## _wp04164

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---

### I. Introduction — motivation and objectives
- Since the beginning of the 21st century, a new era of “price stability” was reached; core CPI inflation reached 1.1 percent by end-2003.
- After the 2001 recession the Fed sharply reduced the federal funds rate target to 1 percentage point, the lowest in 40 years, and kept it at that level for twelve months (between the months of June of 2003 and 2004).
- Recent debates motivating the paper:
  - Use of “unconventional” monetary policy tools if the federal funds rate were to hit zero.
  - A “risk management” approach to monetary policy, advocated by Federal Reserve Chairman Alan Greenspan.
- Paper objectives:
  - Allow asymmetric and nonlinear behavior in Taylor-type monetary policy rules for the United States.
  - Answer two main questions:
    - Have coefficients of the Taylor rule changed over time?
    - Do the coefficients of the Taylor rule change when economic conditions change?
  - Summary answer stated in the text: yes to both questions.
- Main empirical summary:
  - During expansions, the Federal Reserve targets inflation and conducts monetary policy in a highly inertial way.
  - During recessions, the Fed targets output growth and moves the policy rate more sharply.
- Model implication:
  - In a small-scale macroeconomic model, a state-dependent rule can further reduce output volatility when a cost-push shock hits the economy, but at the cost of higher inflation.

### II. How well does the Taylor rule fit the data?
- Description of Taylor (1993) parameterization (notation preserved from source):
  - it = federal funds rate (specified in annualized terms).
  - πt = measure of inflation (last four quarters of the GDP deflator inflation).
  - yt = output gap (percent deviations of actual output from long-term (trend) output).
  - Taylor’s parameterization:
    - steady state real interest rate *r = 2 percent.
    - inflation target *tπ = 2 percent → steady state nominal interest rate = 4 percent.
    - coefficient on inflation deviation πγ = 0.5.
    - coefficient on output deviation yγ = 0.5.
- Empirical illustration:
  - Figure 1 simulated Taylor rule using quarterly data between 1982:Q4 and 2003:Q4, with trend growth assumed 2.8 percent (sample mean).
  - Taylor rule fits well between 1987 and 1994; poor fit in other periods.
  - Specific numeric example: For the last quarter of 2003, the Taylor rule would suggest a value of 3.1 percent for the federal funds rate, while the actual value is 1 percent.
  - In the past three quarters (relative to publication), the distance between rule and actual exceeded 150 basis points.

### III. Time-varying parameters — specification and smoothed results
- Model specification highlights:
  - Estimated a time-varying Taylor-type rule with interest rate smoothing coefficient ρt, time-varying long-run elasticities γπ,t and γg,t with respect to inflation and output growth, output growth Δyt targeted instead of unobservable output gap, and lagged interest rate included.
  - Time-varying intercept c*t,gt,* derived as .)1( *, , *, ** ty tg tt t gr c γ π γ π − − −= (notation preserved).
  - Parameters follow random walks; maximum likelihood used for shock variances and Kalman filter for smoothed coefficients (Hamilton (1994) approach).
- Smoothed-series results and notable episodes:
  - Elasticity to inflation and interest-rate smoothing swing substantially over time.
  - Volcker disinflation: inflation elasticity rises above 2, peaking at 2.3 in 1984:Q3.
  - Interest-rate smoothing declined during the 1980s, became almost zero during the Volcker disinflation, increased afterwards, then declined during the last three years of the sample.
  - During or after recessions (e.g., 1990–1991 and 2001), the inflation elasticity generally decreases.
  - After 2001:Q1 the inflation coefficient declined markedly and became negative (interpreted as preemptive easing).
  - Fluctuations in the intercept and the reaction to output growth are quantitatively less important.
  - When the Fed faced other high risks (1987 stock crash, fall 1990 Kuwait invasion, fall 1998 Russian default, September 11, 2001, and recent deflation risk), the elasticity with respect to inflation falls.

### IV. Regime-switching (state-dependent) Taylor rule — method and estimated state properties
- Two-stage estimation:
  1. Hamilton (1994) switching regime model for output growth Δyt with state st ∈ {1,2} where st=1 is expansion (high mean) and st=2 is recession (low mean).
  2. Use smoothed probabilities of each state as weights in weighted least squares to estimate Taylor-rule coefficients in each state (weighted OLS using square roots of probabilities as weights).
- Estimated state properties (Table 1 / text):
  - Average growth rate in booms (μ1 = c(st=1)): 4.62 percent (Std. Error: 0.38).
  - Average growth rate in recessions (μ2 = c(st=2)): -0.57 percent (Std. Error: 0.90).
  - Transition probabilities:
    - p11 = 0.91 (Std. Error: 0.03).
    - p22 = 0.73 (Std. Error: 0.09).
  - Shock standard deviation σ = 3.35 (Std. Error: 0.19).
  - Implied durations and probabilities:
    - Booms last on average 11 quarters (1/(1-p11)).
    - Recessions last on average roughly 4 quarters.
    - Unconditional probability of expansion = 0.75; recession = 0.25.

### V. Taylor-rule estimates by state — short-run and long-run coefficients (quarterly annualized)
- Short-run coefficients (state-specific weighted estimates; standard errors in parentheses):
  - Recession short-run:
    - Constant: -0.22 (0.24)
    - Inflation: 0.10 (0.04)
    - Output: 0.13 (0.03)
    - Smoothing (lagged i): 0.90 (0.04)
  - Expansion short-run:
    - Constant: -0.31 (0.17)
    - Inflation: 0.17 (0.03)
    - Output: 0.03 (0.02)
    - Smoothing (lagged i): 0.95 (0.03)
- OLS (pooled) short-run for comparison:
  - Constant: -0.34 (0.20)
  - Inflation: 0.14 (0.04)
  - Output: 0.09 (0.02)
  - Smoothing: 0.92 (0.03)
- Wald tests:
  - Short-run: statistic 31.60 (p-value 0.00) — reject equality of coefficients across states.
  - Long-run: statistic 12.07 (p-value 0.02) — reject equality across states.
- Key contrasts and interpretation:
  - Inflation coefficient: expansion 0.17 vs. recession 0.10 (both significant at the 5 percent level).
  - Output-growth coefficient: recession 0.13 vs. expansion 0.03 (in expansion significant at 10 percent only).
  - Interest-rate smoothing larger in expansions (0.95) than in recessions (0.90).
  - OLS coefficients lie between the two regime-specific rules; individually only output growth coefficient is significantly different across regimes, but joint Wald test rejects equality.

- Long-run coefficients (Table 2; OLS / Recession / Expansion; Std. Errors in parentheses where reported):
  - Constant: OLS -4.41 (3.85); Recession -2.14 (2.86); Expansion -6.29 (5.85).
  - Inflation long-run: OLS 1.77 (0.54); Recession 1.00 (0.34); Expansion 3.45 (1.48).
  - Output long-run: OLS 1.16 (0.59); Recession 1.25 (0.61); Expansion 0.71 (0.58).
  - Smoothing long-run: OLS 0.92 (0.03); Recession 0.90 (0.04); Expansion 0.95 (0.03).
- Long-run interpretation:
  - Reaction to inflation much stronger in expansions (3.45) than in recessions (1.00).
  - In recessions the “Taylor principle” is not implemented in the point estimate (reported as 0.9972 in text).
  - Long-run output response increases in recessions (reported as 1.25 in table), making the real interest rate countercyclical.

### VI. Annual-rate specification (Table 3) — main differences
- Annual-rate short-run and long-run estimates (selected):
  - Short-run smoothing: OLS 0.92; Recession 0.90; Expansion 0.94.
  - Short-run inflation: OLS 0.15; Recession 0.12; Expansion 0.17.
  - Short-run output: OLS 0.18; Recession 0.28; Expansion 0.09.
  - Long-run inflation: OLS 1.80; Recession 1.14; Expansion 3.14.
  - Long-run output: OLS 2.13; Recession 2.72; Expansion 1.69.
- Wald tests (annual):
  - Short-run: 30.67 (p-value 0.001) — reject equality across states.
  - Long-run: 16.48 (p-value 0.002) — reject equality across states.
- Notable: coefficients on output reaction are higher in annual specification and significant at 10 percent for long-run; in recessions Taylor principle respected with point estimate 1.14.

### VII. Relation to Federal Reserve chairmen (1960–2003) and dummy results
- Chairmanship/dummy periods used:
  - Burns and Miller: 1970:Q1–1979:Q2
  - Volcker non-borrowed reserves targeting: 1979:Q3–1982:Q4
  - Volcker and Greenspan interest-rate target: 1983:Q1–2003:Q4
  - 1960s used as baseline without dummies.
- Wald test summary (Table 4) highlights:
  - With Burns-Miller, Nonborrowed reserves, post-83 dummies:
    - All coefficients equal (null): Wald 26.42; p-value 0.04 — reject.
    - Dummies equal across states: Wald 16.39; p-value 0.17 — cannot reject.
    - Once dummies equal, levels equal: Wald 9.91; p-value 0.04 — reject.
  - With Burns-Miller and Volcker-Greenspan dummies:
    - All coefficients equal: Wald 30.88; p-value 0.002 — reject.
    - Dummies equal across states: Wald 7.61; p-value 0.47 — cannot reject.
    - Once dummies equal, levels equal: Wald 23.46; p-value 0.001 — reject.
- Interpretation:
  - Coefficients shift between expansions and recessions even when accounting for different chairmen.
  - Dummy effects across chairmen are similar in magnitude in expansions and recessions (cannot reject equality).
  - Burns-Miller period stands out with much smaller reaction to inflation and larger reaction to lagged interest rates relative to 1960s and post-1983.

- Selected short-run coefficients with dummies (Table 5; state-specific):
  - Recession / Expansion (selected):
    - Constant: -0.45 / -0.56 (Std. Errors (0.25) / (0.16))
    - Inflation: 0.52 / 0.51 (Std. Errors (0.07) / (0.05))
    - Output: 0.28 / 0.15 (Std. Errors (0.05) / (0.03))
    - Smoothing: 0.61 / 0.71 (Std. Errors (0.06) / (0.05))
  - Interactions for Inflation*Burns-Miller and Smoothing*Burns-Miller and Smoothing*Volcker-Greenspan are reported in the table (coefficients and Std. Errors).

### VIII. Small-scale New Keynesian model with regime-switching policy — calibration and policy rules
- Model structure:
  - New Phillips Curve (AS): πt = γb πt−1 + (1−γb) Et πt+1 + λ yt + ut (notation preserved).
  - Output (IS) equation with habit formation parameter b and forward-looking expectations.
- Calibration:
  - γb reported in text with preserved notation; calibration references Fuhrer and Moore (1995) for equal weights.
  - λ = 0.2
  - b = 0.5
- Policy-rule coefficients used (rounded but within 95 percent CI) from Table 3 (notation preserved in text):
  - Recession: γπ = 1.5, γg = 1.0, ρ = 0.9 (text notation: 5.1 ,1 ,9.0=== y γ γ ρ π).
  - Expansion: γπ = 3.5, γg = 0.0, ρ = 0.95 (text notation: 0 ,5.3 ,95.0===).
  - Steady state (weighted average: 3/4 expansion, 1/4 recession): γπ = 2.5, γg = 0.6, ρ = 0.9375 (text notation: 6.0 ,5.2 ,9375.0===).
- Simulation setup:
  - Impulse responses to demand shocks and cost-push shocks under three rules: "recession" rule, "expansion" rule, and "steady-state" rule.
  - Shock autoregressive coefficient set to 0.8.
  - Agents form expectations including return to the long-run "average" or steady-state rule.

### IX. Impulse-response dynamics under state-dependent rules — descriptive highlights
- Demand shock (Figure 4 summary):
  - "Recession" rule produces more volatility in output and inflation; not reacting enough to inflation creates higher volatility in nominal interest rate.
  - "Steady-state" dynamics closer to the "expansion" rule due to substantial inflation-targeting component.
- Cost-push shock (Figure 5 summary):
  - Output volatility similar across "recession" and "expansion" rules; inflation more volatile under "recession" rule.
  - "Recession" rule reduces initial drop in output but the maximum loss is similar and delayed by one quarter.
  - "Steady-state" rule yields inflation more volatile but output much less volatile than the other two rules—illustrating stabilizing properties of strong inflation targeting with scope for output stabilization.
- State-dependent responses (Figures 6 and 7 summary):
  - When shocks cause drops in output, the central bank switching to recession mode alters dynamics: output and inflation fall for one additional period and recovery takes about four periods; the rebound is larger under state-dependent rule.
  - With a positive cost-push shock: interest rates increase on impact then decrease afterwards; output recovers faster initially but inflation picks up later leading to oscillatory behavior.

### X. Second moments: simulation design and key volatility statistics (Table 6)
- Simulation design:
  - 1000 simulations of length 180 each (corresponding to 45 years).
- Volatility statistics (Percent deviations / volatility measures from Table 6; entries taken exactly):
  - Inflation under Cost Push Shock:
    - Rec.: 3.10
    - Exp.: 1.99
    - Steady State: 2.32
    - State Dependent: 3.14
  - Output under Cost Push Shock:
    - Rec.: 6.43
    - Exp.: 6.58
    - Steady State: 6.47
    - State Dependent: 6.15
  - Int. Rates under Cost Push Shock:
    - Rec.: 0.35
    - Exp.: 0.82
    - Steady State: 0.65
    - State Dependent: 0.18
  - Inflation under Demand Shock:
    - Rec.: 0.53
    - Exp.: 0.29
    - Steady State: 0.32
    - State Dependent: 0.45
  - Output under Demand Shock:
    - Rec.: 0.59
    - Exp.: 0.38
    - Steady State: 0.41
    - State Dependent: 0.50
  - Int. Rates under Demand Shock:
    - Rec.: 0.18
    - Exp.: 0.15
    - Steady State: 0.15
    - State Dependent: 0.16
- Key second-moment findings:
  - Cost-push shocks:
    - Expansion rule stabilizes inflation best.
    - Recession rule stabilizes output volatility better.
    - Steady state rule yields intermediate results.
    - State-dependent rule yields highest inflation volatility (marginally larger than recession rule), lowest output volatility, and least nominal interest-rate volatility.
    - Low nominal interest-rate volatility under state-dependent rules suggests policy announcements and expectation management can be as powerful as adjusting the policy interest rate.
  - Demand shocks:
    - Expansion (strict inflation targeting) rule delivers the smallest variability in all variables.
    - Steady state rule performs well.
    - Switching rule and recession rule perform worse (in that order).

### XI. Robustness checks and model variants
- Robustness exercises under alternative model specifications produced qualitatively similar results:
  - Purely forward looking model: (0=b, )0= b γ.
  - Forward-looking with parameters: (5.0=b, )0= b γ.
  - Only backward looking behavior in inflation equation: (.0=b, )5.0= b γ.
- Qualitative outcomes were very similar across these variants.

### XII. Conclusions and policy implications
- Empirical evidence: coefficients of the Taylor rule change over time and with economic conditions.
  - During expansions: the Fed operates as a strong inflation targeter with a high degree of policy inertia.
  - During recessions: the Fed targets output growth and moves faster.
- Historical association:
  - Burns-Miller period: response of the nominal interest rate to inflation was very low and interest rate inertia was much higher than during the 1960s or the post-1983 period.
- Policy recommendations from simulations:
  - In the context of the model, strict inflation targeting is the best policy rule overall.
  - During recessionary supply shocks, a switching regime rule may be worthwhile because it minimizes output variability in recessions, potentially at the cost of higher inflation variability.
- Research agenda:
  - Introducing nonlinear elements into the model is a promising area for future research.
  - Optimality of switching regime rules should be explored when either the loss function is not quadratic or constraints are nonlinear.

*Source: Fund staff estimates and simulations as presented in the source PDF.*

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

### References

### I. INTRODUCTION — A. Motivation
- Since the beginning of the 21st century, a new era of “price stability” was reached; core CPI inflation reached 1.1 percent by end-2003.
- After the 2001 recession the Fed sharply reduced the federal funds rate target to 1 percentage point, the lowest in 40 years, and kept it at that level for twelve months (between the months of June of 2003 and 2004).
- Recent events raised two debates:
  - Use of “unconventional” monetary policy tools if the federal funds rate were to hit zero.
  - A “risk management” approach to monetary policy, advocated by Federal Reserve Chairman Alan Greenspan.
- Consequence: the Taylor rule appears to have lost power as an approximation to actual policymaking.
- Paper objectives:
  - Allow asymmetric and nonlinear behavior in Taylor-type monetary policy rules for the United States.
  - Answer two main questions:
    - Have coefficients of the Taylor rule changed over time?
    - Do the coefficients of the Taylor rule change when economic conditions change?
  - Summary answer stated in the text: yes to both questions.
- Main empirical result highlighted:
  - During expansions, the Federal Reserve targets inflation and conducts monetary policy in a highly inertial way.
  - During recessions, the Fed targets output growth and moves the policy rate more sharply.
- Model implication:
  - In a small-scale macroeconomic model, a state-dependent rule can further reduce output volatility when a cost-push shock hits the economy, but at the cost of higher inflation.

### I. INTRODUCTION — B. How Well Does the Taylor Rule Fit the Data?
- Theoretical context:
  - Rules vs. discretion literature: Kydland and Prescott (1977); Barro and Gordon (1983).
  - Interest-rate-rule literature: Woodford (2003); Clarida, Galí, and Gertler (1999).
- Practical/policy context:
  - Central bankers emphasize judgment and risk management; simple mechanical rules may perform well under normal conditions but not under high-risk or abnormally large shocks.
  - Chairman Greenspan’s view: apply judgment and principles of risk management rather than follow a mechanical rule.
- Taylor (1993) rule specification described (notation preserved from source):
  - it = federal funds rate (specified in annualized terms).
  - πt = measure of inflation (last four quarters of the GDP deflator inflation).
  - yt = output gap (percent deviations of actual output from long-term (trend) output).
  - Equation form (as given in text): )( ** t* t t t yy r i − + − + + = γ π π γ π π t (equation text preserved exactly as in source).
  - Taylor’s parameterization (as stated in text):
    - steady state real interest rate *r = 2 percent.
    - inflation target *tπ = 2 percent → steady state nominal interest rate = 4 percent.
    - coefficient on inflation deviation πγ = 0.5.
    - coefficient on output deviation yγ = 0.5.
- Practical limitations of Taylor’s rule for policymaking (as stated):
  - Need estimates of steady-state real interest rate to obtain steady-state nominal interest rate.
  - Potential output is unobservable; Taylor assumed a trend growth rate of 2.2 percent between 1984:Q1 and 1992:Q3.
- Empirical illustration from the source:
  - Figure 1 simulated Taylor rule using quarterly data between 1982:Q4 and 2003:Q4, using Taylor’s parameterization except trend growth assumed to be 2.8 percent (the sample mean for the period).
  - Taylor rule fits well between 1987 and 1994 (extending Taylor’s original study by two years).
  - In other periods, fit is poor; during and after the 2001 recession the Fed eased policy much faster than the rule would suggest.
  - Specific numeric example: For the last quarter of 2003, the Taylor rule would suggest a value of 3.1 percent for the federal funds rate, while the actual value is 1 percent.
  - In the past three quarters (relative to publication), the distance between rule and actual exceeded 150 basis points.
- Paper’s methodological approach:
  - Reconcile policymakers’ nuanced characterization of policy with rules used in academic analysis by estimating a policy rule with nonconstant parameters and examining whether monetary responses vary systematically over the business cycle.
  - Such behavior would capture policymakers’ consideration of factors not in typical response regressions and capture nonlinearities in the Taylor rule.
- Roadmap of the paper (as given):
  - Section II: estimates a model with time-varying coefficients; result noted that the Fed’s tolerance to deviations of inflation from its target increases during recessions or when other risks appear.
  - Section III: estimates the Taylor rule in a switching regime framework; finds the Fed behaves differently in booms and recessions.
  - Section IV: relates switching-regime estimates to Fed chairmen since 1960.
  - Section V: explores implications in a small-scale macro model.
  - Section VI: concludes and suggests directions for future work.

### II. TIME-VARYING PARAMETERS
- Literature context:
  - Empirical work often studies structural breaks in the Taylor rule, focusing on pre- and post-Greenspan years.
  - Typical result (e.g., Clarida, Galí, and Gertler (2000)): the policy rule places more weight on inflation stabilization after 1982.
- Identification challenge highlighted:
  - Structural-break approaches cannot easily discern whether different Fed behavior is due to chairmen preferences or different economic events/shocks that required different (possibly nonlinear and asymmetric) policy actions.
- Section’s raised empirical question (verbatim from source):
  - 1. Have the parameters of the Taylor rule changed significantly over time?

*Italic source: _wp04164 - References..............................................................................................................*

### 2.  Does the Fed behave differently when the state of the economy changes?

### 2.  Does the Fed behave differently when the state of the economy changes?

### Model specification and estimation approach
- Estimated time-varying Taylor-type rule (equation (1)) with:
  - interest rate smoothing coefficient ρt,
  - time-varying long-run elasticities γπ,t and γg,t with respect to inflation and output growth,
  - output growth Δyt targeted instead of an unobservable output gap,
  - lagged interest rate included to capture policy "smoothness".
- Time-varying intercept c*t,gt,* derived as .)1( *, , *, ** ty tg tt t gr c γ π γ π − − −= (text notation preserved).
- All parameters assumed to follow a random walk; estimation via maximum likelihood for shock variances and Kalman filter for smoothed coefficients (Hamilton (1994) approach).

### Time-varying coefficient results (smoothed series)
- Elasticity to inflation and interest-rate smoothing swing substantially over time.
- Notable episodes:
  - Volcker disinflation: inflation elasticity rises above 2, peaking at 2.3 in 1984:Q3.
  - Interest-rate smoothing declined during the 1980s, became almost zero during the Volcker disinflation, increased afterwards, then declined during the last three years of the sample.
- During or after recessions (e.g., 1990–1991 and 2001), the inflation elasticity generally decreases.
- In episodes where the Fed faced other high risks (1987 stock crash, fall 1990 Kuwait invasion, fall 1998 Russian default, September 11, 2001, and recent deflation risk), the elasticity of the Fed Funds rate with respect to inflation falls—interpreted as greater tolerance for inflation when other risks dominate.
- After 2001:Q1 the inflation coefficient declined markedly and became negative, interpreted as preemptive easing to avoid recession and a decline in the inflation rate.
- Fluctuations in the intercept and the reaction to output growth are quantitatively less important.

### Regime-switching (state-dependent) Taylor rule: method
- Two-stage estimation:
  1. Hamilton (1994) switching regime model for output growth Δyt with state st ∈ {1,2} where st=1 is expansion (high mean) and st=2 is recession (low mean). Only the mean of growth changes with state. Transition probabilities pij = Pr(st = j | st−1 = i).
  2. Use smoothed probabilities of each state as weights in weighted least squares to estimate Taylor-rule coefficients in each state (weighted OLS using square roots of probabilities as weights, equivalently maximizing mixture log-likelihood).
- Model for Δyt: εt ~ iid N(0, σ2) and st follows first-order Markov.

### Switching-regime model for output: estimated state properties (Table 1 / text)
- Average growth rate in booms (μ1 = c(st=1)): 4.62 percent (annualized); Std. Error: 0.38
- Average growth rate in recessions (μ2 = c(st=2)): -0.57 percent (annualized); Std. Error: 0.90
- Transition probabilities:
  - p11 = 0.91; Std. Error: 0.03
  - p22 = 0.73; Std. Error: 0.09
- Shock standard deviation σ = 3.35; Std. Error: 0.19
- Implied durations and probabilities:
  - Booms last on average 11 quarters (1/(1-p11)).
  - Recessions last on average roughly 4 quarters.
  - Unconditional probability of expansion = 0.75; recession = 0.25.

### Taylor-rule estimates by state (short-run and long-run coefficients, quarterly annualized rates; Table 2)
- Short-run coefficients (OLS; Recession / Expansion):
  - Constant: -0.34 / -0.31 (Std. Errors: (0.20) / (0.17))
  - Inflation: 0.14 / 0.17 (Std. Errors: (0.04) / (0.03))
  - Output: 0.09 / 0.03 (Std. Errors: (0.02) / (0.02))
  - Smoothing (lagged i): 0.92 / 0.95 (Std. Errors: (0.03) / (0.03))
- Recession / Expansion reported separately (weighted estimates):
  - Recession short-run: Constant -0.22 (0.24), Inflation 0.10 (0.04), Output 0.13 (0.03), Smoothing 0.90 (0.04)
  - Expansion short-run: Constant -0.31 (0.17), Inflation 0.17 (0.03), Output 0.03 (0.02), Smoothing 0.95 (0.03)
- Wald Tests (short-run): statistic 31.60 (p-value 0.00); long-run: 12.07 (p-value 0.02)
- Key short-run contrasts (text interpretation preserved):
  - Inflation coefficient: expansion 0.17 vs. recession 0.10 (both significant at the 5 percent level).
  - Output-growth coefficient: recession 0.13 vs. expansion 0.03 (in expansion significant at 10 percent only).
  - Interest-rate smoothing larger in expansions (0.95) than in recessions (0.90).
  - OLS coefficients lie between the two regime-specific rules; individually only output growth coefficient is significantly different across regimes, but Wald test rejects equality of all coefficients.
- Long-run coefficients (Table 2; OLS / Recession / Expansion):
  - Constant: OLS -4.41; Recession -2.14; Expansion -6.29 (Std. Errors: (3.85), (2.86), (5.85))
  - Inflation long-run: OLS 1.77; Recession 1.00; Expansion 3.45 (Std. Errors: (0.54), (0.34), (1.48))
  - Output long-run: OLS 1.16; Recession 1.25; Expansion 0.71 (Std. Errors: (0.59), (0.61), (0.58))
  - Smoothing long-run: OLS 0.92; Recession 0.90; Expansion 0.95 (Std. Errors: (0.03), (0.04), (0.03))
- Long-run interpretation:
  - Reaction to inflation much stronger in expansions (3.45) than in recessions (1.00).
  - In recessions the “Taylor principle” is not implemented in the point estimate (reported as 0.9972 in text).
  - Long-run output response increases in recessions (1.26 in one reporting) indicating stronger easing to negative output growth; makes real interest rate countercyclical.

### Annual-rate specification (Table 3) — main differences
- Using annual inflation and growth rates:
  - Short-run and long-run smoothing similar: smoothing OLS 0.92, Recession 0.90, Expansion 0.94.
  - Inflation short-run: OLS 0.15; Recession 0.12; Expansion 0.17.
  - Output short-run: OLS 0.18; Recession 0.28; Expansion 0.09.
  - Long-run inflation: OLS 1.80; Recession 1.14; Expansion 3.14.
  - Long-run output: OLS 2.13; Recession 2.72; Expansion 1.69.
- Differences: coefficients on output reaction are higher and significant at 10 percent for long-run; in recessions Taylor principle respected with point estimate 1.14.
- Wald tests (annual): short-run 30.67 (p-value 0.001); long-run 16.48 (p-value 0.002) — reject equality across states.

### Relation to Federal Reserve chairmen (1960–2003)
- Three chairmanship/dummy periods used in tests:
  - Burns and Miller: 1970:Q1–1979:Q2
  - Volcker non-borrowed reserves targeting: 1979:Q3–1982:Q4
  - Volcker and Greenspan interest-rate target: 1983:Q1–2003:Q4
  - 1960s used as baseline without dummies.
- Wald test summary (Table 4):
  - Using dummies for Burns-Miller, Nonborrowed reserves targeting, post-83 periods:
    - All coefficients equal (null): Wald 26.42; p-value 0.04 (quarterly rates) — reject.
    - Dummies equal across states: Wald 16.39; p-value 0.17 — cannot reject.
    - Once dummies equal, levels equal: Wald 9.91; p-value 0.04 — reject.
  - Using dummies for Burns-Miller, Volcker-Greenspan periods:
    - All coefficients equal: Wald 30.88; p-value 0.002 — reject.
    - Dummies equal across states: Wald 7.61; p-value 0.47 — cannot reject.
    - Once dummies equal, levels equal: Wald 23.46; p-value 0.001 — reject.
- Interpretation:
  - Coefficients of the rule shift between expansions and recessions even when accounting for different chairmen.
  - Dummy effects across chairmen are similar in magnitude in expansions and recessions (cannot reject equality).
  - Burns-Miller period stands out: much smaller reaction to inflation and larger reaction to lagged interest rates relative to 1960s and post-1983.

### Estimates with dummies (selected results from Table 5; short-run coefficients)
- Recession / Expansion (selected):
  - Constant: -0.45 / -0.56 (Std. Errors (0.25) / (0.16))
  - Inflation: 0.52 / 0.51 (Std. Errors (0.07) / (0.05))
  - Output: 0.28 / 0.15 (Std. Errors (0.05) / (0.03))
  - Smoothing: 0.61 / 0.71 (Std. Errors (0.06) / (0.05))
- Additional reported interactions:
  - Inflation*Burns-Miller and Smoothing*Burns-Miller and Smoothing*Volcker-Greenspan included (coefficients and Std. Errors reported in table).

### Small-scale New Keynesian model with regime-switching policy
- Model structure:
  - New Phillips Curve (AS): πt = γb πt−1 + (1−γb) Et πt+1 + λ yt + ut (text variant notation preserved).
  - Output equation (IS / demand): relation (3) with habit formation parameter b and forward-looking expectations.
  - Calibration:
    - γb = 0.5 (text: “coefficient of 5.0 = γb” preserved as 5.0=γb in text; interpreted in narrative as equal weights forward/backward per Fuhrer and Moore (1995)).
    - λ = 0.2
    - b = 0.5
  - Policy-rule coefficients used (rounded but within 95 percent CI) from Table 3:
    - Recession: γπ = 1.5, γg = 1.0, ρ = 0.9 (text: "5.1 ,1 ,9.0=== y γ γ ρ π" preserved notation)
    - Expansion: γπ = 3.5, γg = 0.0, ρ = 0.95 (text: "0 ,5.3 ,95.0===" preserved)
    - Steady state (weighted average: 3/4 expansion, 1/4 recession): γπ = 2.5, γg = 0.6, ρ = 0.9375 (text: "6.0 ,5.2 ,9375.0===" preserved)
- Simulation scenarios:
  - Impulse responses to demand shocks and cost-push shocks under three rules: "recession" rule, "expansion" rule, and "steady-state" rule.
  - Shock autoregressive coefficient set to 0.8.

### Impulse response findings (simulations; descriptive highlights)
- Demand shock (Figure 4):
  - "Recession" rule produces more volatility in output and inflation; not reacting enough to inflation creates higher volatility in nominal interest rate.
  - "Steady-state" dynamics closer to the "expansion" rule due to substantial inflation-targeting component.
- Cost-push shock (Figure 5):
  - Output volatility similar across "recession" and "expansion" rules; inflation more volatile under "recession" rule.
  - "Recession" rule reduces initial drop in output (it targets output growth when below potential) but the maximum loss is similar and delayed by one quarter.
  - "Steady-state" rule yields inflation more volatile but output much less volatile than the other two rules—shows stabilizing properties of strong inflation targeting with scope for output stabilization.
- Asymmetry and state-dependent switching (Figures 6 and 7 context):
  - When shocks cause drops in output (negative demand shock or positive cost-push), the state-dependent rule (central bank switching to recession mode) alters dynamics relative to symmetric, non-switching cases.
  - Agents form expectations including return to the long-run "average" or steady-state rule; short-run abandonment of anti-inflation stance need not imply permanent abandonment if credibility holds.

### Key substantive conclusions
- Empirical evidence supports a two-state (expansion vs. recession) characterization of the Fed's Taylor rule: coefficients differ materially across states.
- In expansions the Fed places relatively greater weight on inflation stabilization; in recessions it shifts focus towards output stabilization and eases faster.
- Interest-rate smoothing is generally higher in expansions than recessions (Fed acts quicker in recessions).
- The Fed’s effective policy rule changed across chairmen, notably the Burns-Miller period with smaller inflation response and larger lagged-rate inertia; however, the cyclical shift (difference between expansion and recession) is similar across chairmen.
- Regime-dependent policy rules have important implications in a New Keynesian framework: recession-mode rules can increase inflation volatility and nominal rate volatility, while steady-state (average) rules tend to balance inflation targeting with output stabilization.

*Source: Fund staff estimates and simulations as presented in the source PDF.*

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### Impulse-response dynamics under state-dependent rules
- The “recession” rule creates more volatility; the state-dependent rule under recession also creates more volatility in inflation and output.
- Both output and inflation fall for one additional period after the negative shock hits the economy.
- The recovery takes the same time under both rules: about four periods.
- The rebound is larger under the state-dependent rule than otherwise.
- With a positive cost-push shock (Figure 7):
  - Interest rates increase on impact (because of the initial focus on inflation) but decrease afterwards (because of the focus on output growth).
  - Output seems to recover faster from the shock, but eventually inflation picks up, the central bank tightens again, and output displays an oscillatory behavior.
- Figures referenced (impulse responses):
  - Figure 6: Impulse Response to a Demand Shock with a State-Dependent Rule — responses shown for Inflation, Output, Nom. Int., and Demand over Quarters After Shock (0–25).
  - Figure 7: Impulse Response to a Cost-Push Shock with a State-Dependent Rule — responses shown for Inflation, Output, Nom. Int., and Cost-Push over Quarters After Shock (0–25).
- Source for figures: Fund staff simulations.

### Second moments: simulation design and key volatility statistics
- Simulation design:
  - We run 1000 simulations of length 180 each (corresponding to 45 years).
  - Results are displayed in Table 6.
- Table 6. Volatility of Inflation, Output and Interest Rates (depending on each rule)
  - Columns indicate shock type (Cost Push Shock, Demand Shock) and policy rules (Rec., Exp., Steady State, State Dependent).
  - Reported numeric entries (Percent deviations / volatility measures) taken exactly from Table 6:
    - Inflation under Cost Push Shock:
      - Rec.: 3.10
      - Exp.: 1.99
      - Steady State: 2.32
      - State Dependent: 3.14
    - Output under Cost Push Shock:
      - Rec.: 6.43
      - Exp.: 6.58
      - Steady State: 6.47
      - State Dependent: 6.15
    - Int. Rates under Cost Push Shock:
      - Rec.: 0.35
      - Exp.: 0.82
      - Steady State: 0.65
      - State Dependent: 0.18
    - Inflation under Demand Shock:
      - Rec.: 0.53
      - Exp.: 0.29
      - Steady State: 0.32
      - State Dependent: 0.45
    - Output under Demand Shock:
      - Rec.: 0.59
      - Exp.: 0.38
      - Steady State: 0.41
      - State Dependent: 0.50
    - Int. Rates under Demand Shock:
      - Rec.: 0.18
      - Exp.: 0.15
      - Steady State: 0.15
      - State Dependent: 0.16
- Key findings from second-moment analysis:
  - With cost-push shocks:
    - The expansion rule (inflation targeting) stabilizes inflation best.
    - The recession rule stabilizes output volatility better.
    - The steady state rule yields intermediate results.
    - The state-dependent rule yields:
      - The highest inflation volatility (marginally larger than the recession rule).
      - The lowest output volatility.
      - The least volatility of the nominal interest rate under cost-push shocks.
    - Policy implication highlighted: central bank (or society) must decide whether accepting higher inflation volatility to achieve lower output volatility in recessions is worthwhile.
    - The low nominal interest-rate volatility under state-dependent rules suggests policy announcements and expectation management can be as powerful as adjusting the policy interest rate.
  - With demand shocks:
    - The expansion (strict inflation targeting) rule delivers the smallest variability in all variables.
    - The steady state rule performs well.
    - The switching (state-dependent) rule and the recession rule perform, in this order, much worse.
    - This outcome aligns with the model class where the optimal rule under demand shocks can stabilize both inflation and output perfectly; the trade-off between inflation and output arises from cost-push shocks.

### Robustness checks and model variants
- Exercises repeated under alternative model specifications yielded qualitatively similar results:
  - Purely forward looking model: (0=b, )0= b γ, with only habit formation but pure forward looking inflation.
  - Forward-looking with parameters: (5.0=b, )0= b γ.
  - Only backward looking behavior in inflation equation: (.0=b, )5.0= b γ.
- The paper notes these robustness checks produced very similar qualitative outcomes.

### Concluding remarks and policy implications
- Empirical evidence: coefficients of the Taylor rule change over time and with economic conditions.
  - During expansions: the Fed operates as a strong inflation targeter with a high degree of policy inertia.
  - During recessions: the Fed targets output growth and moves faster.
- Historical association:
  - During the Burns-Miller period: the response of the nominal interest rate to inflation was very low and interest rate inertia was much higher than during the 1960s or the post-1983 period.
- Policy recommendations from simulations:
  - In the context of the model, strict inflation targeting is the best policy rule overall.
  - During recessionary supply shocks, a switching regime rule may be worthwhile because it minimizes output variability in recessions, potentially at the cost of higher inflation variability.
- Research agenda:
  - Introducing nonlinear elements into the model is a promising area for future research.
  - Optimality of switching regime rules should be explored when either the loss function is not quadratic or constraints are nonlinear.
  - References to related work on nonlinear rules and asymmetric preferences or nonlinear Phillips curves are noted in the conclusion.

*Source: Fund staff simulations.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp04164.pdf_