## _wp15134

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---

### Major objectives and scope
- Illustrate the value of an optimal control approach to help policymakers decide on a path for the policy interest rate, arguing that minimization of a loss function reflecting policymakers’ preferences should be added to approaches that use reaction functions.
- Suggest improvements to the Federal Reserve’s communications strategy, including publication of a staff forecast of output, inflation and the policy rate, with the latter based on an endogenous interest rate convention that would bring inflation back towards its target and output back towards potential over the policy horizon.
- Emphasize avoiding “dark corners” (outcomes that could lead to very unfortunate movements in inflation and/or output), focusing on avoiding deflationary spirals and very weak economies when the central bank is limited by the zero lower bound (ZLB).
- Highlight implications of important nonlinearities in the economy and models (for example, a nonlinear Phillips curve) that cause results from linear models not to carry over.

### Key background and context
- The Federal Reserve’s dual mandate—maximum employment and stable prices—is functionally very close to flexible inflation targeting or inflation-forecast targeting (IFT).
- The FOMC’s January 25, 2012 statement announced a 2 percent rate of increase of the PCE deflator as the long-term goal of monetary policy.
- The paper’s approach reflects central bank practice and aims to be illustrative rather than exhaustive of the technical literature on IFT/flexible IT.

### Findings from model experiments and policy implications
- Loss-minimizing strategy (quadratic loss function penalizing deviations of inflation from target and output from potential):
  - Produces a more active policy response under weak demand, unduly low inflation, and interest rates near the lower bound, compared with regular linear policy rules.
  - Implies an overshoot of the inflation target and an outcome in which the federal funds rate is held at zero until inflation and output gaps are almost closed.
  - Shows greater aversion to dark corners because the quadratic loss function penalizes large deviations heavily.
- Ordinary linear policy rules (e.g., Taylor rule) do not display the pronounced risk-avoidance property found under the loss-minimizing approach.
- The loss-function approach can generate alternative scenarios reflecting different preference weightings among FOMC members; scenarios help select a preferred federal funds rate path consistent with longer-term objectives.

### Role of expectations and communication
- Effectiveness of loss-minimizing policy critically depends on public expectations; its impact on aggregate spending relative to policy rules comes from a drop in the real interest rate over an extended horizon.
- Long-term real interest rates fall due to a longer period at the ZLB and a temporary increase in expected inflation under the loss-minimizing strategy.
- Because describing a quadratic loss function to the public is complex, publishable central bank forecasts and clear communications are essential policy instruments.
- Proposed communication reforms include publication of:
  - A baseline staff forecast including output, inflation, and a future path for the short-term interest rate consistent with the FOMC’s longer-run goals.
  - Alternative scenarios based on different policy reaction functions and outlook assumptions to clarify rationale for interest rate settings and the majority view of the FOMC, while allowing dissenting views to be shown.

### Institutional recommendations
- Publish a full Federal Reserve staff forecast, including the forecast for the future path of the short-term interest rate, at the same time as policy decisions are announced, to provide a systematic and specific basis for forward guidance—replacing the current Summary of Economic Projections (“dots”).
- The published forecast should be that of the economic staff, rather than the official forecast of the FOMC; the majority of the FOMC could endorse the published forecast while dissenting members present alternative scenarios.
- Use loss-function based policy analysis as a complement to reaction-function approaches, especially when the ZLB constraint is binding.

### Model outline and policy reaction specifications
- Model type and features:
  - Simple closed-economy, inflation-targeting model with forward-looking output gap (IS curve) and an expectations-augmented Phillips curve.
  - Output gap equation contains a bank-lending-tightening variable (BLT_t).
  - Demand shocks in output gap; supply shocks in inflation equation.
- Three policy reaction frameworks examined:
  - Taylor rule (variant used):
    - i_t = r_t* + 4*pi_t^4 + 0.5*(pi_t^4 - pi^*) + 0.5*y_t + epsilon_i_t
    - Definitions preserved: i_t is the federal funds rate; r_t is the equilibrium real interest rate; pi_t^4 is average annualized inflation over the past four quarters; pi^* is the inflation objective, assumed to be 2 percent; i_t_epsilon is a policy deviation shock.
  - Inflation-forecast-based (IFB) reaction function:
    - i_t = 0.71*i_{t-1} + 0.33*[r_t + 1.0*PCE_inflation_forecast_{t+3}] + 0.71*[0.91*(pi_t^4 - pi^*)] + 0.21*y_t + epsilon_i_t
    - Includes lagged nominal federal funds rate to smooth reactions; uses equilibrium nominal interest rate measured by sum of equilibrium real rate and projected year-on-year core PCE inflation; ignores inflation shocks expected to reverse within the three-quarter horizon.
  - Minimizing a quadratic loss function:
    - L_t = sum_{t=1 to infinity} β^{t-1}[α*(i_t - i^*)^2 + (pi_t^4 - pi^*)^2 + γ*y_t^2]
    - Baseline calibration: equal weight (1.0) on inflation and output gaps; coefficient for change in nominal interest rate set to 0.5 (interest rate smoothing).

### Transmission mechanism
- An exogenous increase in the policy interest rate:
  - Raises the short-term real interest rate.
  - Feeds into the long-term real rate with lags.
  - Decreases aggregate demand and output via the output gap equation.
  - Greater excess capacity gradually puts downward pressure on inflation.
- Feedback loops: shocks reverberate and evoke the IFT policy response; policy eventually stabilizes inflation at target and output at potential for a flexible IT central bank.

### Illustrative simulation results (baseline initial conditions and neutral rate path)
- Baseline initial conditions (as of 2014Q4):
  - Output gap: -2 percent.
  - PCE inflation: about 1.5 percent.
  - Federal funds rate: about zero.
- Neutral (equilibrium real) funds rate assumption:
  - Neutral rate rises from 0 in 2014Q4 to about 1.3 percent in 2020Q4.
  - With a two percent inflation target, equilibrium nominal rate rises from 2 percent in 2014Q4 to 3.3 percent in 2020Q4.
  - Neutral rate assumption noted as close to, but below, median FOMC long-term real fed funds rate forecast (~1.75 percent).
- Key comparative outcomes:
  - IFB: federal funds rate rises gradually starting almost immediately; slower closing of output gap; inflation below target for a longer period.
  - DM1 (loss-function, equal weights): federal funds rate remains at zero until mid-2016 before rising; output gap overshoots to a positive 0.3 percent maximum; inflation overshoots to 2.2 percent maximum. Overshoot reflects deliberate stimulative stance to avoid downside risks given ZLB.

### Sensitivity analysis and nonlinearities
- DM2 calibration: half the weight on the output gap as on the inflation gap — DM1 and DM2 lines sit almost on top of each other; halving the weight on output gap makes hardly any economic difference.
- Comparison across four approaches (Taylor, IFB, DM1, DM2):
  - IFB and Taylor produce very similar results.
  - Major difference between loss-function minimizations (DM1/DM2) versus policy rules (IFB/Taylor): loss-function cases produce a much lower market real rate, stronger growth and higher inflation over time.
  - Loss-function planned overshooting insures against costly downside risks given rising marginal costs and ZLB constraints.
- Nonlinear Phillips curve and overheating scenarios:
  - Linear Phillips curve baseline parameters preserved: λ1 = 0.10; λ2 = 0.70.
  - Non-linear Phillips curve calibration preserved: λ1 = 0.10; λ2 = 0.70 with nonlinear terms; calibration implies inflation cannot go below -5 percent and as output gap approaches +5 percent, inflation can go to infinity.
  - Overheating shocks hitting in 2017q1 and 2017q2 produce sharply higher inflation under some specifications; loss-function approach responds more aggressively to expansionary shocks, limiting inflation and output gap relative to IFB.
  - Under loss-function approach, different Phillips specifications make hardly any economic difference because the response keeps the economy away from regions where non-linearities dominate.

### Forecast confidence bands and probabilistic findings
- Confidence interval construction: shaded bands show confidence intervals from 10 to 90 percent in 10 percentage-point increments.
- Key quantitative findings:
  - Overshoots are very small relative to, for example, the 30 to 70 percent confidence bands.
  - Under IFB, there is a 15 to 20 percent probability of deflation one year into the future.
  - Loss-function approach yields wider confidence bands for the interest rate (skewed because of the zero floor) and much narrower bands for output and inflation than IFB.
  - Volatility is somewhat greater under an initial output gap of -4 percent versus -2 percent due to higher likelihood that the ZLB will impede stabilizing policy.

### Publication of endogenous interest rate forecasts — motivation, benefits, and practicalities
- Rationale:
  - Forecasts with endogenous interest rates are internally coherent and anchor expectations by showing policy reactions consistent with the dual mandate (inflation returning to 2 percent and economy reaching full employment).
  - Publishing a single baseline endogenous interest rate scenario consistent with the baseline economic outlook is recommended: “the only right thing is to explicitly discuss the interest rate path and to choose a particular path as the main forecast.”
- Limitations of the Fed’s “dots”:
  - Do not necessarily represent the FOMC’s majority view.
  - May obfuscate or contradict the majority view.
  - Hard to judge internal consistency because the public cannot identify which dots belong to which individual.
  - The median of the dots may be far from internally consistent with the median macroeconomic forecast.
- Perceived risks and conditionality:
  - Risk that markets interpret a published path as an unconditional commitment; communications must emphasize conditionality: forecasts are conditional upon information available at the time and will change as new information arrives.
  - Experience in several countries suggests market participants learn the conditional nature of forecast interest rate paths.
- Institutional options for publishing forecasts:
  - Option one: publish complete forecasts for each FOMC member (technically demanding).
  - Option two (preferred): staff produce and publish the baseline forecast that the majority of the FOMC could endorse, accompanied by alternative scenarios to show dispersion of views.
  - Rationale for staff ownership: staff can maintain coherence and consistency in projections; disagreements by committee members should be transparently explained with solid economic arguments.

### Conclusion: policy interpretation and recommendations
- A loss-function minimization approach, combined with publication of a staff baseline interest rate path and related uncertainties (replacing or augmenting the “dot” plot), can enhance the effectiveness of policy actions when the ZLB binds.
- Mechanism: planned overshooting of inflation arises because expectations that the short-term policy rate will be held at zero for an extended period boost expected inflation and reduce current and expected real interest rates, helping avoid deflationary traps.
- Publishing a staff-produced baseline interest rate path, endorsed by the majority of the FOMC and accompanied by alternative scenarios and clear conditionality, would:
  - clarify the interest rate path underlying macroeconomic forecasts,
  - guide public expectations about future inflation and policy rates,
  - strengthen the effectiveness of monetary policy,
  - allow the Fed’s goals to be reached more quickly and with “smaller and shorter-lasting deviations over time.”

### Appendix 1 — Key variables and definitions (preserved notation and numeric weights)
- i_t: Federal funds rate (%)
- r_t: Real interest rate (%) with relation: 1 + r_t = i_t − π_{t+1} (expressed in source notation)
- r*_t: Steady-state level of real interest rate (%)
- r^e_t: Equilibrium real interest rate (%) with process: r^e_t = 0.95 * r^e_{t-1} + 0.05 * r^*_t + r^ε_t
- r4_t: Four-quarter average real interest rate (%) defined as r4_t = (r_t + r_{t-1} + r_{t-2} + r_{t-3})/4
- r4^e_t: Four-quarter average equilibrium real interest rate (%) (analogous four-quarter averaging)
- r^l_t: Long-term real interest rate (%) defined as a weighted average of current short-term real rate and a four-quarter average over current year and next 16 quarters with the weighted structure preserved from source expressions (numeric weights and structure preserved in source)
- P_t: PCE price index
- p_t: Log of PCE price index with relation: p_t = )log(*100 t P p = (source formatting preserved)
- π_t: Quarterly PCE inflation (%) with relation: π_t = p_t − p_{t-1}
- π_{4,t}: Year-on-year PCE inflation (%) defined as a four-quarter aggregation (source four-quarter/year-on-year expression preserved)
- π^*: Inflation objective (2%)
- y_t: Output gap
- BLT_t: Bank Lending Tightening condition
- i_t^ε: Deviation from the interest rate reaction function
- π_t^ε: Supply shock
- y_t^ε: Shock to the output gap
- Error processes and BLT construction preserved as in source; BLT constructed from Senior Loan Officer Survey responses.

*Source: IMF staff paper content from _wp15134 (Appendices and excerpts provided).*

### Appendix 1. Definitions ...................................................................................38

### Appendix 1. Definitions ...................................................................................38

### Major objectives and scope
- Illustrate the value of an optimal control approach to help policymakers decide on a path for the policy interest rate, arguing that minimization of a loss function reflecting policymakers’ preferences should be added to approaches that use reaction functions.
- Suggest improvements to the Federal Reserve’s communications strategy, expanding on recent Fed changes, including publication of a staff forecast of output, inflation and the policy rate, with the latter based on an endogenous interest rate convention that would bring inflation back towards its target and output back towards potential over the policy horizon.
- Emphasize the importance of avoiding “dark corners” (outcomes that could lead to very unfortunate movements in inflation and/or output), with a focus on avoiding deflationary spirals and very weak economies when the central bank is limited by the zero lower bound (ZLB) on the policy interest rate.
- Highlight implications of important nonlinearities in the economy and models (for example, a nonlinear Phillips curve) that cause results from linear models not to carry over, with major implications for policy-rate paths.

### Key background and context
- The Federal Reserve’s dual mandate—maximum employment and stable prices—has evolved and is functionally very close to flexible inflation targeting or inflation-forecast targeting (IFT), although official Fed statements avoid that terminology.
- The FOMC’s January 25, 2012 statement announced a 2 percent rate of increase of the PCE deflator as the long-term goal of monetary policy; the statement also emphasized the full employment goal and commitment to transparency.
- The paper’s approach reflects central bank practice and aims to be illustrative rather than exhaustive of technical literature on IFT/flexible IT.

### Findings from model experiments and policy implications
- A loss-minimizing strategy (minimizing a quadratic loss function that penalizes deviations of inflation from target and output from potential) yields:
  - A more active policy response under weak demand, unduly low inflation, and interest rates near the lower bound, compared with regular linear policy rules.
  - An implied overshoot of the inflation target and an outcome in which the federal funds rate is held at zero until inflation and output gaps are almost closed.
  - Greater aversion to dark corners because the quadratic loss function penalizes large deviations heavily, favoring actions that quickly move the economy away from deflation at the ZLB.
- Ordinary linear policy rules (e.g., the Taylor rule) do not display the pronounced risk-avoidance property found under the loss-minimizing approach.
- The loss function approach can generate alternative scenarios reflecting different preference weightings among FOMC members; these scenarios help select a preferred federal funds rate path consistent with longer-term objectives.

### Role of expectations and communication
- The effectiveness of the loss-minimizing policy critically depends on public expectations; its impact on aggregate spending relative to policy rules comes from a drop in the real interest rate over an extended horizon.
- Long-term real interest rates fall due to a longer period at the ZLB and a temporary increase in expected inflation under the loss-minimizing strategy.
- Because describing a quadratic loss function to the public is complex, effective communication of central bank forecasts is a key policy instrument.
- Proposed communication reforms include publication of:
  - A baseline staff forecast including output, inflation, and a future path for the short-term interest rate consistent with the FOMC’s longer-run goals.
  - Alternative scenarios based on different policy reaction functions and different outlook assumptions to clarify the rationale for interest rate settings and the majority view of the FOMC, while allowing dissenting views to be shown.

### Institutional recommendations
- Publish a full Federal Reserve staff forecast, including the forecast for the future path of the short-term interest rate, at the same time as policy decisions are announced, to provide a systematic and specific basis for forward guidance—replacing the current Summary of Economic Projections (“dots”).
- The published forecast should be that of the economic staff, rather than the official forecast of the FOMC; the majority of the FOMC could endorse the published forecast while dissenting members present alternative scenarios.
- Use loss-function based policy analysis as a complement to reaction-function approaches, especially during abnormal times when the ZLB constraint is binding.

### Modeling and technical notes (as presented)
- The model used has a forward-looking component in expectations of the public and policymakers.
- The paper provides an example of a nonlinear Phillips curve to illustrate how nonlinearities can change policy implications relative to linear models.
- The approach is illustrative and does not attempt to incorporate all technical elements from the broader literature on these subjects.

*Source: Appendix 1. Definitions, _wp15134 - Appendix 1. Definitions ...................................................................................38*

### Appendix 2 sets out the complete model. Appendix 3 provides a technical description of the

### _wp15134 - Appendix 2 sets out the complete model. Appendix 3 provides a technical description of the

### Recent developments at the Fed: communications, guidance, and balance-sheet policy
- Since 2011 the FOMC has taken several steps to shed light on its policy actions and strategy; the 2012 announcement of the two percent inflation objective was the most striking.
- The Summary of Economic Projections (SEP), released four times per year shortly after an FOMC meeting, was expanded to provide information about members’ projections for:
  - the future path of the federal funds rate (presented in greater detail as the “dot plot”),
  - GDP growth,
  - the unemployment rate, and
  - inflation.
- The SEP gives a range of the individual forecasts for each variable; the range for the federal funds rate is presented as dots for each member’s forecast.
- Forward guidance categories and timeline (as summarized in Table 1):
  - December 2008 to present — Forward guidance: Qualitative (Dec 2008-Aug 2011), date-based (Aug 2011-Dec 2012), threshold-based (Dec 2012-March 2014), qualitative (March 2014-March 2015).
  - November 2008 to present — Balance sheet guidance: Volume of purchases, pace of purchases, assets purchased, criteria for revising asset purchases, reinvestment and shrinking of the balance sheet.
  - April 2011 — Post-meeting press conference: More comprehensive and timely information on the FOMC policy decision and views, including Summary of Economic Projections.
  - January 2012 — Statement on longer-run goals and policy strategy: Clarify the Federal Reserve’s objectives and policy strategy, including the introduction of a long-run two percent inflation goal.
  - January 2012 — Policy rate projections: Individual FOMC members’ policy rate projections were added to the quarterly Selected Economic Projections published following FOMC meetings.
- Communication was especially important when the policy interest rate hit its lower bound near zero; unconventional instruments (quantitative easing and forward guidance) were introduced and required extensive explanation to connect them to the Fed’s employment and inflation objectives.
- The “taper tantrum” of May 2013 highlighted the sensitivity of markets to communications: concerns about tapering of purchases caused jumps in expected future federal funds rate and in longer-term rates and risk premiums, larger than implied by modest FOMC envisaged policy-rate increases.
- The main near-term communications challenge for the FOMC is to explain progress toward gradual tightening (a gradual upward path in the fed funds rate and scaling down of the Fed’s large balance sheet) without triggering market overreaction.
- Transparency enhancements discussed and their rationales:
  - Publishing the inflation objective, tentative estimates of the equilibrium (or natural) rate of unemployment, and forecasts of the policy rate path have supported stimulation efforts.
  - Proposals for further enhancements include publishing a quarterly monetary policy report and alternative policy interest-rate paths (e.g., “a few Taylor-like rules”) alongside the FOMC central forecast to shed light on risks and trade-offs.
  - Releasing a complete forecast for each FOMC member (with coherent connection between economic variables and expected policy interest-rate path) and individual confidence bands was discussed as a way to convey uncertainty and median views more accurately.
- The inflation forecast in policy reports can serve as an intermediate target under flexible inflation targeting: the forecast incorporates the expected policy actions of the central bank and allows the central bank to show an operating target for the short and medium term that depends on shocks disturbing the economy.

### The role of forward guidance, credibility, and predictability
- For forward guidance to be effective it should be credible and predictable: the announced path for the policy instrument should be consistent with policy objectives and the economic outlook.
- The policy interest rate in the forecast must be endogenously determined along with other key macroeconomic variables so that the expectation of convergence to the official target (the nominal anchor) is established.
- The basis for subsequent modifications to the policy rate path, in response to changes in the outlook, should be well understood.
- Flexible inflation targeting implies monetary policy takes account of short-run impacts of interest-rate changes on output as well as inflation, and the official target remains a long-run reference.

### Loss-minimizing monetary policy based on a simple model — approach and key insights
- Purpose: Provide an example of optimal-control approaches applied to a simple model of the U.S. economy that captures the trade-off between inflation and output, set in a stylized version of the current economic situation.
- The model:
  - Main equations are presented in Appendices 1 and 2; the full model is set out in Appendix 2.
  - The model is based on a stripped-down version of the Fund’s Global Projection Model (GPM).
- Three policy reaction frameworks examined in the same model and context (each deriving a projected path for the federal funds rate that will achieve the official target over the medium term while taking account of welfare costs of output gaps):
  - a simple Taylor rule;
  - an inflation-forecast-based (IFB) reaction function; and
  - minimization of a loss function that reflects policymakers’ preferences (loss-minimizing policy).
- Important result: In a situation near the zero lower bound for the interest rate, with a negative output gap and undesired disinflationary pressure, the loss-minimization approach could result in an ex ante planned overshoot of the inflation target. This corresponds to a risk management strategy where policymakers place a high priority on avoiding deflationary dynamics.
- The loss-minimization framework illustrates how models and alternative policy rules or optimal-control techniques can provide different policy insights and improve policy dialogue.

*Source: Excerpt from the IMF working paper content unit provided.*

### references for further detail on the GPM family of models.

### _wp15134 - references for further detail on the GPM family of models.

### III.2 Outline of the model
- Model type: simple closed-economy, inflation-targeting model with:
  - Output (IS curve) equation (forward-looking output gap).
  - Inflation formation equation based on an expectations-augmented Phillips curve.
  - Forward-looking expectations and policy reactions driven in part by the model’s own future solved values; long-run convergence to steady state paths.
  - Output gap equation contains a bank-lending-tightening variable capturing exogenous changes in credit conditions.
  - Demand shocks represented by the stochastic term in the output gap equation; supply shocks by the stochastic term in the inflation equation.
- Policy decision focus: three alternative federal funds rate reaction specifications; common feature is that the real interest rate rises and falls with the inflation rate and the output gap (excess demand).

### Policy reaction specifications (exact functional forms and calibration)
- Taylor rule (variant used in the paper):
  - i_t = r_t* + 4*pi_t^4 + 0.5*(pi_t^4 - pi^*) + 0.5*y_t + epsilon_i_t
  - Definitions and parameters preserved as in the source:
    - i_t is the federal funds rate.
    - r_t is the equilibrium real interest rate.
    - pi_t^4 is average annualized inflation over the past four quarters.
    - pi^* is the inflation objective, assumed to be 2 percent.
    - i_t_epsilon is a policy deviation shock.
  - Interpretation: nominal federal funds rate is a function of equilibrium real rate, inflation rate, deviation of inflation from target with coefficient 0.5, and the output gap with coefficient 0.5.
  - Assumption: exogenous increase over time in equilibrium real interest rate (neutral rate) based on dissipation of contractionary forces and heightened uncertainties from the crisis.

- Inflation-forecast-based (IFB) reaction function (focus on year-on-year PCE inflation three quarters ahead):
  - Exact functional form preserved from source:
    - i_t = 0.71*i_{t-1} + 0.33*[r_t + 1.0*PCE_inflation_forecast_{t+3}] + 0.71*[0.91*(pi_t^4 - pi^*)] + 0.21*y_t + epsilon_i_t
    - (Formulation as presented: i_tttt_t ttt yrii ε π π π ... with coefficients and forecast horizon preserved in textual description.)
  - Features:
    - Includes lagged nominal federal funds rate to smooth reactions.
    - Uses equilibrium nominal interest rate measured by sum of equilibrium real rate and projected year-on-year core PCE inflation.
    - Ignores inflation shocks expected to reverse within the three-quarter policy horizon.
    - Allows central bank to take into account known developments affecting inflation over the three-quarter horizon, including lagged effects of policy.

- Minimizing a quadratic loss function (loss-function approach):
  - Loss function:
    - L_t = sum_{t=1 to infinity} β^{t-1}[α*(i_t - i^*)^2 + (pi_t^4 - pi^*)^2 + γ*y_t^2]
    - Baseline calibration puts equal weight (1.0) on inflation and output gaps; coefficient for change in nominal interest rate set to 0.5 (interest rate smoothing).
  - Rationale:
    - Quadratic formulation: large deviations weigh disproportionately more (rising marginal cost of inflation deviations from target, output gaps, and interest rate volatility).
    - The squared change of federal funds rate represents aversion to interest rate volatility and smooths policy response.
    - Designed to avoid “dark corners” such as the ZLB where recovery from shocks becomes much more difficult.
  - Sensitivity analysis: consider variants where weights on inflation and output differ (e.g., DM2 with half the weight on output gap relative to inflation gap).

### Transmission mechanism (policy rate to output and inflation)
- An exogenous increase in the policy interest rate:
  - Raises the short-term real interest rate.
  - Feeds into the long-term real rate with lags.
  - Decreases aggregate demand and output via the output gap equation.
  - Increased excess capacity (more negative output gap) gradually puts downward pressure on inflation.
- Exogenous shocks in aggregate demand and inflation equations have direct and lagged effects on both output and inflation.
- Feedback loops: shocks reverberate through the system and evoke the IFT policy response; policy eventually stabilizes inflation at target and output at potential for a flexible IT central bank with a dual mandate.

### III.3 Illustrative simulation results (baseline initial conditions and neutral rate path)
- Baseline initial conditions (as of 2014Q4):
  - Output gap: -2 percent.
  - PCE inflation: about 1.5 percent.
  - Federal funds rate: about zero.
- Neutral (equilibrium real) funds rate assumption in simulations:
  - Neutral rate rises from 0 in 2014Q4 to about 1.3 percent in 2020Q4.
  - With a two percent inflation target, equilibrium nominal rate rises from 2 percent in 2014Q4 to 3.3 percent in 2020Q4.
  - Neutral rate assumption noted as close to, but below, median FOMC long-term real fed funds rate forecast (~1.75 percent).
- Key simulation comparisons:
  - IFB reaction function:
    - Federal funds rate rises gradually starting almost immediately.
    - Anticipated increase in equilibrium real rate prompts earlier tightening.
    - Slower closing of the output gap; inflation below target for a longer period.
  - DM1 (loss-function with equal weights):
    - Federal funds rate remains at zero lower bound until mid-2016 before beginning to rise.
    - Output gap and inflation overshoot somewhat: output gap reaches a positive 0.3 percent maximum; inflation reaches 2.2 percent maximum.
    - Overshoot reflects deliberate stimulative policy stance to avoid deep downside risks given ZLB constraints.

### III.4 Sensitivity analysis: variants and comparative outcomes
- DM2 calibration: half the weight on the output gap as on the inflation gap.
  - Empirical result: DM1 and DM2 lines sit almost on top of each other; halving weight on output gap makes hardly any economic difference.
- Comparison across four approaches (Taylor, IFB, DM1, DM2):
  - IFB and Taylor rule produce very similar results to each other.
  - Major difference is between the pair of loss-function minimizations (DM1/DM2) versus the pair of policy rules (IFB/Taylor).
  - Loss-function cases produce a much lower market real rate, yielding stronger growth (output gap) and higher inflation over time compared to reaction functions.
  - Planned/predicted overshooting under DM1/DM2 is intended to insure against costly downside risks given rising marginal costs and ZLB constraints.
- Nonlinear Phillips curve and overheating scenarios:
  - Linear Phillips curve baseline specification:
    - Parameters preserved as in source: λ1 = 0.10; λ2 = 0.70; equations as given for the linear case.
  - Non-linear Phillips curve specification (calibration preserved from source):
    - Parameters preserved as in source: λ1 = 0.10; λ2 = 0.70 with nonlinear terms; calibration implies inflation cannot go below -5 percent and as output gap approaches +5 percent, inflation can go to infinity.
  - Overheating scenarios include large expansionary (financial conditions) shocks hitting in 2017q1 and 2017q2.
  - Results and implications:
    - Overshooting risks: inflation can increase sharply if expansionary shocks hit when inflation overshoots.
    - Under the loss-function approach, central bank raises interest rates much more aggressively in response to expansionary shocks than under IFB; thus inflation is lower and output gap smaller under loss-function approach.
    - Under IFB, non-linear versus linear Phillips curve produces different policy reactions and inflation outcomes; under loss-function approach, the different Phillips specifications make hardly any economic difference because the loss-function response “depends on everything” and keeps the economy away from regions where non-linearities dominate.
    - Loss-function approach yields narrower forecast confidence bands compared to IFB.

### III.5 Forecast confidence bands and alternative simulations
- Confidence interval construction:
  - Shaded bands show confidence intervals from 10 to 90 percent in 10 percentage-point increments.
  - Left-hand panels in Figures 8–10: confidence intervals for forecast federal funds rate, output gap, and year-on-year inflation for IFB and two loss-function calibrations.
  - Right-hand panels: results if initial output gap believed to be -4 percent rather than -2 percent.
- Key quantitative and probabilistic findings:
  - Overshoots are very small relative to, for example, the 30 to 70 percent confidence bands.
  - Under IFB, there is a 15 to 20 percent probability of deflation one year into the future.
  - Loss-function approach yields wider confidence bands for the interest rate (skewed because of the zero-interest rate floor) and much narrower bands for output and inflation than IFB.
  - Volatility is somewhat greater under the -4 percent initial output gap case versus -2 percent due to higher likelihood that the ZLB will impede stabilizing policy response.

### III.6 Summing up: policy implications and risk management
- When policy rates are at or near the zero lower bound (ZLB):
  - Quadratic loss-function approach appears to give better results because it implies extended commitment to hold the rate at the floor and stronger response to contractionary shocks to avoid deflation/dark corners.
  - Holding the short-term rate at zero long enough that inflation rises (perhaps temporarily above the long-run target) boosts inflation expectations, reducing the real medium-term interest rate even if the nominal short rate cannot be lowered.
  - Under circumstances with high risk of being stuck in a deflation trap and little risk of sustained inflationary pressure, this approach represents prudent risk management.
- Complementarity of approaches:
  - Information from Taylor rule and IFB approaches should not be ignored; they can serve as useful crosschecks when specification of the loss-function model is uncertain.
- Nonlinearities:
  - Benefits of focusing on a quadratic loss function are stronger when additional nonlinearities (e.g., nonlinear Phillips curve, nonlinear credibility responses) exist.

### IV. Publication of endogenous interest rate forecasts (motivation and benefits)
- Rationale for endogenous interest rate forecasts:
  - Monetary policy is forward-looking and pre-emptive; central bank’s views on future output and inflation are central to decision making and communications.
  - Forecasts with endogenous interest rates are internally coherent: they anchor the system and avoid logical inconsistency where inflation paths do not evoke offsetting monetary policy reactions.
  - For the Fed, the endogenous interest rate path will be determined by its dual mandate: inflation returns to its 2 percent objective and the economy reaches full employment over the policy horizon.
  - Within feasible paths that satisfy the mandate, the FOMC will choose a path resulting in output close to potential at the end of the policy horizon, inflation approaching target, and movements of key variables not being unduly volatile.
- Communication benefits:
  - Publishing the Fed’s outlook with an endogenous interest rate scenario aids transparency and helps manage expectations of future interest rate movements.
  - An endogenous interest rate convention allows the Fed to tell a logical and coherent story about policy reactions to pressures on output and inflation.

*Italicized source attribution: IMF staff paper content from _wp15134 - references for further detail on the GPM family of models.*

### conclusion that the only right thing is to explicitly discuss the interest rate path and to choose

### _wp15134 - conclusion that the only right thing is to explicitly discuss the interest rate path and to choose

### Recommendation: publish an explicit interest rate path as the main forecast
- “the only right thing is to explicitly discuss the interest rate path and to choose a particular path as the main forecast, as well as publishing the interest rate path and justifying its selection.”
- Not publishing the interest rate forecast “would be to hide the most important information.”
- Publishing a single baseline endogenous interest rate scenario that is consistent with the baseline economic outlook is the recommended practice to avoid leaving the public unclear about the interest rate path underlying output and inflation projections.

### Limitations of the Fed’s “dots” and current practice
- The Fed’s summary of economic projections and FOMC members’ interest rate projections (“the dots”) provide useful information about individual policymakers’ views.
- Shortcomings of the “dots”:
  - “they do not necessarily represent the FOMC’s majority view of the expected policy rate path.”
  - They may be “counterproductive in obfuscating (or at time contradicting) the majority view of the FOMC.”
  - It is “hard to judge internal consistency” because for each “dot” there is presumably a consistent view of growth, interest rate, and inflation paths, but the public cannot identify which dots belong to which individual.
  - “the median of the dots may be far from internally consistent with the median of the macroeconomic forecast.”

### Perceived risks and market learning about conditional forecasts
- Principal perceived risk: some market participants might interpret a published path as a commitment to bring about that path regardless of new developments.
- Important communication requirement: emphasize conditionality — the forecast “is conditional upon information available at the time of writing and will almost certainly change as new information (and new interpretations) become available.”
- Example communications practice: Riksbank Governor (Ingves, 2007) noted that publishing the repo rate path is a forecast “that currently appears most likely given the information available. We are not making any promises.”
- Empirical observation: despite initial learning periods, experience in New Zealand since June 1997, Norway since November 2005, Sweden since February 2007, Israel since July 2007 and the Czech Republic since February 2008 “suggests that the financial market participants learn fairly quickly about the conditional nature of the forecast interest rate path.”

### Practical and institutional difficulties in committee settings
- Forging agreement on an expected path is difficult when the decision-making committee is large and diverse.
- Arguments against committee-produced single path:
  - “attempting to agree on a path in a committee setting could complicate the process of deciding on the next change in the policy rate.”
  - “Diverse views on the economic outlook and the appropriate policy rate path among committee members would likely make it difficult to come to agreement on an FOMC endorsed policy rate path.”
  - The decentralized nature of the Federal Reserve system adds complications for formulating model-based policy.
- Specific logistical issue: The Fed with twelve voting members “might have to publish up to twelve (or more if non-voting FOMC member’s projections are also included) complete projections” if the committee were to take direct ownership of the projection.

### Whose forecast should be published: two options and the preferred approach
- Option one: baseline forecast and alternative scenarios supplied by FOMC members and aggregated by staff — “technically demanding” and requiring transparency in assumptions; could be achieved by releasing a complete forecast for each FOMC member (Mester, 2015).
- Option two (preferred in many central banks): staff produce and publish the baseline forecast that the majority of the FOMC could endorse through the monetary policy report, with alternative scenarios to show dispersion of views.
- Reasons to prefer staff ownership:
  - Practical judgment: staff can maintain coherence and consistency in projections, especially for central banks with large policymaking bodies and divergent views.
  - “Maintaining the mechanics of the projection within the staff creates an efficient mechanism for producing a coherent projection,” which supports credibility among expert observers.
  - The staff baseline is an important input but “represents only one input” and policymakers may disagree with it; alternative scenarios and discussion of uncertainties can reflect policymakers’ views.
- Important safeguard: committee members who disagree with the staff should have “solid economic arguments” and differences should be transparently explained in the monetary policy report to help market understanding and potentially increase credibility.

### Model evidence and policy implications
- A simple model shows that a loss-function minimization approach that includes publication of a baseline forecast and related uncertainties, replacing the “dot” plot, can enhance the effectiveness of policy actions.
- Simulation under conditions of:
  - a negative output gap,
  - below-target inflation,
  - a binding zero interest floor,
  results in a modest, planned overshooting of inflation.
- Mechanism: monetary stimulus arises from expectations that the short-term policy interest rate will be held at zero for an extended period and the boost to expected inflation, which reduces current and expected real interest rates.
- Policy interpretation: by getting the economy away from the “dark corner of the deflation spiral” more quickly, this approach can be viewed as a risk management strategy.

### Conclusion: benefits of publishing a staff baseline interest rate path
- Publishing a staff-produced baseline interest rate path, endorsed by the majority of the FOMC and accompanied by alternative scenarios and clear explanations of conditionality, would:
  - clarify the interest rate path underlying macroeconomic forecasts,
  - guide public expectations about future inflation and policy rates,
  - strengthen the effectiveness of monetary policy,
  - allow the Fed’s goals to be reached more quickly and with “smaller and shorter-lasting deviations over time.”

*Source: _wp15134 - conclusion that the only right thing is to explicitly discuss the interest rate path and to choose*

### Appendix 1. Definitions

### Appendix 1. Definitions

### Key variables and variable transformations
- i_t: Federal funds rate (%)
- r_t: Real interest rate (%) with relation: 1 + r_t = i_t − π_{t+1} (expressed in the source as 1+ − = t t t ir π — interpreted as r_t = i_t − π_{t+1})
- r*_t: Steady-state level of real interest rate (%)
- r^e_t: Equilibrium real interest rate (%) with process: r^e_t = 0.95 * r^e_{t-1} + 0.05 * r^*_t + r^ε_t (source shows r_t r_{t-1} r ε + + = *05.0*0.95, preserving numeric weights 0.95 and 0.05)
- r4_t: Four-quarter average real interest rate (%) defined as 4/)(4 321 + + + + + + = t t t t t r r r r r (explicitly: r4_t = (r_t + r_{t-1} + r_{t-2} + r_{t-3})/4 with the source notation preserved)
- r4^e_t: Four-quarter average equilibrium real interest rate (%) with analogous four-quarter averaging
- r^l_t: Long-term real interest rate (%) defined as a weighted average of current short-term real rate and a four-quarter average over current year and next 16 quarters with exact weights preserved from source expression:
  - weights in source expression: )/544444(*0.2+     )/3444(*0.35+4 *0.35+ *0.1 161284 84 + + + + + + + + = (preserve structure and numeric weights as presented)
- r^{l,e}_t: Equilibrium long-term real interest rate (%) with identical weighting structure to r^l_t (source repeats the same weighted-average form)
- P_t: PCE price index
- p_t: Log of PCE price index with relation: p_t = )log(*100 t P p = (preserve the source formatting)
- π_t: Quarterly PCE inflation (%) with relation: π_t = p_t − p_{t-1} (source: 1− − = t t t p p π)
- π_{4,t}: Year-on-year PCE inflation (%) defined as 4/)(4 321 − − − + + + = t t t t t π π π π π (preserve source four-quarter/year-on-year expression)
- π^*: Inflation objectives (2%)
- y_t: Output gap
- BLT_t: Bank Lending Tightening condition
- i_t^ε: Deviation from the interest rate reaction function
- π_t^ε: Supply shock
- y_t^ε: Shock to the output gap

### Averaging and forecast notations (as presented)
- Four-quarter averages and year-on-year aggregates are denoted by 4/)(4 style expressions in the source; future-quarter forecasting notation appears as 3/4+ etc. (source uses expressions such as 4/)(4 and 3 4 + t π)
- Forecasted and monitored inflation components enter model equations as combinations of observable/monitored inflation and model forecasts, indicated in the source by text and the multi-term expressions with indices for leads and lags (preserve the source characterization that future rates are “partly based on observable or monitored inflation and partly on the model forecast”)

### Error and shock terms
- The definitions list explicit error/shock variables:
  - r^ε_t: error term in equilibrium short-term real interest rate process
  - y_t^ε: shock to the output gap
  - π_t^ε: supply shock in Phillips curve
  - BLT_t^ε: shock to bank lending tightening condition
- The BLT variable is constructed from Senior Loan Officer Survey responses; changes in BLT_t (a reduction in t η in the main text) correspond to easing/tightening financial conditions affecting aggregate demand and the output gap.

*Source: Appendix 1. Definitions (page extract provided).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15134.pdf_