## _wp0577

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---

### Introduction and research question
- Over the past 25 years, inflation in the United States declined from double digits in the 1970s to close to 1 percent by the early 2000s.
- Core question: how has the Federal Reserve conducted monetary policy during this broadly successful period?
- Literature context:
  - Since 1979, the FOMC has consistently responded to increases in inflation above the target level by raising the real federal funds rate above its natural rate (Taylor principle).
  - The Fed has responded to deviations of output from potential: when output falls below potential, the Fed lowers the real federal funds rate below its natural rate.
  - Most policy-rules literature assumes the natural rate of interest and the target level of inflation are constant; example: Clarida et al. (1998) estimate under a constant natural rate of interest of 3.5 percent and conclude the target has been 4 percent over 1979-94.

### Motivation for time variation in key parameters
- Evidence motivating time variation:
  - Laubach and Williams (2003): substantial variation in the natural rate of interest over the past four decades; natural rate varies about one-for-one with changes in the growth rate of potential GDP.
  - Maccini et al. (2004): identify long-run changes (regime shifts) in the natural rate with low real rates in the 1970s and high rates in the early 1980s.
  - The Federal Reserve lacks an explicit inflation target and policymaker statements suggest the inflation objective has varied; assuming a constant target is overly restrictive.

### Methodology overview
- Two-step estimation approach relaxing constancy of natural rate and inflation objective:
  1. Estimate a time-varying natural rate of interest using the Kalman filter and a model linking the natural rate to changes in trend productivity growth and to a random component (following Laubach and Williams (2003)).
  2. Use the estimated natural rate to estimate a time-varying implicit inflation target in the context of a forward-looking Taylor rule, modeling the implicit inflation target as a random walk and conducting estimation with the Kalman filter and the median-unbiased estimator of Stock and Watson (1998).
- Taylor-rule specification and state dynamics:
  - Target federal funds rate: i* t = r n t + π e t + (β-1)(π e t – π* t ) + γ t y~.
  - Interest rate smoothing: i t = (1-ρ)i* t + ρ i t-1 + ε 0,t, with ε 0,t mean zero i.i.d. normal.
  - Inflation target dynamics: π* t = π* t-1 + ε 3,t, with ε 3,t mean zero i.i.d. normal.
- Distinguishing features:
  - (i) allow π* to vary over time; (ii) allow r n to vary over time; (iii) estimate jointly with the Kalman filter and maximum likelihood.
- Treatment of low signal-to-noise problem for π* innovations:
  - Estimate λ = σ0^2 / σ3^2 using Stock and Watson (1998) median-unbiased inversion of sup-Wald structural-break test; fix λ at obtained value when estimating full system.
- Initialization:
  - Baseline initial implicit inflation target in 1979Q3 set to π* 0 = 4 percent; robustness checks use π* 0 = 2.5 percent and π* 0 = 5.5 percent.

### Data
- Sample period: 1979Q3 to 2004Q1.
- Inflation:
  - Measure: annualized quarterly growth rate of the price index for personal consumption expenditures excluding food and energy (core PCE inflation).
  - Expected inflation, π e t : expectation of average inflation over the four quarters ahead formed by out-of-sample forecasts from a univariate AR(3) with a 40-quarter rolling-regression window (Laubach and Williams (2003) procedure).
  - Data source: Federal Reserve Bank of St. Louis.
- Nominal interest rate:
  - Policy rate: annualized federal funds rate.
  - Data source: Federal Reserve Bank of St. Louis.
- Output gap:
  - Real-time output gap series from the Greenbooks of the Federal Reserve Board of Governors (real-time estimates available up to 1995Q4).
  - For 1996-2004, supplement Greenbook series with Congressional Budget Office (CBO) output gap estimates.
  - Rationale: Greenbook output gaps represent information available to policymakers at decision time and differ from retrospective ex post gap measures.

### Results: parameter estimates and inferred implicit inflation target path
- Signal-to-noise ratio:
  - Median-unbiased estimate of λ = 0.15 with 90 percent confidence interval (0.06, 0.51).
  - Null H0: λ = 0 rejected at the 1 percent level, indicating statistically significant time variation in π* t over the sample.
- Taylor rule parameter estimates (Table 1):
  - β = 3.12 (0.70) — estimate of the inflation response; β > 1 satisfies the Taylor principle.
  - γ = 0.68 (0.22) — positive output-gap response.
  - ρ = 0.74 (0.07) — significant interest rate inertia.
  - 0 2 σ = 1.09 (0.12)
  - 3 2 σ = 0.17 (--)
  - Log likelihood = -63.7
- Fit of the rule:
  - Time-varying parameter model’s estimated federal funds target tracks the actual federal funds rate closely over 1979-2004; fitted interest rate matches actual rate almost exactly due to high ρ.
  - Canonical constant-parameter Taylor rule produces similar response coefficients but a target that tracks actual rates less closely and exhibits more pronounced temporary deviations.
- Estimated trajectory of implicit inflation target, π* t (four historical phases):
  - (i) Volcker disinflation (1979 until the early 1980s): implicit target near 3 percent during early 1980s.
  - (ii) Opportunistic approach to disinflation (mid-1980s to early 1990s): implicit target falls into 1-2 percent range; minimum of 1.3 percent in 1996Q2.
  - (iii) Low-inflation equilibrium (late 1990s): implicit target and actual inflation remain in the 1-2 percent range.
  - (iv) Deflation scare (2001-2004): implicit target drifts upward into the 2 to 3 percent range consistent with a policy to avoid deflation by committing to keep short rates low longer.
- Confidence intervals:
  - 95 percent confidence intervals for π* t obtained from Kalman smoother variance corrected for parameter uncertainty (Ansley and Kohn (1986)).

### Robustness analysis
- Robustness to initial π* 0 (Table 2; λ fixed at baseline 0.15):
  - Baseline π* 0 = 4% versus alternative π* 0 = 2.5% and π* 0 = 5.5%:
    - Estimated Taylor rule parameters remain similar across initializations.
    - Paths of π* t converge by the late 1980s.
  - Selected parameter comparisons (baseline / low / high):
    - β: 3.12 (0.70) / 2.80 (0.54) / 3.36 (1.08)
    - γ: 0.68 (0.22) / 0.74 (0.21) / 0.58 (0.29)
    - ρ: 0.74 (0.07) / 0.71 (0.06) / 0.81 (0.07)
    - 0 2 σ: 1.09 (0.12) / 1.04 (0.11) / 1.24 (0.12)
    - 3 2 σ: 0.17 (--) / 0.16 (--) / 0.19 (--)
    - Log likelihood: -63.7 / -60.8 / -67.9
- Robustness to λ (Table 3; π* 0 fixed at 4%):
  - Baseline λ = 0.15 versus λ = 0.06 and λ = 0.51:
    - Taylor rule parameters similar across λ choices; estimated targets follow similar paths.
  - Selected parameter comparisons (λ = 0.15 / 0.06 / 0.51):
    - β: 3.12 (0.70) / 3.57 (0.76) / 2.49 (0.69)
    - γ: 0.68 (0.22) / 0.76 (0.24) / 0.56 (0.22)
    - ρ: 0.74 (0.07) / 0.77 (0.06) / 0.72 (0.08)
    - 0 2 σ: 1.09 (0.12) / 1.16 (0.12) / 1.06 (0.13)
    - 3 2 σ: 0.17 (--) / 0.07 (--) / 0.54 (--)
    - Log likelihood: -63.7 / -62.8 / -66.3

### Key findings and implications
- Empirical findings:
  - The implicit inflation target of the Federal Reserve varied substantially over 1979Q3–2004Q1.
  - Estimated inflation response β = 3.12 indicates the Fed responded actively to inflation gaps, satisfying the Taylor principle (β > 1).
  - Output stabilization motive present (γ = 0.68) and significant interest rate smoothing (ρ = 0.74).
  - Median-unbiased λ = 0.15 (90 percent CI (0.06, 0.51)) implies statistically significant time variation in π* t.
  - Four historical phases of implied target: Volcker disinflation; opportunistic disinflation; low-inflation equilibrium; deflation scare.
- Methodological contribution:
  - Joint estimation of time-varying r n t and π* t using Kalman filter and maximum likelihood provides a framework to infer a central bank’s evolving implicit inflation objective from observed policy and real-time data.
- Robustness:
  - Results are robust to alternative initializations of π* 0 and to alternative values of λ within the 90 percent confidence interval.
- Broader implications and future research:
  - The finding that implicit inflation targets vary during a broadly successful monetary history suggests potential advantages to allowing for a time-varying inflation target.
  - Framework adaptable to other countries; recommended future research: whether implicit targets are more stable under explicit inflation-targeting regimes.

### Appendix I — The natural rate of interest
- Model structure and identifying assumptions:
  - Two identifying assumptions:
    - (i) the output gap converges to zero if the real rate gap is zero.
    - (ii) the change in inflation converges to zero if the output gap is zero.
  - Output equation (I.S. equation) (A.1):
    - yt = y*t + Ay(L)(yt-1 – y*t-1) + Ar(L)(rt-1 – rn t-1) + ε1,t
    - yt is the log of GDP; y*t is the log of potential GDP; yt – y*t is the output gap.
    - ε1,t denotes a mean zero i.i.d. normal shock to output.
  - Phillips curve (A.2):
    - πt = Bπ(L)πt-1 + By(L)(yt-1 – y*t-1) + Bπ(L)xt + ε2,t
    - xt denotes the data matrix containing the relative oil and non-oil import price inflation series.
    - ε2,t denotes a mean zero i.i.d. normal shock to output.
  - Stable inflation is consistent with both the real interest rate and output equaling their respective natural rates.
- Unobserved state variables and dynamics:
  - Natural rate specification (A.3) and stochastic drift (A.4):
    - rn t = c g t + z t
    - z t = Dz(L) z t-1 + ε4,t
  - Baseline simplifying assumption:
    - LW baseline: z t is a random walk, so that ∆z t = ε4,t; present analysis uses this simpler baseline; natural rate of interest follows a random walk.
  - Potential output (A.5):
    - y*t = y*t-1 + g t-1 + ε5,t
  - Trend growth (A.6):
    - For parsimony, LW assume g t = g t-1 + ε6,t (random walk).
- Estimation approach:
  - Equations (A.1) through (A.6) estimated using maximum likelihood and the Kalman filter.
  - Estimation yields (a) estimates of model parameters, and (b) time-varying paths of unobserved state variables.
  - Lag selection:
    - Include two lags of the output and real interest rate gap in the output equation.
    - Include eight lags of inflation and one lag of output in the inflation equation.
- Sample and implementation notes:
  - LW apply approach to 1961Q1 to 2002Q1; present estimation extends to 2004Q1 and estimates using 1961Q1 to 2004Q1.
  - Advantage of longer sample: initialization of rn t in 1979Q3 is not needed.
- Key model features:
  - Observable variables: yt (log GDP), πt (inflation), rt (real rate), xt (relative oil and non-oil import price inflation series).
  - Unobservables: y*t (potential GDP), rn t (natural rate of interest), g t (trend in productivity/trend growth), z t (stochastic drift).
  - Error terms: ε1,t, ε2,t, ε4,t, ε5,t, ε6,t — all specified as mean zero i.i.d. normal shocks.

*Source: _wp0577 (PDF chapter/section).*

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

### _wp0577 - References

### Introduction and research question
- Over the past 25 years, inflation in the United States has declined from double digits in the 1970s to close to 1 percent by the early 2000s.
- Core question: how has the Federal Reserve conducted monetary policy during this broadly successful period?
- Literature context:
  - Since 1979, the Federal Open Market Committee (FOMC) has consistently responded to increases in inflation above the target level by raising the real federal funds rate above its natural rate, in accordance with the Taylor principle.
  - The Fed has responded to deviations of output from potential: when output falls below potential, the Fed lowers the real federal funds rate below its natural rate.
  - Most policy-rules literature assumes the natural rate of interest and the target level of inflation are constant over the sample period (example: Clarida et al. (1998) estimate under a constant natural rate of interest of 3.5 percent and conclude the target has been 4 percent over 1979-94).

### Motivation for time variation in key parameters
- Evidence indicates the natural rate of interest varies:
  - Laubach and Williams (2003) find substantial variation in the natural rate of interest over the past four decades in the U.S. and suggest the natural rate varies about one-for-one with changes in the growth rate of potential GDP.
  - Maccini et al. (2004) identify long-run changes (regime shifts) in the natural rate with low real rates in the 1970s and high rates in the early 1980s.
- The Federal Reserve does not have an explicit inflation target and statements by policymakers suggest the inflation objective has varied; assuming a constant target is overly restrictive.

### Methodology overview
- Two-step estimation approach relaxing constancy of natural rate and inflation objective:
  1. Estimate a time-varying natural rate of interest using the Kalman filter and a model linking the natural rate to changes in trend productivity growth and to a random component, following Laubach and Williams (2003).
  2. Use the estimated natural rate to estimate a time-varying implicit inflation target in the context of a forward-looking Taylor rule, modeling the implicit inflation target as a random walk and conducting estimation with the Kalman filter and the median-unbiased estimator proposed by Stock and Watson (1998).

### Main findings (four points)
- (i) Stability tests indicate significant variation in the Federal Reserve's implicit target over the 1979-2004 period.
- (ii) In the early 1980s, the inflation target estimate is near 3 percent, indicating that the Federal Reserve under Volcker sought to substantially reduce inflation from its double digit level.
- (iii) In the late 1980s and early 1990s, the target is close to actual inflation of 3-4 percent and declines to 1-2 percent only after the 1990-91 recession reduces inflation, corroborating qualitative historical evidence of an “opportunistic approach to disinflation” at the Fed.
- (iv) Over 2001-04, the target rises to 2-3 percent, interpretable as a response by the FOMC to the risks of hitting the zero bound on nominal interest rates.

### Organization of the paper (as provided)
- The rest of the paper is organized as follows. Section II describes the methodology.

*Source: _wp0577 - References*

### Section III describes the data used in the analysis, Section IV discusses the results, Section V

### _wp0577 - Section III describes the data used in the analysis, Section IV discusses the results, Section V

### Methodology: model and estimation approach
- Taylor rule target with time-varying natural rate and inflation target:
  - Target federal funds rate specification: i* t = r n t + π e t + (β-1)(π e t – π* t ) + γ t y~.
  - Interest rate smoothing: i t = (1-ρ)i* t + ρ i t-1 + ε 0,t, with ε 0,t mean zero i.i.d. normal.
  - Inflation target dynamics: π* t = π* t-1 + ε 3,t, with ε 3,t mean zero i.i.d. normal.
- Distinguishing features of the approach:
  - (i) allow π* to vary over time; (ii) allow r n to vary over time; (iii) estimate jointly with the Kalman filter and maximum likelihood.
- Estimation of the time-varying natural rate, r n t:
  - Follow Laubach and Williams (2003) baseline random-walk specification for r n t.
  - Use one-sided (Kalman filter) estimates for real-time analysis; two-sided (smoother) estimates also produced for comparison.
- Treatment of the low signal-to-noise (“pile-up”) problem for π* innovations:
  - Estimate the signal-to-noise ratio λ = σ0^2 / σ3^2 using the Stock and Watson (1998) median-unbiased inversion of sup-Wald structural-break test.
  - Fix λ at the obtained value when estimating the full system by maximum likelihood with the Kalman filter.
- Initialization:
  - Baseline initial implicit inflation target in 1979Q3 set to π* 0 = 4 percent based on historical narrative; robustness checks use π* 0 = 2.5 percent and π* 0 = 5.5 percent.

### Data
- Sample period: 1979Q3 to 2004Q1.
- Inflation:
  - Measure: annualized quarterly growth rate of the price index for personal consumption expenditures excluding food and energy (core PCE inflation).
  - Expected inflation, π e t : expectation of average inflation over the four quarters ahead formed by out-of-sample forecasts from a univariate AR(3) with a 40-quarter rolling-regression window (Laubach and Williams (2003) procedure).
  - Data source: Federal Reserve Bank of St. Louis.
- Nominal interest rate:
  - Policy rate: annualized federal funds rate.
  - Data source: Federal Reserve Bank of St. Louis.
- Output gap:
  - Real-time output gap series from the Greenbooks of the Federal Reserve Board of Governors (real-time estimates available up to 1995Q4).
  - For 1996-2004, supplement Greenbook series with Congressional Budget Office (CBO) output gap estimates.
  - Rationale: Greenbook output gaps represent information available to policymakers at decision time and differ from retrospective ex post gap measures.

### Results: parameter estimates and inferred target path
- Signal-to-noise ratio estimate:
  - Median-unbiased estimate of λ = 0.15 with 90 percent confidence interval (0.06, 0.51).
  - Null H0: λ = 0 rejected at the 1 percent level, indicating statistically significant time variation in π* t over the sample.
- Taylor rule parameter estimates (Table 1):
  - β = 3.12 (0.70)
    - Interpretation: estimate of the inflation response; β > 1 satisfies the Taylor principle.
  - γ = 0.68 (0.22)
    - Interpretation: positive output-gap response (leaning against the wind).
  - ρ = 0.74 (0.07)
    - Interpretation: significant interest rate inertia.
  - 0 2 σ = 1.09 (0.12)
  - 3 2 σ = 0.17 (--)
  - Log likelihood = -63.7
- Fit of the rule:
  - The time-varying parameter model’s estimated federal funds target tracks the actual federal funds rate closely over 1979-2004; fitted interest rate matches actual rate almost exactly due to high ρ.
  - Canonical constant-parameter Taylor rule produces similar response coefficients but a target that tracks actual rates less closely and exhibits more pronounced temporary deviations.
- Estimates of the implicit inflation target, π* t :
  - Trajectory divided into four periods corroborated by historical evidence:
    - (i) Volcker disinflation (1979 until the early 1980s): implicit target near 3 percent during early 1980s.
    - (ii) Opportunistic approach to disinflation (mid-1980s to early 1990s): implicit target falls into 1-2 percent range; minimum of 1.3 percent in 1996Q2.
    - (iii) Low-inflation equilibrium (late 1990s): implicit target and actual inflation remain in the 1-2 percent range.
    - (iv) Deflation scare (2001-2004): implicit target drifts upward into the 2 to 3 percent range consistent with a policy to avoid deflation by committing to keep short rates low longer.
  - 95 percent confidence intervals for π* t obtained from Kalman smoother variance corrected for parameter uncertainty (Ansley and Kohn (1986)).

### Robustness analysis
- Robustness to initial π* 0 (Table 2; λ fixed at baseline 0.15):
  - Baseline π* 0 = 4% versus alternative π* 0 = 2.5% and π* 0 = 5.5%:
    - Estimated Taylor rule parameters remain similar across initializations.
    - Paths of π* t converge by the late 1980s.
  - Selected parameter comparisons (baseline / low / high):
    - β: 3.12 (0.70) / 2.80 (0.54) / 3.36 (1.08)
    - γ: 0.68 (0.22) / 0.74 (0.21) / 0.58 (0.29)
    - ρ: 0.74 (0.07) / 0.71 (0.06) / 0.81 (0.07)
    - 0 2 σ: 1.09 (0.12) / 1.04 (0.11) / 1.24 (0.12)
    - 3 2 σ: 0.17 (--) / 0.16 (--) / 0.19 (--)
    - Log likelihood: -63.7 / -60.8 / -67.9
- Robustness to λ (Table 3; π* 0 fixed at 4%):
  - Baseline λ = 0.15 versus λ = 0.06 and λ = 0.51:
    - Taylor rule parameters similar across λ choices; estimated targets follow similar paths.
  - Selected parameter comparisons (λ = 0.15 / 0.06 / 0.51):
    - β: 3.12 (0.70) / 3.57 (0.76) / 2.49 (0.69)
    - γ: 0.68 (0.22) / 0.76 (0.24) / 0.56 (0.22)
    - ρ: 0.74 (0.07) / 0.77 (0.06) / 0.72 (0.08)
    - 0 2 σ: 1.09 (0.12) / 1.16 (0.12) / 1.06 (0.13)
    - 3 2 σ: 0.17 (--) / 0.07 (--) / 0.54 (--)
    - Log likelihood: -63.7 / -62.8 / -66.3

### Key findings and implications
- Empirical findings:
  - The implicit inflation target of the Federal Reserve varied substantially over 1979Q3–2004Q1.
  - Estimated inflation response β = 3.12 indicates the Fed responded actively to inflation gaps, satisfying the Taylor principle (β > 1).
  - Output stabilization motive present (γ = 0.68) and significant interest rate smoothing (ρ = 0.74).
  - Median-unbiased λ = 0.15 (90 percent CI (0.06, 0.51)) implies statistically significant time variation in π* t.
  - Four historical phases of implied target: Volcker disinflation; opportunistic disinflation; low-inflation equilibrium; deflation scare.
- Methodological contribution:
  - Joint estimation of time-varying r n t and π* t using Kalman filter and maximum likelihood provides a framework to infer a central bank’s evolving implicit inflation objective from observed policy and real-time data.
- Robustness:
  - Results are robust to alternative initializations of π* 0 and to alternative values of λ within the 90 percent confidence interval.
- Broader implications and future research:
  - The finding that implicit inflation targets vary during a broadly successful monetary history suggests potential advantages to allowing for a time-varying inflation target.
  - Framework adaptable to other countries; question raised whether implicit targets are more stable under explicit inflation-targeting regimes is recommended for future research.

*Source: _wp0577 (PDF chapter/section).*

### APPENDIX                                                                        I

### THE NATURAL RATE OF INTEREST

### Model structure and identifying assumptions
- Two basic identifying assumptions:
  - (i) the output gap converges to zero if the real rate gap is zero.
  - (ii) the change in inflation converges to zero if the output gap is zero.
- Output equation (I.S. equation) (A.1):
  - yt = y*t + Ay(L)(yt-1 – y*t-1) + Ar(L)(rt-1 – rn t-1) + ε1,t
  - yt is the log of GDP; y*t is the log of potential GDP; yt – y*t is the output gap.
  - ε1,t denotes a mean zero i.i.d. normal shock to output.
- Phillips curve (A.2):
  - πt = Bπ(L)πt-1 + By(L)(yt-1 – y*t-1) + Bπ(L)xt + ε2,t
  - xt denotes the data matrix containing the relative oil and non-oil import price inflation series.
  - The inflation rate depends on lags of inflation with the unity sum restriction on the coefficients, relative oil and non-oil import price inflation, and the output gap.
  - ε2,t denotes a mean zero i.i.d. normal shock to output.
- Stable inflation is consistent with both the real interest rate and output equaling their respective natural rates.

### Unobserved state variables and their dynamics
- Natural rate specification (A.3) and stochastic drift (A.4):
  - rn t = c g t + z t
  - z t = Dz(L) z t-1 + ε4,t
- Baseline simplifying assumption used:
  - LW report results for a baseline case where z t is a random walk, so that ∆z t = ε4,t, and also consider a stationary z t case.
  - The present analysis uses the simpler baseline case; consequently, the natural rate of interest follows a random walk.
- Potential output (A.5):
  - y*t = y*t-1 + g t-1 + ε5,t
- Trend growth (A.6):
  - For parsimony, LW assume the trend growth rate g t follows a random walk:
  - g t = g t-1 + ε6,t

### Estimation approach
- Equations estimated:
  - Equations (A.1) through (A.6) are estimated using maximum likelihood and the Kalman filter.
- Estimation yields:
  - (a) estimates of the model parameters, and
  - (b) estimates of the time-varying paths of the unobserved state variables.
- Lag selection and specification details (as in LW):
  - Include two lags of the output and real interest rate gap in the output equation.
  - Include eight lags of inflation and one lag of output in the inflation equation.
  - The number of lags are determined by the data.

### Sample and implementation notes
- LW apply the approach to the 1961Q1 to 2002Q1 sample.
- The present estimation extends the sample to 2004Q1 and estimates the equations using the 1961Q1 to 2004Q1 period.
- Advantage of the longer sample:
  - By conducting estimation over this long sample, initialization of rn t in 1979Q3 is not needed.

### Key model features (concise bullets)
- Observable variables: yt (log GDP), πt (inflation), rt (real rate), xt (relative oil and non-oil import price inflation series).
- Unobservables: y*t (potential GDP), rn t (natural rate of interest), g t (trend in productivity/trend growth), z t (stochastic drift).
- Error terms: ε1,t, ε2,t, ε4,t, ε5,t, ε6,t — all specified as mean zero i.i.d. normal shocks in the model setup.
- Identification rests on the convergence properties encoded in (A.1) and (A.2).

*Source: APPENDIX I, _wp0577 - APPENDIX I*

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