarter of the sample: the bias amounts to 2.52 percent on average.
  - Interpretation:
    - The bias is predominantly driven by periods of strong macroeconomic turbulence, in particular severe recessions.
    - The bias is relatively weak most of the time but large during deep recessions, implying a danger of policy under-reaction if relying on HP estimates.
- Timing bias (signaling when gap crosses zero):
  - Using the Harding-Pagan business cycle dating algorithm:
    - A switching of the true output gap from negative to positive (upswings) is on average preceded by a switching of the estimated output gap 4.7 quarters earlier.
    - A switching of the true output gap from positive to negative (downturns) is on average followed by a switching of the estimated gap 0.5 quarters later.
  - Consequence: the HP filter tends to signal recovery too early in upswings and signal downturns too late.
- Mechanism linking downward nominal wage rigidity and HP bias:
  - Downward nominal wage rigidity causes negative deviations of output from potential to be larger in aggregate than positive deviations.
  - A filter constrained by the zero-mean property cannot accommodate the asymmetric integral between output and potential produced by such rigidity, and therefore reduces estimated potential (a downward shift) so that the integral of cyclical deviations is zero.
  - This manifests as:
    - Incorrect interpretation of negative demand shocks as negative supply shocks (reducing estimated potential).
    - Understating positive supply shocks by interpreting them as demand shocks.
  - Two-period illustration intuition: asymmetric adjustment (e.g., gaps -2% and +1%) yields net integral -1%, forcing a zero-mean filter to reinterpret part of the negative gap as a fall in potential.
- Policy-relevant implications:
  - Over-reliance on zero-mean filters (like HP) during deep downturns can cause policy under-reaction because estimated potential is biased downward when policy support is most needed.
  - Repeated downward revisions of estimated potential since 2008 are plausibly a consequence of the zero-mean property of filters rather than necessarily evidence of hysteresis.
  - Suggested policy implications:
    - Place greater reliance on observables such as inflation, unemployment and the level of public debt rather than just filter-implied output gaps.
    - Recognize that fiscal policy designed symmetrically around full employment will be expansionary more often than contractionary if the mean output gap is negative; fiscal consolidation may need to begin while the output gap remains negative to avoid explosive debt dynamics.
    - Taylor rules and optimal monetary policy may differ materially if time-series for the output gap are allowed to have negative means.

*Source: wpiea2019183-print-pdf (sections 1.1–1.2, 2.3, 5.1).*

### 1.1  Motivation and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   3

### 1.1 Motivation and Contributions

### Main contributions
- The paper makes and elaborates the claim that, properly defined, the output gap should have a negative mean over the long run.
- It links the New-Keynesian model notion of the output gap (distance from the fully flexible price allocation) to the Friedman-Phelps concept of the output gap (the gap relevant to buildup of inflationary or deflationary pressures).
- It shows that downward nominal wage rigidity implies a negative mean output gap in a New-Keynesian model extended with labor search and matching frictions.
- It demonstrates that standard output gap estimation methods (filters) impose symmetry by constraining the estimated series average to zero, producing systematic bias when the true output gap has a non-zero mean.
- Using the Abbritti and Fahr (2013) model, the paper simulates artificial data where the true output gap is observable and shows that standard filters underestimate potential output relative to the model-implied level.

### Relevance for policy
- The size and evolution of the output gap is central to both monetary and fiscal policy: rules-based interest rate setting (e.g., Taylor Rule) and structural budget balance calculations rely on an estimate of the output gap.
- Biased estimates of potential output caused by symmetric filters can mislead monetary policy committees and fiscal authorities relying on structural budget concepts.

### Summary empirical findings cited
- The International Wage Flexibility Project (IWFP) finds strong evidence of downward nominal wage rigidity with heterogeneity across countries (Dickens et al., 2007).
- In the US, the share of workers with wage changes “frozen” at zero rose from 12% in 2006 to 16% in 2011 (Daly et al., 2013).

---

### 1.2 Output Gaps, Wage Rigidity and Filters

### Conceptual linkage of gaps and frictions
- The Friedman-Phelps “natural rate” conception: the natural rate of unemployment is isomorphic to a zero output gap where inflation is stable; if output/employment rise above this level, inflation accelerates; if below, inflation declines.
- In New Keynesian models, price rigidities cause deviations of output from its natural level; fully flexible-price allocation corresponds to a closed output gap.
- When extended with labor search and matching frictions (Pissarides, 2000), the model’s output gap corresponds closely to the Friedman-Phelps concept: demand shocks coupled with sluggish price/wage adjustment move the economy above/below potential and generate inflation dynamics consistent with Friedman-Phelps.

### Intuition for negative mean output gap from downward nominal wage rigidity
- Downward nominal wage rigidity is empirically strong: histogram of wage changes shows little weight below zero and a sharp spike at zero.
- Because nominal wages are hard or impossible for employers to cut, negative shocks require relatively greater adjustment in quantities (employment) than in prices (wages).
- If positive and negative shocks are symmetric, cumulative negative deviations from potential exceed cumulative positive deviations, producing an asymmetric cycle with a negative mean output gap.
- Downward nominal wage rigidity binds more strongly in deep recessions than in shallow downturns, so the negative mean is driven mainly by periods of elevated volatility.

### Quantitative results from simulations (model-based)
- The mean output gap in the Abbritti and Fahr (2013) model is negative, amounting to about -1% of potential output.
- The negative mean is driven mainly by periods of elevated volatility; the average is closer to zero when turbulent periods are excluded.
- Filters’ bias examples:
  - The Hodrick-Prescott (HP) filter has a mean-zero cyclical component by construction.
  - The IMF’s multivariate filter behaves similarly (approximately zero-mean).
  - Applying filters to simulated model data yields underestimated potential output (i.e., overestimated output gap).
  - The bias of the HP filter is about 1% of potential when averaged over the full generated sample.
  - The HP filter bias increases to about 3% when focusing on the most turbulent quarter of the sample.

### Mechanism of estimation bias
- Filters impose symmetry (zero mean) on estimated output gaps; when the true data-generating process has asymmetric adjustment (negative-mean gap), filters must “shift down” potential output so that estimated negative and positive deviations cancel, producing biased (downward) potential estimates.
- Bias is most prominent in deep recessions when downward nominal wage rigidity is most binding.

### Related empirical and interpretive implications
- Results are consistent with critiques that official potential output estimates move with cyclical shocks that have little long-run impact (Coibion, Gorodnichenko and Ulate, 2019).
- Fitting an HP filter to asymmetric time series generates potential output estimates that track demand shocks and are therefore biased, particularly after deep recessions (e.g., widespread downward revisions to potential output after the Great Recession).

---

### Model summary (selected features relevant to the output gap)
- Model: New-Keynesian DSGE with search and matching frictions and asymmetric wage adjustment costs (Abbritti and Fahr, 2013).
- Labor market: search and matching frictions; matching function with constant returns to scale; contemporaneous hiring.
- Households: representative household with pooled consumption; period utility log(Ct); unemployed earn bt; wages wt for employed.
- Demand shocks: risk-free nominal bonds yield gross nominal interest rate Rt exp(εb,t), with εb,t = ρb εb,t−1 + ηb,t and ηb,t ∼ N(0, σ2,b) — a serially correlated risk premium shock introducing demand fluctuations.
- Production: wholesale firms use capital and labor; productivity exp(Zt) with Zt = ρz Zt−1 + ηz,t and ηz,t ∼ N(0, σ2,z).
- Wage adjustment costs (asymmetric):
  - cw,t(πw,t) = φw −1 /2 (πw,t − γ)2 + 1/ψ2 {exp[−ψ(πw,t − γ)] + ψ(πw,t − γ) − 1}
  - πw,t is wage inflation; γ is long-run deterministic output growth rate in steady state.
  - φw governs convexity; ψ governs asymmetry (ψ > 0: reducing wage inflation below steady state is more costly than raising it).
- Wage determination: Nash bargaining with effective bargaining power ωt = η / [η + (1−η) τt,t+1], where τt,t+1 depends on cw,t and its derivatives and is state-dependent.
- Price setting: retail firms face Dixit-Stiglitz preferences; optimal price setting with convex adjustment costs yields a Phillips Curve: Γ′t Πt = ϵ(φt + Γt) − (ϵ − 1) + Et[βt,t+1 Yt+1/Yt Γ′t+1 Πt+1].
- Monetary policy: standard Taylor rule with interest rate smoothing (monetary policy shocks abstracted from in this paper).
- Output gap definition:
  - gapt = Yrigidityt − Yflext, where Yflext denotes output under price and wage flexibility and Yrigidityt output with price and wage rigidity.
  - This definition corresponds to the Friedman-Phelps conception: gapt = 0 implies stable inflation and unemployment equal to NAIRU.

---

*Source: wpiea2019183-print-pdf (sections 1.1–1.2).*

### 2.3  Calibration

### 2.3  Calibration

### Calibration strategy and targets
- Only parameters for exogenous shocks are adjusted relative to the baseline calibration of Abbritti and Fahr (2013).
- The original shock calibration implies a standard deviation (SD) of gap_t of 0.55 percent of potential output.
- This original SD is considered unreasonably small relative to empirical amplitudes of estimated output gaps for advanced economies (typically 3 to 4 times larger).
- Calibration objective: match (i) the SD of IMF output gap estimates and (ii) the SD of IMF-estimated potential around its long-run trend, while leaving the mean of the output gap endogenously determined by the model.

### Data and empirical moments used
- Time series: IMF’s 2017 World Economic Outlook, 1981–2017.
- Country focus: G7 countries (non-weighted average).
- Target moments for the output gap from WEO (non-weighted G7 average):
  - SD of gap_t = 1.77% of potential output
  - First-order autocorrelation (AC) of gap_t = 0.61

### Demand shock calibration
- Parameters chosen: volatility parameter σ^2_b and AR(1) coefficient ρ_b.
- Calibration mapping: σ^2_b and ρ_b are selected so that the model’s SD and first-order AC of gap_t match the WEO moments above.
- Rationale: the model’s output gap is almost exclusively driven by demand shocks, so these parameters have a clear mapping into the SD and AC of the model’s output gap.

### Productivity (supply) shock calibration
- AR(1) coefficient of productivity shocks: ρ_z = 0.95 (standard value; highly persistent shocks).
- The parameter σ^2_z governs the standard deviation of potential output around its long-run trend.
- Empirical counterpart for σ^2_z:
  - Detrend IMF potential output series using multiple breakpoint test (eViews global information criteria), allowing up to 5 breaks in a regression of potential on a time trend and a constant, and allowing changes in trend growth at detected break points.
  - The standard deviation of the resulting residuals is 0.3% percent of potential (averaged over the G7 countries).

### Implications of calibration for model dynamics
- Productivity shocks: induce comparatively weak but highly persistent movements of output around its long-run trend.
- Demand shocks: give rise to stronger but considerably less persistent fluctuations.
- The calibrated magnitudes imply that output fluctuations in the model align well with empirical data.

### Simulation setup and sample statistics (from subsequent model analysis)
- Simulated sample: 400,000 quarters of data.
- Perturbation method: second order accurate perturbation to account for model asymmetries (following Abbritti and Fahr (2013)).
- Over the full 400,000-period sample, the model-implied output gap averages -1.02 percent of potential output.
- Sub-sample analysis (turbulent vs. calm windows defined by a 3-year moving average threshold):
  - Example: when one third of the sample is counted as turbulent, the output gap averages -1.9% of potential in the turbulent third and -0.6% in the calm two-thirds.
- Table of average output gap in sub-samples (true model-implied values):
  - 10% / 90% split: -3.61 / -0.72
  - 25% / 75% split: -2.68 / -0.46
  - 33% / 67% split: -2.40 / -0.33
  - 50% / 50% split: -1.95 / -0.08
  - Full sample: -1.03
  - Note: output gaps reported in percent of potential output.

### Key interpretation arising from the calibration and model
- Asymmetric wage adjustment costs, together with symmetric shocks, produce a negative mean output gap in the model because large negative demand shocks are amplified more than large positive shocks.
- The calibration ensures that demand shocks are the primary drivers of high-frequency output gap variability, while productivity shocks shape low-frequency, persistent movements in potential output.

*Source: wpiea2019183-print-pdf - 2.3  Calibration*

### 5.1   The Hodrick-Prescott Filter

### 5.1   The Hodrick-Prescott Filter

### Zero-mean property and basic mechanics
- The HP filter decomposes an output series into a trend component (interpreted as potential) and a cyclical component (interpreted as output gap).
- The cyclical component from the HP filter has a mean of zero for any sample size (the appendix proves this).
- For quarterly data the HP filtering in the illustrative exercise uses λ = 1600 (as suggested by Ravn and Uhlig, 2002).
- The HP construction forces estimated potential to “cut through” actual output so that cumulative deviations sum to zero; as a result estimated potential is centered on actual output.

### Quantitative bias in level of potential
- HP-estimated potential tends to lie below true potential when the true output gap is negative and overshoots when the true gap is positive; because negative gaps are more frequent and larger, this creates a negative average true gap relative to the HP estimate.
- Sample results reported:
  - The HP-estimated gap has a mean of zero over the whole sample, while the true gap averages -1.02 percent of potential. The difference, -1.02 percent, is the filter’s average bias.
- Sub-sample (turbulent vs calm) results (as discussed and reported in Table 2):
  - Excluding the most turbulent third of the sample: true output gap averages -0.33 percent while the HP estimates have a mean of 0.09 percent; resulting bias is 0.39 percent on average.
  - Focusing on the most turbulent quarter of the sample: the bias amounts to 2.52 percent on average.
- Interpretation:
  - The bias is predominantly driven by periods of strong macroeconomic turbulence, in particular severe recessions.
  - The bias is relatively weak most of the time but large during deep recessions, implying a danger of policy under-reaction if relying on HP estimates.

### Timing bias (signaling when gap crosses zero)
- Using the Harding-Pagan business cycle dating algorithm to identify upswings and downturns:
  - A switching of the true output gap from negative to positive (upswings) is on average preceded by a switching of the estimated output gap 4.7 quarters earlier.
  - A switching of the true output gap from positive to negative (downturns) is on average followed by a switching of the estimated gap 0.5 quarters later.
- Consequence: the HP filter tends to signal recovery too early in upswings and signal downturns too late.

### Mechanism linking downward nominal wage rigidity and HP bias
- Downward nominal wage rigidity causes negative deviations of output from potential to be larger in aggregate than positive deviations.
- A filter constrained by the zero-mean property cannot accommodate the asymmetric integral between output and potential produced by such rigidity, and therefore reduces estimated potential (a downward shift) so that the integral of cyclical deviations is zero.
- This manifests as:
  - Incorrect interpretation of negative demand shocks as negative supply shocks (reducing estimated potential).
  - Understating positive supply shocks by interpreting them as demand shocks.
- Example intuition (two-period illustration in text): asymmetric adjustment (e.g., gaps -2% and +1%) yields net integral -1%, forcing a zero-mean filter to reinterpret part of the negative gap as a fall in potential.

### Policy-relevant implications from the HP filter bias
- Over-reliance on zero-mean filters (like HP) during deep downturns can cause policy under-reaction because estimated potential is biased downward when policy support is most needed.
- Repeated downward revisions of estimated potential since 2008 are plausibly a consequence of the zero-mean property of filters rather than necessarily evidence of hysteresis.
- Suggested policy implications (developed further in the paper):
  - Place greater reliance on observables such as inflation, unemployment and the level of public debt rather than just filter-implied output gaps.
  - Recognize that fiscal policy designed symmetrically around full employment will be expansionary more often than contractionary if the mean output gap is negative; fiscal consolidation may need to begin while the output gap remains negative to avoid explosive debt dynamics.
  - Taylor rules and optimal monetary policy may differ materially if time-series for the output gap are allowed to have negative means.

*Source: IMF Working Paper chapter section "5.1   The Hodrick-Prescott Filter" (from the supplied PDF content).*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019183-print-pdf.pdf_