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---

### I. Introduction and framing
- Stars and steady state:
  - Steady state described by: natural rate of unemployment (u*), neutral interest rate (r*), level of potential output (y*), growth rate of potential output (g*), central bank inflation target (π*).
  - Deviation of contemporaneous values from steady state defines cyclical position; policy stance measured as policy rate minus neutral rate.
- Empirical tension:
  - Over the last 30 years there has been a persistent shortfall in the inflation gap (π−π*) while existing output gap estimates suggest, on average, lower shortfalls of output and employment than implied by the inflation gap.

### Challenges in estimating the stars
- Sensitivity and instability:
  - FRB/US output gap estimates vary with estimation window, indicating parameter instability.
  - Laubach and Williams (2003) benchmark neutral rate model’s output gap estimates are rarely statistically different from zero in the last three decades.
- Data revisions example:
  - August 2018 revisions to Laubach-Williams: potential GDP growth down by around 0.4 percentage points; neutral rate revised up by around 0.6 percentage points.
- Policy implications of mismeasurement:
  - Incorrect output gap measurement can lead to suboptimal fiscal/monetary policy and affects assessments of external positions.
- Operational policymaker response:
  - Policymakers have often used inertial policy stances and allowed the economy to “probe for the stars.”

### Role of the inflation target
- Inflation target π* is under policymakers’ control; many central banks set low-but-positive π* (usually close to 2 percent in advanced economies).
- Relationship between realized inflation and π* is central to locating other stars.

### Modeling approach and key ingredients
- Four key ingredients (in rough order of importance):
  1. Identifying restrictions:
     - Equilibrium defined as inflation at the FOMC’s inflation target (2 percent for most of the sample).
     - Wage growth = trend productivity growth + long-run inflation expectations.
     - Federal funds rate = neutral rate; unemployment = NAIRU; output = potential.
  2. Broader data set for triangulation:
     - Uses inflation, labor market outcomes, wages, and non-oil current account to narrow statistical uncertainty.
     - Multivariate approaches tend to yield lower errors in cycle estimation.
  3. Estimation window:
     - Sample: 1990Q1-2019Q4 to enhance parameter stability.
  4. Careful specification balancing additional data and identifying restrictions.

### Major empirical findings and calibrated "stars"
- End-2019 estimates and stylized stars:
  - Output gap at end-2019: 0.7 percent.
  - Estimated NAIRU: around, or slightly below, 4 percent.
  - Potential growth rate: around 1.6 percent; fell for nearly 15 years, troughing in 2013, then rebounded slightly.
  - Real neutral policy rate consistent with 2 percent inflation: modestly negative; implies neutral nominal fed funds rate between 1.75 and 2 percent.
- Additional findings:
  - Slack estimates align with recent inflation behavior and consumer sentiment measures.
  - FOMC’s 2019 rate cuts interpretable as reaction to policy becoming slightly tight during 2018 given lower-than-appreciated neutral rate.
  - Model does not suggest sharply overheating mid-2000s economy; housing bubble largely shows up as declining productivity growth.

### Model structure and core equations
- Trend-cycle state space decomposition: observed series = cycle (cyc_t and lags) + time-varying trend + sometimes exogenous drivers (τ_t).
- Core three-equation New Keynesian structure:
  - Business cycle AR with policy deviations:
    - cyc_t = ρ1 cyc_{t−1} + ρ2 cyc_{t−2} + ϑ1 1/2 ∑_{i=1}^{2} (r_{t−i} − r_{t−i}^*)^2 + ε_t^cyc
    - Stationarity: |ρ1 + ρ2|<1.
  - Phillips Curve:
    - π_t − π_t^* = λ1 1/4 ∑_{i=1}^{4} cyc_{t−i} + β_{i1} rpi_{t−1} + β_{i2} Δ2yr(lpg_t^*) + ε_t^π
    - Long-run inflation expectations π_t^* from SPF survey adjusted to PCE; after 1995 equivalent to FOMC target.
  - Inertial Taylor Rule for real policy gap:
    - r_t − r_t^* = ρ3(r_{t−1} − r_{t−1}^*) + ρ4(r_{t−2} − r_{t−2}^*) + ϑ2 cyc_{t−1} + ϑ3 π̃_{t−1} + ε_t^r
    - π̃_t = 1/4 ∑_{i=1}^{4} cyc_{t−i} + ε_t^π̃
    - Neutral rate r_t^* follows a unit root: r_t^* = r_{t−1}^* + ε_t^{r^*}
    - Lag structure allows non-monotonic policy paths; r^* not forced to track potential growth.
- Shadow rate and real rates:
  - Wu-Xia (2015) shadow fed funds rate used for unconventional policy at ELB.
  - Real rates = nominal rate deflated by y/y change in core PCE.
- Additional identities:
  - GDP and GDI: gdp_t = cyc_t + gdpo_t^* + ε_t^{gdp}; gdi_t = cyc_t + gdpo_t^* + ε_t^{gdi}; σ_u^{gdp} > σ_u^{gdi}.
  - Trend output unit root: gdpo_t^* = gdpo_{t−1}^* + g_{t−1}^*.
  - Potential growth: g_t^* = lfg_t^* + lpg_t^* (both follow unit roots).
  - NAIRU (U3) random walk: U3_t^* = U3_{t−1}^* + ε_t^{nairu}.
  - Okun’s laws and wage equations provided linking cyclical movements to unemployment, PAEPOP, and wage growth.
  - Current account relative to trend linked to U.S. business cycle via cabxpo_t − cabxpo_t^* = λ7 ∑_{i=0}^{1} cyc_{t−i} + β3 cyc_t^{ROW} + ε_t^{gtb}.

### Estimation approach and identifying assumptions
- Estimation via Kalman Filter over 1990Q1-2019Q4.
- Additional identifying restrictions:
  - Measurement error for two inflation and wage measures assumed equal; slack coefficients allowed to differ.
  - Standard deviation of measurement error of Wu-Xia shadow rate set to 25 basis points.
  - Cycle/trend volatility ordering: σ(ε_t^{cyc}) > σ(ε_t^{lpg}) > σ(ε_t^{lfg}); σ(ε_t^{r−r^*}) > σ(ε_t^{r^*}).
  - Variance of trend unemployment set equal to CBO long-run NAIRU estimate: σ(ε_t^{u*}) = 0.15.
- Nine independent state variables estimated; richer cyclical information (including wages) helps identify trend productivity and potential growth.

### Key parameter estimates (Table 2)
- Dynamic equations:
  - IS-Equation: ρ1 = 1.67, ρ2 = -0.69, θ = -0.02
  - Phillips Curve: λ1 = 0.15
  - Taylor Rule: ρ3 = 1.6, ρ4 = -0.7, θ1 = 0.048, θ2 = 0.036
- Measurement equations:
  - Okun’s Law – U3: λ2 = -0.1, λ3 = -0.52
  - Okun’s Law – PAEPOP: λ4 = 0.06, λ5 = 0.59
  - Wage Phillips Curve slope: λ6 = 0.19

### Phillips Curve comparison and implications for slack measurement
- Anchored-expectations Phillips Curve:
  - Anchors inflation around long-run expectations π_t^*, yielding tighter inflation-cycle linkage and smaller standard errors for the output gap.
- Contrasts:
  - Laubach-Williams uses accelerationist Phillips Curve with inflation unit root; unexplained inflation noisy and weakly cyclical.
  - FRB/US uses anchored expectations New Keynesian Phillips Curve with lagged deviations and more detailed supply structure.
- Model fit (P.C. standard error, q/q saar, core PCE):
  - Laubach-Williams - Full sample: 0.76
  - FRB/US beginning in 1965: 0.78
  - FRB/US beginning in 1990: 0.5
  - Williams, Abdih, and Kopp: 0.52
- Conclusion: anchoring around expectations improves precision and analytic clarity relative to unit-root inflation specifications.

### Uncertainty, model comparison, and quasi-real-time performance
- Output gap estimation certainty (expanding window, end points 2008Q1-2019Q3):
  - Laubach-Williams (beginning in 1965): Avg. s.e. 2.28, Final s.e. 3.10, RMSE 1.43, Revision Ratio 1.34
  - FRB/US (beginning in 1965): Avg. s.e. 0.85, Final s.e. 0.98, RMSE 0.85, Revision Ratio 0.30
  - FRB/US (beginning in 1990): Avg. s.e. 0.62, Final s.e. 0.8, RMSE 0.66, Revision Ratio 0.24
  - Williams, Abdih, and Kopp (most basic model): Avg. s.e. 0.16, Final s.e. 0.3, RMSE 0.48, Revision Ratio 0.15
  - Full model: Avg. s.e. 0.13, Final s.e. 0.22, RMSE 0.7, Revision Ratio 0.22
- Notes:
  - "s.e." denotes standard error of the output gap.
  - RMSE is root mean squared error between final smoothed estimate and expanding window end-point estimates.
  - Revision ratio = RMSE / standard deviation of the output gap over 2008-2019; scale invariant.
  - FRB/US supply-side model results end in 2015 due to data availability.
  - Estimates of g* and r* are less precise than output gap.

### Main results, interpretation, and policy implications
- Baseline characterization:
  - Estimated "stars" point to a more pessimistic narrative over the last 20 years than most other models.
  - Cyclical component dynamics very slow moving (ρ1 + ρ2 close to 1).
  - Policy rate has only a small effect on the cycle; impact of slack on wage and prices small.
- Output gap, potential growth, and r*:
  - Output gap more negative after the financial crisis than other work suggests.
  - Contemporaneous neutral real federal funds rate (r*) modestly negative and well below most other studies.
  - Long-run decline in r* consistent with term premium model results.
- Labor market:
  - Labor market indicators are central due to stationarity and lower noise; NAIRU is time varying but moves slowly and declines.
- Hysteresis and structural drivers:
  - Hysteresis possible but not strong; trends longer than traditional business cycles complicate extraction of stars.
  - Timing of r* and g* movements point to links with housing boom-and-bust, household debt, or end of IT investment boom.
- Policy implications highlighted:
  - Financial excesses are best addressed by regulatory policy rather than monetary policy focused on inflation and employment.
  - Lower r* implies federal funds rate was closer to neutral in recent years, so monetary policy may have been less accommodative than thought.
  - Low r* increases importance of counter-cyclical fiscal policy and affects assessments of government debt paths.
  - Persistently low rates could foster financial fragility through risk-taking and leverage.
  - Labor-market slack larger than previously thought; integrate extensive margins of slack into business cycle modeling.

*Source: wpiea2020136-print-pdf*

### REFERENCES .............................................................................................................

### REFERENCES

### I. INTRODUCTION
- In standard macroeconomic models, key variables fluctuate around their “stars” in response to short-term shocks and the business cycle. The stars define the economy’s longer-term steady state after cyclical shocks and short-term rigidities dissipate.
- The steady state can be described by: the natural rate of unemployment (u*), the neutral interest rate (r*), the level of potential output (y*), the growth rate of potential output (g*), and the central bank’s inflation target (휋*).
- The deviation of contemporaneous values from the steady state defines the cyclical position of the economy; the difference between the policy rate and the neutral interest rate measures the stance of monetary policy.
- Policymakers use these “stars” to conduct monetary and fiscal policy, but the steady state is not directly observable and changes over time, leading to significant model and definitional uncertainty—especially around cyclical turning points, following data revisions, or during periods of structural change.
- Empirical tension noted: over the last 30 years there has been a persistent shortfall in the inflation gap (휋−휋*) while existing estimates of the output gap suggest, on average, lower shortfalls of output and employment over the period than implied by the inflation gap.

### CHALLENGES IN ESTIMATING THE STARS
- Estimation sensitivity:
  - The Federal Reserve Board’s FRB/US model produces very different output gap estimates depending on the estimation window, indicating parameter instability and sensitivity to changes in inflation behavior and FOMC behavior.
  - The Laubach and Williams (2003) benchmark neutral rate model’s output gap estimates are rarely statistically different from zero in the last three decades, suggesting stable inflation complicates some estimation approaches.
- Data revisions example:
  - August 2018 revisions to the Laubach-Williams model estimates: benchmark historical revisions to U.S. GDP led to a revision of potential GDP growth downwards by around 0.4 percentage points and the neutral rate of interest revised up by around 0.6 percentage points.
- Policy implications:
  - Incorrect measurement of the output gap can lead to suboptimal policies and is particularly problematic for countries aiming to follow a cyclically adjusted fiscal rule or that have a counter-cyclical capital buffer.
  - Assessments of a country’s external position can be sensitive to the measurement of the output gap.
- Operational response by policymakers:
  - Given estimation challenges, policymakers have often adopted relatively inertial policy stances and allowed the economy to “probe for the stars” rather than reacting strongly to preexisting notions about them.

### ROLE OF THE INFLATION TARGET
- Unlike other stars, the inflation target 휋* is under policymakers’ direct control; many central banks have chosen low-but-positive values for 휋* (usually close to 2 percent in advanced economies).
- The relationship between realized inflation and the inflation target (assumed equivalent to equilibrium inflation) is a key determinant in estimating the location of other stars.

### MODELING APPROACH AND KEY INGREDIENTS
- The modeling approach to estimate stars uses four key ingredients (in rough order of importance):
  1. Identifying restrictions:
     - Equilibrium defined as inflation at the FOMC’s inflation target (2 percent for most of the sample).
     - Wage growth equal to trend productivity growth plus long-run inflation expectations.
     - Federal funds rate equal to the neutral rate.
     - Unemployment at the NAIRU.
     - Output equal to potential.
  2. Broader data set for triangulation:
     - Uses inflation, labor market outcomes, wages, and the non-oil current account to narrow statistical uncertainty, at the cost of introducing more parameters and requiring additional identifying restrictions and careful modeling.
     - Multivariate approaches tend to yield lower errors in estimating the cycle (citing Stock and Watson (1998) and Fleischman and Roberts (2011)).
  3. Estimation window:
     - The estimation window is from 1990Q1-2019Q4.
     - This shorter window enhances parameter stability by ensuring the Phillips Curve and reaction function are relatively constant over the sample; extending much further would reduce this stability.
  4. (Implicit) Careful specification to balance additional data and identifying restrictions.

### FIGURES AND TABLES (AS LISTED)
- Tables:
  - 1. Phillips Curve Accuracy
  - 2. Key Parameter Estimates
  - 3. Output Gap Estimation Certainty
- Figures:
  - 1. Inflation and Output Gap Estimates
  - 2. Output Gap Uncertainty
  - 3. Drivers of the Business Cycle
  - 4. What the Phillips Curve Implies about Slack
  - 5. A Comparison of Stars
  - 6. Quasi-Real Time Estimate of the Output Gap
- Notes on figures cited in text:
  - Figure 1 highlights tension between the inflation gap (휋−휋*) shortfall and existing output gap estimates over the last 30 years.
  - Figure 2 illustrates output gap uncertainty: FRB/US output gap estimates vary with estimation window; Laubach-Williams output gap estimates include standard error bands and are often close to zero for the last three decades.
  - FRB/US supply-side variables are estimated in a state space model as outlined in Fleischman and Roberts (2011) and Roberts (2014); recent estimation stops in 2015 due to public data availability for the supply-side model’s separate database.

*Source: wpiea2020136-print-pdf - REFERENCES*

### 4.      The central bank’s policy reaction function is explicitly modeled (as an inertial

### 4.      The central bank’s policy reaction function is explicitly modeled (as an inertial variant of a Taylor rule)

### Major empirical findings and calibrated "stars"
- The U.S. had a relatively modest output gap at end-2019 of 0.7 percent.  
- The estimated NAIRU is around, or slightly below, 4 percent.  
- The potential growth rate is around 1.6 percent; it fell for nearly 15 years, troughing in 2013, and then rebounded slightly from these post-Crisis lows.  
- The real neutral policy rate that is consistent with 2 percent inflation is modestly negative; this implies that the neutral nominal fed funds rate is between 1.75 and 2 percent.  
- The model’s slack estimates align with recent inflation behavior and with measures of consumer (that is, citizen) sentiment.  
- The FOMC’s 2019 rate cuts can be interpreted as a reaction to policy becoming slightly tight during 2018 given the lower-than-commonly-appreciated level of the neutral rate.  
- The model does not suggest a sharply over-heating economy in the mid-2000s; the housing bubble largely shows up as declining productivity growth.

### Model structure and core equations
- Trend-cycle decomposition in a state space framework: observed series = cycle (cyc_t and lags) + time-varying trend (⋆) + sometimes exogenous drivers (τ_t).
- Core three-equation New Keynesian structure linking output and inflation gaps and policy stance:
  - Equation 1 (business cycle AR process including real policy rate deviations):
    - cyc_t = ρ1 cyc_{t−1} + ρ2 cyc_{t−2} + ϑ1 1/2 ∑_{i=1}^{2} (r_{t−i} − r_{t−i}^*)^2 + ε_t^cyc
    - Stationarity ensured by |ρ1 + ρ2|<1.
    - Cycle estimated around GDP and Gross Domestic Income (GDI).
  - Equation 2 (Phillips Curve form used here):
    - π_t − π_t^* = λ1 1/4 ∑_{i=1}^{4} cyc_{t−i} + β_{i1} rpi_{t−1} + β_{i2} Δ2yr(lpg_t^*) + ε_t^π
    - Long-run inflation expectations π_t^* are based on the SPF survey adjusted into PCE equivalent terms; after 1995 equivalent to the FOMC’s target.
    - Includes change in relative import prices and medium-term changes in productivity growth.
  - Equation 3 (inertial Taylor Rule for the real policy gap):
    - r_t − r_t^* = ρ3(r_{t−1} − r_{t−1}^*) + ρ4(r_{t−2} − r_{t−2}^*) + ϑ2 cyc_{t−1} + ϑ3 π̃_{t−1} + ε_t^r
    - π̃_t = 1/4 ∑_{i=1}^{4} cyc_{t−i} + ε_t^π̃
    - Neutral rate r_t^* follows a unit root: r_t^* = r_{t−1}^* + ε_t^r
    - Lag structure allows non-monotonic policy paths; neutral rate not forced to track potential growth.

- Shadow rate and real rates:
  - Wu-Xia (2015) shadow fed funds rate used to incorporate unconventional policy effects at the effective lower bound.
  - Real rates = nominal rate deflated by y/y change in core PCE.

- Additional identities and dynamics:
  - GDP and GDI decomposition: each series = common cycle + trend output + measurement error.
    - gdp_t = cyc_t + gdpo_t^* + ε_t^{gdp}
    - gdi_t = cyc_t + gdpo_t^* + ε_t^{gdi}
    - σ_u^{gdp} > σ_u^{gdi}
  - Trend output unit root: gdpo_t^* = gdpo_{t−1}^* + g_{t−1}^*
  - Potential growth components: g_t^* = lfg_t^* + lpg_t^* (trend labor force growth + trend labor productivity); both follow unit roots.
  - NAIRU (based on U3) follows a random walk: U3_t^* = U3_{t−1}^* + ε_t^{nairu}
  - Okun relationships:
    - U3_t − U3_t^* = λ2 cyc_t + λ3 cyc_{t−1} + ε_t^{u3}
    - Prime-age EPOP: PAEPOP_t − PAEPOP^* = λ4 cyc_t + λ5 1/3 ∑_{i=1}^{3} cyc_{t−i} + ε_t^{PAEPOP}, with PAEPOP^* constant in this specification.
  - Nominal wage growth (ECI and Atlanta Fed tracker):
    - wage growth_t − (π_t^* + lpg_t^*) = λ6 1/4 ∑_{i=0}^{3} cyc_{t−i} + ε_t^{wages}
  - Current account relative to trend linked to U.S. business cycle:
    - cabxpo_t − cabxpo_t^* = λ7 ∑_{i=0}^{1} cyc_{t−i} + β3 cyc_t^{ROW} + ε_t^{gtb}

### Estimation approach and identifying assumptions
- Estimation via Kalman Filter over 1990Q1-2019Q4.
- Rationale for sample: Phillips Curve and central bank reaction function viewed as more stable since late-1980s.
- Use of broader data set and explicit identifying assumptions reduces common estimation problems (e.g., "pile-up" for trend states).
- Additional identifying restrictions:
  - Measurement error for the two inflation and wage measures assumed equal; coefficients on slack allowed to differ.
  - Standard deviation of measurement error of the Wu-Xia shadow rate is set to 25 basis points.
  - Cycles more volatile than trends: σ(ε_t^{cyc}) > σ(ε_t^{lpg}) > σ(ε_t^{lfg}), and σ(ε_t^{r−r^*}) > σ(ε_t^{r^*}).
  - Variance of trend unemployment set equal to CBO’s long-run NAIRU estimate: σ(ε_t^{u*}) = 0.15.
- Nine independent state variables to estimate; richer cyclical information (including wages) helps pin down trend productivity and potential growth.

### Phillips Curve comparison and implications for slack measurement
- This framework’s Phillips Curve is anchored around long-run inflation expectations (π_t^*); in equilibrium (four-quarter average output at potential, import price effects wash out, productivity at long-term value) inflation equals expectations.
- Anchoring inflation around expectations gives tighter linkage between inflation and the cyclical component and yields smaller standard errors for the output gap.
- Contrasts with other approaches:
  - Laubach-Williams: accelerationist Phillips Curve with inflation having a unit root and not anchored to a long-run value; unexplained inflation component is noisy and weakly cyclical, producing output gap estimates not necessarily consistent with Fed target or long-run inflation expectations.
  - FRB/US: anchored expectations New Keynesian Phillips Curve with lagged deviations allowing short-term persistence; more detailed supply structure.
- Model fit for core PCE (P.C. standard error, q/q saar) reported:
  - Laubach-Williams - Full sample: 0.76
  - FRB/US beginning in 1965: 0.78
  - FRB/US beginning in 1990: 0.5
  - Williams, Abdih, and Kopp: 0.52
- Anchored-expectations Phillips Curve used here improves precision and analytic clarity in estimating the unobserved output gap relative to unit-root inflation formulations.

### Conceptual policy implications highlighted
- By linking slack, the neutral rate, and inflation, the model implies financial excesses are best addressed by regulatory policy rather than by monetary policy focused on inflation and employment outcomes.  
- The model’s assessment of a modest output gap and low neutral rate alters the interpretation of monetary policy tightness and the rationale for policy moves such as the FOMC’s 2019 rate cuts.  

*Source: IMF staff authors’ estimates and model documentation (model estimated over 1990Q1–2019Q4).*

### 2. Our baseline model is estimated on y/y core PCE inflation but here we use q/q saar inflation, without allowing for

### 2. Our baseline model is estimated on y/y core PCE inflation but here we use q/q saar inflation, without allowing for

### Model specification and data
- The baseline model in this section is estimated on q/q saar inflation (core PCE) without allowing for any additional shorter-term dynamics, to make estimates comparable with other models.
- Including moving average terms would have improved fit without altering interpretation (see Kim (2009)).
- Sources referenced for data and estimates: Federal Reserve Bank of New York, Federal Reserve Board of Governors, and authors estimates.

### Key parameter estimates (Table 2)
- Dynamic Equations
  - IS-Equation: ρ1 = 1.67, ρ2 = -0.69, θ = -0.02
  - Phillips Curve: λ1 = 0.15
  - Taylor Rule: ρ3 = 1.6, ρ4 = -0.7, θ1 = 0.048, θ2 = 0.036
- Measurement Equations
  - Okun’s Law – U3: λ2 = -0.1, λ3 = -0.52
  - Okun’s Law – PAEPOP: λ4 = 0.06, λ5 = 0.59
  - Wage Phillips Curve slope: λ6 = 0.19
- Sources: Authors Estimates

### Main results and interpretation (V. RESULTS – MORE SLACK AND LOWER RATES)
- General characterization
  - The estimated "stars" generally suggest a more pessimistic narrative for the economy over the last 20 years than most other models.
  - Parameters governing dynamics and internal relationships are consistent with various other studies in the literature.
- Properties of dynamics and transmission
  - The dynamics of the cyclical component is very slow moving (휌1 + 휌2 close to 1).
  - The policy rate has only a small effect on the cycle.
  - The impact of slack on wage and prices is small (consistent with well-anchored expectations).
  - The Okun’s law coefficient is broadly in line with other estimates.
- Output gap, potential growth, and r*
  - The output gap is estimated as being more negative following the financial crisis than in other work.
  - Broader dataset allows extensive margins of labor supply to influence the split between trend and cycle.
  - Output remains below potential for an extended part of the chosen sample period (consistent with below target inflation).
  - Following a secular decline in the potential growth rate between the late 1990s and 2010, there is a modest improvement in potential growth over the past 5 or 6 years.
  - The contemporaneous estimate of the neutral real federal funds rate (r*) is modestly negative (well below most other studies).
  - The long-run decline in r* is consistent with market-based results from most term premium models.
- Labor market and NAIRU
  - The labor market is key to defining the cycle due to stationarity of labor market indicators and lower high-frequency noise relative to national accounts.
  - The NAIRU is time varying but moves slowly and consistently declines.
  - Changes in the unemployment rate map closely into the model’s measure of the cyclical position of the economy.
- Hysteresis and structural drivers
  - Hysteresis is suggested as possible but is not a strong feature of the results.
  - Trends in the stars appear longer than traditional business cycles, complicating estimation of direct links to the output gap.
  - Timing of movements in r* and g* point to possible links between boom-and-bust in housing prices and household debt or persistent effects from the end of the IT investment boom in the 1990s.

### Uncertainty, model comparison, and quasi-real-time performance
- Standard errors and information content
  - Standard errors for full sample estimates indicate greater precision for unobserved parameters and states.
  - The noisiness of Phillips Curve signals is linked to these standard errors.
- Role of data and model complexity
  - The most basic model (core PCE, GDP, ECI, unemployment rate) has smaller standard errors than alternative approaches.
  - Inclusion of the unemployment rate improves precision of cycle estimation relative to using only activity data.
  - Estimating models over the post-1990 sample period reduces standard errors.
  - Additional data contributes incremental improvement in precision.
- Trade-offs in model complexity and real-time performance
  - The simpler version of the model has better quasi-real-time performance than the full model, highlighting a trade-off between more data and more parameters to estimate.
  - The full model’s quasi-real-time performance compares favorably with other methodologies.
- Inertia and revisions
  - The output gap has a strong degree of inertia, which helps describe the data but can lead to revisions around cyclical turning points.
- Table 3 — Output Gap Estimation Certainty (expanding window, end points from 2008Q1-2019Q3)
  - Laubach-Williams (beginning in 1965): Avg. s.e. 2.28, Final s.e. 3.10, RMSE 1.43, Revision Ratio 1.34
  - FRB/US (beginning in 1965): Avg. s.e. 0.85, Final s.e. 0.98, RMSE 0.85, Revision Ratio 0.30
  - FRB/US (beginning in 1990): Avg. s.e. 0.62, Final s.e. 0.8, RMSE 0.66, Revision Ratio 0.24
  - Williams, Abdih, and Kopp (GDP, core PCE, ECI, U3 and interest rates — most basic model): Avg. s.e. 0.16, Final s.e. 0.3, RMSE 0.48, Revision Ratio 0.15
  - Full model: Avg. s.e. 0.13, Final s.e. 0.22, RMSE 0.7, Revision Ratio 0.22
  - Notes:
    - "s.e." denotes standard error of the output gap.
    - RMSE is the root mean squared error between the final smoothed estimate and the end points of the expanding window estimates.
    - Revision ratio is the ratio of the RMSE to the standard deviation of the output gap over 2008-2019; it adjusts for relative volatility and is preferred as it is scale invariant.
    - FRB/US results end in 2015 as that is the last date available in the supply-side model database.
    - Estimates of g* and r* are less precise than the output gap because they are less directly observed.

### Policy implications and conclusions (VI. CONCLUSIONS)
- Decline in equilibrium real interest rate
  - Estimates of the equilibrium real interest rate (r*) have fallen significantly over time; r* is now modestly negative.
  - A lower r* implies that the federal funds rate was close to its neutral level over the past few years and monetary policy was less accommodative than generally thought.
- Broader ramifications of lower r*
  - Implications for appropriate tools for monetary policy, the size of the term premium, and the speed of recovery in risk assets after recessions.
  - Low levels of r* may hamper economic stabilization as the effective lower bound becomes more binding.
  - Persistently low rates could create financial fragility by incentivizing risk-taking and higher leverage.
  - With less room for monetary policy to respond, counter-cyclical fiscal policy will become more important and judgments on the future path of government debt must account for much lower interest rates.
- Potential growth, hysteresis, and measurement challenges
  - Secular decline in potential growth suggests negative effects from the housing boom and the global financial crisis, with partial reversal in recent years.
  - These effects complicate extraction of the stars from data and triangulating changes in potential growth and NAIRU.
  - Policymakers need to incorporate the possibility of such effects in assessments and policy setting.
- Inflation, Phillips Curve, and labor-market slack
  - Persistently low inflation over the last decade suggests a flat Phillips Curve, a falling NAIRU, and a relatively modest output gap at end-2019 despite a decade of expansion and very low unemployment.
  - Labor market slack appears to have been larger than previously thought; inflation, wages, and alternative measures of labor market slack were consistent with a large output gap and a low NAIRU.
  - Extensive margins of slack should be directly integrated into business cycle modeling.
- Framework utility
  - The proposed framework helps policymakers learn about the changing structure of the economy in a manner consistent with inflation targets and can incorporate a wide variety of measures of economic activity.

*Source: Federal Reserve Bank of New York, Federal Reserve Board of Governors, and authors estimates.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020136-print-pdf.pdf_