## _wp04213

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---

### I. Purpose and interpretation of “forward looking” monetary policy
- Two interpretations of “forward looking”:
  - Responding to the public’s expectations which affect contemporaneous fundamentals.
  - Responding to shocks about which policymakers have some information.
- With rational expectations and no private information the two interpretations coincide; with private information asymmetries they may diverge.
- Paper uses Greenbook forecasts (produced before each FOMC meeting by Board Research staff) to represent the Federal Reserve’s expectations and provides statistical evidence that policymakers respond to shocks (expected or not).

### II. Advantages of Greenbook forecasts and real-time output-gap estimation
- Twofold advantage of Greenbook forecasts:
  - Allows testing for the rule’s “direction.”
  - Allows calculation of more precise real-time estimates of lagged potential output available to policymakers.
- Contrast with common practice:
  - Most authors use ex post output-gap estimates from detrending revised data.
  - This paper includes Greenbook forecasts in the information set for output-gap estimation for greater realism.

### III. Reestimating lag-based rules — output-gap specifications and errors
- Three output-gap specifications:
  - "Lags and Leads": ex post HP detrending using full path 1947:Q1—1999:Q2 (one-sided leads and lags).
  - "Lags Only": real-time HP detrending using lagged observations to t-1 with unrevised data (Philadelphia Fed real-time dataset).
  - "Lags and Forecasts": detrend series of observed lagged values 1947:Q1—t-1 plus forecasts for t through t+4 (Greenbook forecasts).
- Real-time errors (Lags-and-Leads minus Lags-Only):
  - Mean: 0.53
  - St. Deviation: 1.45
  - Max: 3.33
  - Min: -1.54
- Real-time errors (Lags-and-Leads minus Lags-and-Forecasts):
  - Mean: 0.24
  - St. Deviation: 0.87
  - Max: 2.09
  - Min: -0.97
- Inclusion of Greenbook forecasts roughly halves mean and standard deviation of errors relative to Lags-Only; Lags-and-Forecasts estimates mostly lie between Lags-and-Leads and Lags-Only.

### IV. Estimating lag-based Taylor rules — regression results (sample 1987:Q3–1994:Q4)
- Estimated backward-looking rule: i_t = C + C_π π_{t-1} + C_y y_{t-1} (i = federal funds rate; π = annual growth of GNP/GDP deflator; y = output gap).
- Table 2 outcomes (coefficients with standard errors in parentheses; Adj R² and SSR):
  - Lags and Leads:
    - C_π = 1.58 (0.31)
    - C_y = 1.25 (0.13)
    - CNST = 0.32 (1.05)
    - Adj R² = 0.86
    - SSR = 19.88
  - Lags and Forecasts:
    - C_π = 2.33 (0.23)
    - C_y = 1.02 (0.14)
    - CNST = -1.39 (0.94)
    - Adj R² = 0.82
    - SSR = 24.58
  - Lags Only:
    - C_π = 2.90 (0.27)
    - C_y = 0.83 (0.17)
    - CNST = -2.97 (1.11)
    - Adj R² = 0.73
    - SSR = 38.01
- Key inferences:
  - Lags-and-Forecasts yields a strong inflation response C_π ≈ 2.3 and output-gap response C_y ≈ 1.0.
  - Lags-and-Leads estimates are closer to canonical Taylor values (C_π ≈ 1.5, C_y ≈ 1.25).
  - Hypothesis tests:
    - Cannot reject that C_y (lags-and-forecasts) = 1.25 (lags-and-leads value).
    - Reject C_π = 1.58 at 0.05 level for lags-and-forecasts.
    - Reject joint hypothesis C = 0.32, C_π = 1.58, C_y = 1.25 at 0.05 level.
  - Conclusion: inclusion of forecasts materially affects estimated parameters, especially the inflation coefficient.

### V. Theoretical forward-looking rule — model and identification
- Model (annual):
  - Inflation (accelerationist Phillips curve): π_{t+1} = π_t + α y_{t+1} + ε_{t+1}
  - IS curve (inertial output gap): y_{t+1} = y_t - λ (r_t - r^*) + η_{t+1}
  - Optimal real rate: r_t = r^* + C_π (E_t π_{t+1} - π^*) + C_y E_t y_{t+1} + terms in expected shocks.
- Derived parameter relationships:
  - C_π = q/β
  - C_y = (λ + qα)/β
- Identification:
  - Greenbook forecasts provide E_t π_{t+3} and E_t y_{t+3}; residuals from forecasting equations can proxy expected inflation shock E_t ε_{t+1}.
  - Expected output shock E_t η_{t+1} cannot be recovered from Greenbook because it depends on unobserved assumed interest-rate paths.

### VI. Empirical construction of expected inflation shocks and forward-looking rule estimates (quarterly)
- Quarterly notation:
  - Lagged quarterly inflation: π_{t-1}^q
  - Lagged output gap: y_{t-1}^q (based on lags-and-forecasts)
  - Inflation forecast one year ahead: E_t^q π_{t+3}
- Estimated forecast Phillips curve (equation (5)):
  - E_t^q π_{t+3} = 1.004 + 0.34 y_{t-1}^q + 0.08 π_{t-1}^q  with R² = 0.72
  - Reported standard errors: (0.08) on π_{t-1}^q and (0.04) on y_{t-1}^q
  - Interpretation: coefficient on lagged inflation close to 1; output-gap coefficient 0.34 corresponds to a sacrifice ratio of 3.
- Define expected inflation shock residual:
  - E_t^q ε_{t+3} = E_t^q π_{t+3} - (1.004 + 0.34 y_{t-1}^q + 0.08 π_{t-1}^q)
  - These shocks have standard deviation 0.48; peaks ≥ 1.0 percent at 1987:H2, 1988.2, and 1992.2.
- Estimated quarterly policy rules:
  - Forward-looking (includes expected inflation shock):
    - i_t = 3.20 + 2.03 π_{t-1}^q + 1.52 y_{t-1}^q + 0.70 E_t^q ε_{t+3}
    - Adj R² = 0.89
    - SSR = 14.94
    - Reported standard errors: (0.32) on constant, (0.28) on π_{t-1}^q, (0.29) on y_{t-1}^q, (0.70) on E_t^q ε_{t+3}
  - Backward-looking (lags only):
    - i_t = 1.39 + 3.23 π_{t-1}^q + 0.94 y_{t-1}^q
    - Adj R² = 0.82
    - SSR = 24.58
    - Reported standard errors: (0.94) on constant, (0.23) on π_{t-1}^q, (0.14) on y_{t-1}^q
- Empirical findings:
  - Inclusion of expected inflation shock raises Adj R² from 0.82 to 0.89 and lowers SSR from 24.58 to 14.94.
  - Difference between coefficient on lagged inflation and coefficient on expected inflation shock ≈ 1, consistent with model prediction.
  - Forward-looking specification explains about 90 percent of movements in the Federal Funds Rate with lagged inflation, lagged output gap, and expected inflation shocks; remaining 10 percent may be due to expected output-gap shocks not recoverable from Greenbook.

### VII. Implied long-run targets and parameter calculations
- Algebraic relations used in source to infer π^* and r^* from estimated coefficients.
- Reported calculations:
  - Assuming r^* = 2.00 implies π^* = 3.22.
  - Assuming r^* equal to the sample's average of 2.48 implies π^* = 3.61.
  - Assuming π^* = 2.00 implies r^* = 0.47.
  - Assuming π^* equal to the sample's average of 3.48 implies r^* = 2.32.
- Implication: a reasonable real interest rate target of 2.0–2.3 percent implies an appropriate inflation target around 3.2–3.5 percent, which translates into a nominal interest-rate target of 5.2–5.8 percent; sample average nominal interest rate is 5.97 percent.

### VIII. Historical evidence — fit, episodes, and unexplained movements
- Comparison of actual federal funds rate with fitted backward- and forward-looking rules:
  - Most of the time differences are small; including expected inflation shock brings forward-looking fit closer to actual path.
  - Several episodes where backward-looking rule differs substantially (~1 percent or more) from actual rate; forward-looking rule aligns more closely when expected inflation shocks are high.
- Specific episodes (as reported):
  - 1987:Q3 and 1987:Q4:
    - Backward rule: 6.07 percent and 5.49 percent.
    - Forward-looking rule: 7.03 percent and 6.76 percent.
    - Actual federal funds rates: 6.84 percent and 6.92 percent.
  - 1988:Q2–1988:Q3: peak expected inflation shock linked to liquidity after October 1987 stock market crash, later prompting tightening.
  - 1992:Q2: peak in expectations reflects PPI rebound after low 1991 PPI due to oil-price decline post-1990:Q3 Middle East conflict.
- Remaining unexplained movements:
  - From 1988:Q3 through 1989:Q1, actual policy was much tighter than the forward-looking rule suggests (differences reported up to 1.36 percent).

### IX. How “tough” was Paul Volcker? — time-varying targets and reinterpretation
- Misleading inference from constant-target Taylor rule:
  - Standard Taylor-rule regressions assume constant r^* and π^*, so a falling actual inflation series during Volcker’s tenure produces large measured inflation gaps and mechanically small estimated C_π.
  - Estimating constant-target specification for 1979:Q3–1987:Q2 yields an inflation response roughly equal to 1, output-gap response roughly 0, real interest rate set equal to 4.23; regression R² = 0.77.
- Alternative approach: allow a time-varying intercept Vt:
  - Assume Volcker’s policy coefficients matched Greenspan-era estimates (C_π = 2.5; C_y = 1.0) and compute implied time-varying intercept Vt that rationalizes observed interest rates.
  - With choice r^* = 4.32 the implied inflation target π*_t and actual inflation π_t move similarly; a break around mid-1983 is evident, after which both move around a flatter trend.
- Inflation-gap statistics during Volcker’s tenure (reported):
  - Full period: range π*_t − π_t from -2.09 to 3.56; mean = 0.44; standard deviation = 1.47.
  - After 1983:Q3: absolute inflation gap ≤ 1.25 percent; mean = 0.06; standard deviation = 0.89.
- Interpretation:
  - A constant-target Taylor rule can mischaracterize an aggressive, time-varying-target policy as “passive.”
  - Allowing for a time-varying inflation target reconciles observed interest-rate setting with an aggressive policy responding strongly to inflation; Volcker could be as “tough” as Greenspan under this interpretation.

*Source: _wp04213 (excerpt and reported tables/figures as provided).*

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

### _wp04213 - References

### I. INTRODUCTION — purpose and interpretation of “forward looking” monetary policy
- Policymaking is argued to be forward looking, but the term permits various interpretations.
- Two main interpretations discussed:
  - Forward looking in the sense of aiming at the public's expectations, which in turn affect contemporaneous fundamentals.
  - Forward looking in the sense of responding to shocks, about which policymakers have some information.
- In environments with rational expectations but without private information, the two interpretations coincide; with private information asymmetries they may diverge.
- Romer and Romer (2000) are cited as providing evidence of asymmetry in information available to the public versus the Federal Reserve.
- This study provides statistical evidence that policymakers respond to shocks (whether expected by the public or not), using Greenbook forecasts of inflation and output to represent the Federal Reserve's expectations.
- Greenbook forecasts are produced before each meeting of the Federal Open Market Committee by the Research staff at the Board of Governors (footnote marker 2).

### Advantages of using Greenbook forecasts and real-time output gap estimation
- Twofold advantage of Greenbook forecasts:
  - Allows testing for the rule's “direction.”
  - Allows calculation of more precise real-time estimates of lagged potential output available to policymakers.
- Contrast with common practice:
  - Most authors use ex post estimates of output gap obtained by detrending revised output data.
  - Tchaidze (2001) uses a more realistic specification of output gap based on preliminary estimates of actual output using only lagged observations.
  - This paper advances realism by including Greenbook forecasts into the information set for output gap estimation.

### Empirical approach and model framing
- Section II: Estimates backward-looking rules using different estimates of the lagged output gap:
  - Starts with a commonly used specification based on both lags and leads of output.
  - Proceeds to more realistic specifications based first on lags only, and then on lags plus forecasts reflecting additional information policymakers have.
- Section III: Uses a simple model to derive a forward-looking rule that responds to:
  - Inertial components of inflation and output gap.
  - Expected shocks.
  - Discusses empirical results and argues that the inflation response coefficient is higher than usually suggested values, at about 2.5.
- Section IV: Provides historical evidence justifying Federal Reserve behavior consistent with the model.
- Section V: Attempts to explain why a similar exercise applied to Paul Volcker’s tenure does not produce results consistent with his inflation-hawk reputation.
- Paper contains six sections in total.

### Empirical findings highlighted in the text
- Calculated shocks based on Greenbook forecasts are statistically significant in the estimated policy rule.
- The analysis confirms that policymakers take into account future developments, and that policy is more preemptive than a simple response to inertial components alone would suggest.
- Among other results, the inflation response coefficient is argued to be about 2.5.

### Supporting material listed in the excerpt
- Tables:
  - Table 1. Errors in Real-Time Estimates of Output Gap
  - Table 2. Rules with Various Output Gap Specifications
- Figures:
  - Figure 1. Output Gap Estimates
  - Figure 2. Expected t+ 3 Inflation Shock as of t
  - Figure 3. Federal Funds Rate: Actual and Fitted Values
  - Figure 4. Expected t+3 Output Gap Shock as of t
  - Figure 5. Actual Inflation and Implied Inflation Target During Volcker's Tenure

*Source: _wp04213 - References (excerpt) — pages and sections as provided in the source content.*

### Section VI concludes.

### _wp04213 - Section VI concludes.

### II. REESTIMATING LAG-BASED RULES — specification and data
- Backward-looking Taylor rule estimated: it responds only to movements in lagged fundamentals and is specified as i_t = C + C_π π_{t-1} + C_y y_{t-1}, where:
  - i is the overnight interest rate (federal funds rate);
  - π is inflation, measured as an annual growth of the GNP/GDP deflator;
  - y is the output gap (difference between log-output and its trend).
- All data are quarterly. Sample runs from 1987:Q3 to 1994:Q4.
- Lagging fundamentals reflects policymakers' most up-to-date information.

### II.A Three sets of output gap estimates — construction and errors
- Three output-gap specifications used:
  - "Lags and Leads": ex post, detrend full path log-output 1947:Q1—1999:Q2 using Hodrick-Prescott (HP) filter (one-sided leads and lags included).
  - "Lags Only": real-time, detrend lagged observations from 1947:Q1 until t-1 using one-sided HP filter; uses unrevised data (Philadelphia Fed real-time dataset).
  - "Lags and Forecasts": detrend series consisting of observed lagged values (1947:Q1 to t-1) plus forecasts for t through t+4 (Greenbook forecasts) — intended to approximate policymakers' information set.
- Real-time errors (Lags-and-Leads minus Lags-Only):
  - Mean: 0.53
  - St. Deviation: 1.45
  - Max: 3.33
  - Min: -1.54
- Real-time errors (Lags-and-Leads minus Lags-and-Forecasts):
  - Mean: 0.24
  - St. Deviation: 0.87
  - Max: 2.09
  - Min: -0.97
- Inclusion of Greenbook forecasts significantly reduces errors: both mean and standard deviation roughly halved relative to Lags-Only; Lags-and-Forecasts estimates mostly lie between Lags-and-Leads and Lags-Only.

### II.B Estimating the lag-based rule — results
- Estimations use OLS/Newey-West procedure (accounting for heteroscedasticity and autocorrelation).
- Table 2 regression outcomes (coefficients reported with standard errors in parentheses; Adj R² and SSR reported):
  - Lags and Leads:
    - C_π = 1.58 (0.31)
    - C_y = 1.25 (0.13)
    - CNST = 0.32 (1.05)
    - Adj R² = 0.86
    - SSR = 19.88
  - Lags and Forecasts:
    - C_π = 2.33 (0.23)
    - C_y = 1.02 (0.14)
    - CNST = -1.39 (0.94)
    - Adj R² = 0.82
    - SSR = 24.58
  - Lags Only:
    - C_π = 2.90 (0.27)
    - C_y = 0.83 (0.17)
    - CNST = -2.97 (1.11)
    - Adj R² = 0.73
    - SSR = 38.01
- Findings:
  - Strong estimated inflation response of 2.3 (lags-and-forecasts), higher than commonly suggested ranges 1.4–2.0.
  - Strong output-gap response around 1.0 (lags-and-forecasts), at the right end of commonly suggested range 0.5–1.0.
  - Lags-and-leads estimates are closer to suggested Taylor-type values (1.5 for inflation, 1 for output gap, constant 1).
- Hypothesis tests:
  - Cannot reject that the output gap coefficient for lags-and-forecasts equals 1.25 (value from lags-and-leads).
  - Reject C_π = 1.58 at 0.05 significance level for lags-and-forecasts.
  - Reject joint hypothesis C = 0.32, C_π = 1.58, and C_y = 1.25 at 0.05 significance level.
- Conclusion: confidence intervals for parameters across lags-and-leads and lags-and-forecasts intersect, but tests indicate substantial differences—particularly that the inflation coefficient in lags-and-forecasts is much higher than 1.5.

### III. FORWARD-LOOKING RULES — theoretical basis
- Model framework: standard closed-economy model (Romer, 2000; Ball, 1999; Svensson, 1997; Rudebusch and Svensson, 1999).
- Annual model equations:
  - Inflation (accelerationist Phillips curve): π_{t+1} = π_t + α y_{t+1} + ε_{t+1}
  - IS curve (inertial output gap): y_{t+1} = y_t - λ (r_t - r^*) + η_{t+1}
  - Policymakers set real interest rate r_t and minimize weighted sum of inflation and output-gap variances → optimal rule: r_t = r^* + C_π (E_t π_{t+1} - π^*) + C_y E_t y_{t+1} + terms in expected shocks.
- Derived relationships:
  - C_π = q/β and C_y = (λ + qα)/β, where q reflects relative weight on inflation variance.
  - Reformulated nominal interest-rate rule includes response to expected inflation shock E_t ε_{t+1} and expected output shock E_t η_{t+1}.
- Identification issues:
  - Greenbook provides forecasts for output growth and inflation; residuals from forecasting equations can proxy expected inflation shocks E_t ε_{t+1}.
  - Expected output shock E_t η_{t+1} cannot be recovered because it depends on assumed interest-rate paths not observed in the Greenbook.
- Empirical test implications:
  - Inclusion of expected inflation shock in estimated rule should reduce SSR and increase fit if policy is forward looking.
  - Model implies coefficient difference between lagged inflation and expected inflation shock should equal 1.

### III.B Empirical results — construction of expected inflation shock and rule estimates
- Quarterly notation:
  - Lagged quarterly inflation: π_{t-1}^q
  - Lagged output gap: y_{t-1}^q (based on lags-and-forecasts)
  - Inflation forecast one year ahead corresponds to the third-quarter horizon forecast E_t^q π_{t+3}
- Estimated forecast Phillips curve (equation (5)):
  - E_t^q π_{t+3} = 1.004 + 0.34 y_{t-1}^q + 0.08 π_{t-1}^q  with R² = 0.72
  - Coefficients in parentheses: (0.08) on π_{t-1}^q and (0.04) on y_{t-1}^q as reported.
  - Interpretation: coefficient on lagged inflation close to 1; output-gap coefficient 0.34 corresponds to a sacrifice ratio of 3 (output gap of -3 percent causes 1 percent decrease in inflation), close to Mankiw (1997) and Sachs (1985) estimates 2.8 and 2.9.
- Define expected inflation shock as residual:
  - E_t^q ε_{t+3} = E_t^q π_{t+3} - (1.004 + 0.34 y_{t-1}^q + 0.08 π_{t-1}^q)
  - These shocks have standard deviation 0.48; notable peaks ≥ 1.0 percent at 1987:H2, 1988.2, and 1992.2.
- Estimated policy rules (quarterly, forward- and backward-looking specifications):
  - Forward-looking (includes expected inflation shock):
    - i_t = 3.20 + 2.03 π_{t-1}^q + 1.52 y_{t-1}^q + 0.70 E_t^q ε_{t+3}
    - Adj R² = 0.89; SSR = 14.94
    - Reported standard errors: (0.32) on constant, (0.70) on E_t^q ε_{t+3}, (0.28) on π_{t-1}^q, (0.29) on y_{t-1}^q as shown in source.
  - Backward-looking (lags only):
    - i_t = 1.39 + 3.23 π_{t-1}^q + 0.94 y_{t-1}^q
    - Adj R² = 0.82; SSR = 24.58
    - Reported standard errors: (0.94) on constant, (0.14) on y_{t-1}^q, (0.23) on π_{t-1}^q as shown in source.
- Key empirical findings:
  - Inclusion of expected inflation shock increases fit (Adj R² from 0.82 to 0.89) and lowers SSR (from 24.58 to 14.94).
  - The difference between coefficients on lagged inflation and expected inflation shock is close to 1, consistent with model prediction.
  - Evidence supports that Federal Reserve policy is forward looking (incorporates expected shocks), not purely mechanical.
  - Estimated inflation-response coefficients remain higher than commonly suggested values (e.g., Orphanides (1998a) reported 2.51 when using forecasts).

### III.B Implied targets and long-run parameter calculations
- Formulas used to infer targets:
  - r^* = Ĉ - (Ĉ_π / Ĉ) π^*  (as described in source in algebraic form)
  - π^* = Ĉ_π + Ĉ r^*  (as described in source)
- Calculations reported:
  - Assuming r^* = 2.00 implies π^* = 3.22.
  - Assuming r^* equal to the sample's average of 2.48 implies π^* = 3.61.
  - Assuming π^* = 2.00 implies r^* = 0.47 (described as unrealistically low).
  - Assuming π^* equal to the sample's average of 3.48 implies r^* = 2.32.
- Implication: a reasonable real interest rate target of 2.0–2.3 percent implies an appropriate inflation target around 3.2–3.5 percent, translating into a nominal interest-rate target of 5.2–5.8 percent; sample average nominal interest rate is 5.97 percent.

### IV. HISTORICAL EVIDENCE — implications and examples
- Comparison of actual federal funds rate with fitted values from backward-looking and forward-looking rules (Figure 3):
  - Most of the time, difference between fitted series is small; including expected inflation shock brings forward-looking fit closer to actual path.
  - Several episodes where backward-looking rule differs substantially (~1 percent or more) from actual rate, whereas forward-looking rule aligns more closely—these divergences are associated with high expected inflation shocks.
- Specific historical episodes:
  - 1987:Q3 and 1987:Q4:
    - Backward rule suggested rates: 6.07 percent and 5.49 percent respectively.
    - Forward-looking rule suggested: 7.03 percent and 6.76 percent respectively.
    - Actual federal funds rates: 6.84 percent and 6.92 percent respectively.
    - Context: sharp increase in inflationary expectations; 1988 Economic Report of the President cites potential for greater inflation linked to dollar weakness and building inflation expectations as rationale for tightening.
  - 1988:Q2–1988:Q3:
    - Peak in expected inflation shock linked to excessive liquidity after October 1987 stock market crash; later led to policy reversal and tightening when it became “evident that the stock market crash would not seriously affect spending growth.”
  - 1992:Q2:
    - Peak in inflation expectations reflects producer-price inflation rebound after unusually low 1991 PPI driven by declining oil prices post-1990:Q3 Middle East conflict.
- Remaining unexplained movements:
  - Inclusion of expected inflation shock does not explain all movements of the federal funds rate.
  - From 1988:Q3 through 1989:Q1, actual policy was much tighter than the forward-looking rule suggests (differences reported up to 1.36 percent in the source).

*Source: _wp04213 - Section VI concludes.*

### 1.52 percent and 1.38 percent), while in 1989:Q2 and from 1993:Q2 through 1994:Q1, the

### V. HOW “TOUGH” WAS PAUL VOLCKER?

### Expected output gap shocks and residuals
- Equation (4) implies residuals from the forward-looking specification of the rule are proportional to the expected output gap shock, given by qttE3 1+ − ηβ, where β is a parameter of the IS curve (2) reflecting the sensitivity of output gap to changes in real interest rate.
- Ball (1999) uses β = 1 in his calibrations.
- The paper defines an empirical variable εt equal to a function of expected future inflation and output-gap forecasts and plots it in Figure 4 to identify periods when policy deviated from the rule (e.g., 1988:Q3–1989:Q1 showing much tighter policy; 1989:Q2 and 1993:Q2–1994:Q1 suggesting expected declines in output).

### Misspecification of a constant inflation target and implications for Volcker-era estimates
- The standard Taylor-rule specification assumes constant targets r* and π*, so an intercept from regressing it on lagged inflation πt-1 and output gap yt-1 reflects those constant targets.
- During Volcker’s tenure inflation peaked near 10 percent in early 1981 and fell to 2.3 percent in 1986; assuming a constant inflation target implies very large inflation gaps (πt-1 − π*), which mechanically produces a small estimated inflation response coefficient Cπ.
- Estimating the constant-target specification for 1979:Q3–1987:Q2 yields (HAC standard errors in parentheses):
  - Estimated rule (reported): inflation response is roughly equal to 1 and output gap response is roughly 0; the real interest rate is set equal to 4.23.
  - The regression R² = 0.77 (as reported).

### Reconciliation: time-varying inflation target and constructing Vt
- The paper proposes an alternative: assume Volcker’s policy coefficients matched Greenspan’s estimated coefficients (inflation response = 2.5; output gap response = 1.0) and compute a time-varying intercept Vt that rationalizes observed interest rates under that rule.
- Under equation (8) and this assumption, Vt should equal 5.1/) (r*επ+−−1t t and oscillate around a negatively sloped trend; oscillations reflect εt and average to zero in the author’s interpretation.
- Choosing r* = 4.32 equalizes average πt and implied π*t for the 1983:Q3–1987:Q2 period; with this choice:
  - The implied inflation target π*t has a decreasing trend similar to actual inflation πt.
  - Both series show a clear break around mid-1983, after which they move around a flatter trend.

### Inflation gap statistics during Volcker’s tenure (as reported)
- The inflation gap (π*t − πt) ranges from -2.09 to 3.56.
- Mean of the inflation gap over the full period is 0.44.
- Standard deviation of the inflation gap over the full period is 1.47.
- After 1983:Q3:
  - The inflation gap does not exceed 1.25 percent in absolute value.
  - Mean = 0.06.
  - Standard deviation = 0.89.
- The paper notes that if εt could be calculated precisely the inflation gap might have been even smaller.

### Interpretation and substantive findings
- A simple Taylor rule that assumes a constant inflation target can mischaracterize an aggressive, time-varying-target policy as “passive.”
- The exercise suggests Paul Volcker could have been as “tough” as Alan Greenspan (i.e., inflation response = 2.5 and output gap response = 1.0) if policymakers operated with a time-varying, declining inflation target.
- The Greenspan-era estimated rule (using Greenbook forecasts and forward-looking elements) shows an inflation response coefficient of 2.5 and output gap response of 1.0.
- Together, the three variables used in the author's forward-looking Taylor rule (lagged inflation, lagged output gap, and expected future inflation shocks from Greenbook forecasts) explain about 90 percent of movements in the Federal Funds Rate; the remaining 10 percent may represent expected output gap shocks that could not be recovered from the Greenbook forecasts.

### Conclusion relevant to Volcker’s evaluation
- The apparent passivity of monetary policy during Volcker’s chairmanship, as suggested by a simple Taylor rule with constant targets, may reflect misspecification rather than true policy slack.
- Allowing for a time-varying inflation target reconciles observed interest-rate setting with an aggressive policy responding strongly to inflation, consistent with the Fed being “tough” on inflation during Volcker’s tenure.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp04213.pdf*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp04213.pdf_