## wpiea2025128-print-pdf

## Source details

**Canonical URL:** [wpiea2025128-print-pdf](https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025128-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2025/english/wpiea2025128-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2025/english/wpiea2025128-print-pdf.pdf.json)

---

### 2 A Motivating Example — core insight
- Core message:
  - For almost all models there exists some term structure of monetary policy news that can rationalize any given EMPS; without empirical discipline on the term structure, EMPS cannot be used to falsify theory.
- Textbook New Keynesian representation (as presented):
  - New Keynesian Phillips curve: π_t = β E[π_{t+1}] + κ y_t
  - Euler equation: i_t = E_t[γ(y_{t+1} − y_t) + π_{t+1}]
  - Taylor rule: i_t = φ_y y_t + φ_π π_t + ν_t
  - Policy shock decomposition: ν_t = ν_{0,t} + ν_{1,t−1} + ν_{2,t−2} + ...
    - ν_{h,t−h} are i.i.d.; ν_{h,t} ∼ N(0, σ_h^2)
    - ν_{0,t} is the surprise at time t; ν_{h,t−h} is news at t−h about policy h periods ahead
- Empirical illustration:
  - Gertler and Karadi (2015) EMPS produces a gradual deflation over 18 months.
  - Standard surprise ν_{0,t} causes immediate deflation then rapid stabilization.
  - News shocks ν_{h,t} at different h produce linearly independent price IRFs; anticipated future tightenings cause smooth deflations.
  - Linear combinations of surprise and news shocks can approximate any empirical IRF:
    - Using the first n news horizons improves approximation as n increases.
    - With 24 shocks the Gertler-Karadi IRF is reproduced perfectly.
  - Implication: empirical discipline on the term structure is required to distinguish news from surprise interpretations.

### 3 Methodology — framework and estimators
- Monetary policy framework and notation:
  - Policy rule: y_t = x_t φ + r_t  (equation (1))
    - y_t: policy instrument; r_t: exogenous monetary policy residual; x_t: row vector of endogenous inputs; φ: coefficient vector
  - Residual dynamics with smoothing/delays:
    - r_t = Σ_{ℓ=1}^L (ρ_{y,ℓ} y_{t−ℓ} + x_{t−ℓ} φ_ℓ) + ν_t  (equation (2))
  - Innovation decomposition (H_ν horizons):
    - ν_t = ν_{0,t} + ν_{1,t−1} + ... + ν_{H_ν,t−H_ν}  (equation (3))
    - ν_{h,t} ∼ N(0, σ_h^2)
  - EMPS DGP (for EMPS indexed j ∈ J):
    - w^j_t = Σ_{h=0}^{H_w} β^j_h ν_{h,t} + ξ^j_t  (equation (4)), with ξ_t orthogonal to ν_{t+h}
  - Term structure definition:
    - γ^j_h ≡ dE[ν_{t+h} | w^j_t] / dw^j_t
    - Linear representation: ν_t = Σ_{h=0}^{H_w} γ^j_h w^j_{t−h} + u_t  (equation (5))
    - Relationship: γ^j_h = β^j_h Var(ν_{h,t}) / Var(w^j_t)  (equation (6))
- Four-stage estimation strategy (addresses unobserved ν_t and endogeneity):
  1. Instrument endogenous x_t with exogenous shocks z_t orthogonal to u_t and w^j_t.
  2. Estimate φ̂ by 2SLS.
  3. Compute residuals r̂_t = y_t − x_t φ̂ (equation (7)), whiten via r̂_t = x_t ρ + ν_t using L lags (equations (8)–(9)).
  4. Estimate term structure by regressing ν̂_t on stacked EMPS lags w_t: ν̂_t = w_t γ + u_t (equation (10)).
- Closed-form one-step estimator (Proposition 1):
  - γ̂ = (W′W)^{−1} W′ M_X (I − X(X′ P_Z X)^{−1} X′ P_Z) y
  - φ̂ = (X′ P_Z X)^{−1} X′ P_Z y  (equation (11))
  - Definitions include x_t stacked lags, w_t stacking lags of EMPS, M_X ≡ I − X(X′X)^{−1} X′, P_Z ≡ Z(Z′Z)^{−1} Z′.
- Validity requirements:
  - Correct policy rule specification and adequate whitening lag length.
  - Valid instruments z_t satisfying orthogonality restrictions.
- Smoothing generalization (stage 4):
  - Penalized (smoothed) local projections use ridge penalty with matrix P_B and penalty λ (selected by cross-validation).
  - Smoothed estimator (Proposition 4): γ̂_λ = [ W′W + λ P_B ]^{−1} W′ M_X (I − X(X′ P_Z X)^{−1} X′ P_Z) y
  - The smoothed estimator nests the unsmoothed case when λ = 0.
- Theoretical properties:
  - Unbiasedness (Proposition 2): If Z′W = 0 and E[Z′u] = 0, the 4-stage estimator is unbiased.
  - Conditional variance (Proposition 3) given Z′W = 0 and E[Z′u] = 0:
    - Var(γ̂ | W, X, Z) = (W′W)^{−1} W′ M_X [ I − X(X′ P_Z X)^{−1} X′ P_Z ] Ω [ I − X(X′ P_Z X)^{−1} X′ P_Z ]′ M_X W (W′W)^{−1}
    - Ω = E[uu′]; standard errors require consistent Ω estimate using û_t from equation (10).

### 4 Estimated Term Structures — data and main empirical results
- Data for EMPS (selected entries; date ranges preserved in source):
  - Gertler and Karadi (2015): HFI, 30 min. window, 1990:M1-2007:M12
  - Jarociński and Karadi (2020): HFI, 1990:M1-2016:M12
  - Bundick and Smith (2020): HFI, 1994:M2-2019:M06
  - Miranda-Agrippino and Ricco (2021): HFI, 1991:M1-2009:M12
  - Bu et al. (2021): HFI, 1994:M2-2024:M12
  - Bauer and Swanson (2023): HFI, 1988:M2-2023:M12
  - Swanson (2024): HFI, 1988:M2-2023:M12
  - Jarociński (2024): HFI, 1990:M2-2024:M09
  - Romer and Romer (2004): Narrative, 1983:M1-2007:M12
  - Aruoba and Drechsel (2024): Narrative, 1982:M10-2008:M10
- Instruments for policy rule estimation (monthly series):
  - Government spending shocks: Romer and Romer (2016); Fieldhouse et al. (2018)
  - Oil shocks: Känzig (2021); Baumeister and Hamilton (2019)
  - Other shocks: Kim et al. (2025); Adams and Barrett (2024)
- 4.2.1 Estimated Taylor Rules — key numerical findings and diagnostics:
  - Estimation choices:
    - Term structure horizon H_w = 24 (two years)
    - Whitening lag L = 6
    - Include six lags of IVs
  - Inflation coefficient φ_π roughly 1.5 across specifications, largely satisfying the Taylor principle (φ_π > 1).
  - Baseline regressors: 12-month PCE inflation (demeaned) and detrended private non-farm employment (demeaned).
  - Selected parameter values (standard errors in parentheses):
    - 12-month PCE Inflation, demeaned (baseline): 1.517*** (0.035)
    - Alternative inflation coefficients: 1.536*** (0.035); 1.541*** (0.034); 1.528*** (0.034); 1.512*** (0.037); 1.187*** (0.033); 1.512*** (0.033); -0.249*** (0.057); 2.294*** (0.106)
    - Private nonfarm payroll employment, departures from quadratic log trend, demeaned: 0.370*** (0.010); alternates: 0.377*** (0.010); -0.033** (0.014); 0.408*** (0.013)
    - GDP, CF-low-pass: 0.697*** (0.076)
    - GDP growth, annualized, demeaned: 0.031*** (0.010)
    - Industrial Production, CF low-pass: 0.058** (0.026)
    - Unemployment rate: 0.265*** (0.005)
    - Excess bond premium, demeaned: 0.440*** (0.090)
    - Cleveland Fed inflation expectations, 1 year, demeaned: 2.180*** (0.052)
    - Michigan inflation expectations, 1 year, demeaned: -1.170*** (0.141)
  - Goodness-of-fit and diagnostics (selected):
    - Baseline residual autocorrelation: 0.95; R^2: 0.58; Observations: 569
    - Add π^e_1, Clev.: residual autocorrelation 0.89; R^2: 0.79; Observations: 485
    - 1-month PCE Inflation, demeaned coefficient: 0.598*** (0.076); residual autocorrelation 0.96; R^2 0.46; Observations 569
    - 12-month Core PCE Inflation, demeaned: 2.031*** (0.037)
    - 12-month CPI inflation, demeaned: 1.125*** (0.029)
- 4.2.2 Estimated Term Structures — heterogeneity and R^2_k decomposition:
  - Estimated γ_jh plotted with 95 percent confidence intervals (Figure 2 referenced).
  - All shocks display non-trivial term structure at horizons longer than h = 0.
  - Summary statistic R^2_k (Proposition 5):
    - R^2_k ≡ Var(ν_t | w^j_{t−k}) / Var(ν_t | {w^j_{t−h}}_{h=0}^{H_w}) = (γ^j_k)^2 / Σ_{h=0}^{H_w} (γ^j_h)^2
    - Use R^2_{0:1} to capture immediate change in monetary policy innovation.
  - Decomposition by horizons (monthly sums R^2_{0:1}, R^2_{2:6}, R^2_{7:12}, R^2_{13:24}) — selected ordered entries (shocks ordered by R^2_{0:1}):
    - Swanson FG: 0.01 ; 0.22 ; 0.59 ; 0.18
    - Bu−Rogers−Wu: 0.02 ; 0.11 ; 0.16 ; 0.71
    - Jarocinski FG: 0.04 ; 0.35 ; 0.36 ; 0.25
    - Jarocinski Info: 0.18 ; 0.11 ; 0.32 ; 0.40
    - Bundick−Herriford−Smith Slope: 0.21 ; 0.17 ; 0.43 ; 0.19
    - Swanson LSAP: 0.23 ; 0.50 ; 0.22 ; 0.05
    - Bundick−Herriford−Smith Level: 0.24 ; 0.22 ; 0.25 ; 0.29
    - Bauer−Swanson: 0.27 ; 0.36 ; 0.26 ; 0.11
    - Jarocinski LSAP: 0.28 ; 0.45 ; 0.07 ; 0.20
    - Jarocinski−Karadi MPS: 0.28 ; 0.25 ; 0.28 ; 0.19
    - Miranda−Agrippino−Ricco: 0.34 ; 0.38 ; 0.16 ; 0.12
    - Jarocinski FFR: 0.43 ; 0.27 ; 0.11 ; 0.18
    - Gertler−Karadi: 0.44 ; 0.35 ; 0.04 ; 0.17
    - Aruoba−Drechsel: 0.47 ; 0.48 ; 0.02 ; 0.04
    - Swanson FFR: 0.49 ; 0.40 ; 0.02 ; 0.09
    - Romer−Romer MPS: 0.52 ; 0.36 ; 0.05 ; 0.07
  - Interpretation:
    - Shocks targeting surprises have large shares at short horizons (e.g., Swanson FFR, Jarocinski FFR, Aruoba−Drechsel, Romer−Romer, Gertler−Karadi).
    - No shock perfectly isolates surprises; even short-run-targeted shocks contain substantial longer-horizon information.
    - Bu−Rogers−Wu heavily concentrates in long-run news: R^2_{13:24} = 0.71.

### 4.3 Validation — Monte Carlo summary
- Monte Carlo conclusions:
  - The 4-stage estimator delivers unbiased estimates of Taylor rule coefficients and EMPS term structures in small samples.
  - Confidence intervals for term structures are accurate even with weak instruments.
  - OLS performs poorly in small samples: biases small but confidence intervals spuriously tight, leading to unreliable inference for term structures when using OLS.
- Relation to literature:
  - Carvalho et al. (2021) find OLS bias small in some contexts; here structural IV isolates non-monetary shock variation to correct endogeneity and deliver unbiased estimates even in small samples and with weak instruments.

### 5 Synthetic Monetary Policy Shocks — theory and construction (start)
- Definitions and algebraic results:
  - Normalized term structure vector ⃗γ_j with Var(w^j_t) = 1.
  - Proposition 6: For w^c_t = λ_a w^a_t + λ_b w^b_t, γ⃗_c ∝ λ_a γ⃗_a + λ_b γ⃗_b when components normalized.
  - Proposition 7 and Corollary 1: If a set J of EMPS contains H_w + 1 linearly independent term structures, any term structure up to horizon H_w can be constructed in span{γ⃗_j}_{j∈J}.
  - Proposition 8: Closest attainable synthetic term structure to target γ⃗_i is γ⃗_s = Γ_J (Γ_J′ Γ_J)^{−1} Γ_J′ γ⃗_i.
- Practical implications:
  - Construct synthetic MPS to approximate pure surprise (γ⃗_0 = [1 0 0 ...]′) or pure 1-period-ahead news (γ⃗_1 = [0 1 0 ...]′).
  - If target outside span, approximate using projection (Proposition 8).

### 5.2 Synthetic Surprise and News — empirical implementation and IRFs
- Selected EMPS subset for parsimony (five shocks):
  - Aruoba-Drechsel; Miranda-Agrippino and Ricco; Swanson FFR; Swanson FG; Swanson LSAP.
  - These five shocks are orthogonalized to purge information effects.
- Constructed three synthetic MPS (equally weighted collections of news shocks by horizon):
  - Immediate interest rate tightening: news about current month and 1 month ahead.
  - Short-term forward guidance: news in the 2–6-month-ahead window.
  - Long-term forward guidance: news about months 7−24.
  - Targets cannot be matched exactly with five EMPS but actual γ⃗_s approximately match targets; Appendix D repeats with alternative subsets.
- Estimation approach for macro effects:
  - Proxy VAR (Gertler and Karadi (2015) style) with variables: 1-year Treasury yields, log CPI, log industrial production, excess bond premium (EBP).
  - Constructed responses by projecting each synthetic shock onto reduced-form residuals; VAR lag length = 11 (AIC).
  - Confidence intervals: shaded 90% CI conditional on impact; dotted lines include impact uncertainty.
- Impulse response findings (IRFs):
  - Immediate synthetic shock:
    - Rates rise immediately; contraction in real activity.
    - Price response perhaps deflationary but not statistically different from zero.
    - Driven largely by Swanson FFR and Aruoba-Drechsel EMPS (closest to true surprises).
  - Short-term news shock (2−6 months):
    - Increases rates but is expansionary in initial half year, becoming contractionary later.
    - No large effect on prices at any horizon.
    - Produces an “output puzzle” (short-run expansion despite rate increases) difficult to reconcile with textbook New Keynesian model.
  - Long-run forward guidance shock (months 7−24):
    - Deflationary; decreases rates in short run (endogenous Taylor rule response because long-run shock is contractionary).
  - Decomposition insight:
    - Standard rate increases are driven by immediate and short-run news; standard deflation driven by long-run news; rapid contractions driven by immediate news.
- Tensions with theory:
  - Some alignment: surprise tightening is contractive; future contractions can cause anticipatory rate declines.
  - Puzzles:
    - Short-run expansionary effect of a short-run news shock that raises rates is hard to explain with textbook New Keynesian channels.
    - Immediate shock causing sharp contraction with near-zero price effect is also challenging for standard mechanisms.
  - The analysis indicates large deflationary effects in EMPS are driven by long-run forward guidance components rather than by surprises.
- Robustness checks:
  - Term structure variance decomposition robust to alternative x_t choices, sample periods, instrument lags, and smoothing choices.
  - Smoothing parameter λ tested at λ = 10 (“low smoothing”) and λ = 0 (“no smoothing”); small λ values do not meaningfully change results; removing smoothing reallocates effects from longer horizons to shorter horizons (making EMPS appear more surprise-like).
  - Including the ZLB period materially alters estimates, plausibly because the Taylor rule breaks down when ZLB binds.
- Key takeaways and methodological guidance:
  - Identified monetary policy shocks can be viewed as a common exogenous disturbance varying in anticipated timing.
  - Decomposition shows most EMPS have large news components across horizons.
  - Synthetic shock approach allows construction of shocks with targeted term structures and provides benchmarks:
    - Positive monetary surprises are contractionary and deflationary.
    - News at longer horizons tends to increase output, employment, and prices; at very long horizons effects are negligible.
  - Further work needed to better isolate true surprises from multi-horizon policy news.

### Appendix E — Monte Carlo design and key numeric parameters (selected exact values)
- Simulation parameters (Table 10 exact values):
  - 1 − α Returns to scale: 0.67
  - β Utility discount factor: 0.997
  - ρ_a Technology shock persistence: 0.98
  - ρ_r Interest rate smoothing in Taylor rule: 0.49
  - ρ_z Demand shock persistence: 0.56
  - ρ_zpc Supply shock persistence: 0.49
  - σ Risk aversion: 1.0
  - φ Labor supply elasticity: 1.0
  - φ_π Taylor rule inflation coefficient: 1.5
  - φ_y Taylor rule output gap coefficient: 0.125
  - θ Calvo parameter: 0.75
  - Demand elasticity: 6.0
  - {β_i}_{i=0}^{L−1} EMPS lag structure: 1
  - L_EMPS lag length: 4
  - var ξ_MPS measurement error variance: 0.003^2
  - var ν_0 EMPS surprise variance: 0.003^2
  - var ν_1 EMPS 1-period news variance: 0.0008^2
  - var ν_3 EMPS 1-period news variance: 0.0001^2
  - var ν_4 EMPS 1-period news variance: 0.00003^2
- Large-sample Monte Carlo example (Table 11, four-stage IV with strong instruments):
  - Taylor rule φ_π: 1.500 (0.003)
  - Taylor rule φ_y: 0.125 (0.002)
  - Term-structure γ_0: 0.435 (0.004)
  - γ_1: 0.109 (0.003)
  - γ_2: 0.017 (0.004)
  - γ_3: 0.004 (0.004)
  - F-test first stage π_t: 374.2; F-test first stage y_t: 102.2
- Short-sample experiments:
  - Four-stage IV with strong instruments: unbiased distributions concentrated near true values for sample sizes 250–750.
  - OLS: tight but biased distributions; poor coverage for φ_π, φ_y, γ_0 at practical sample sizes.
  - Practical implication: four-stage IV performs well with strong instruments and larger samples; including lags of instruments mitigates weakness; IV often outperforms OLS for term-structure estimation.

*Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025128-print-pdf.pdf*

### conclusions.  Already-motivated readers can skip to Section 3, which describes our method

### conclusions.  Already-motivated readers can skip to Section 3, which describes our method

### 2 A Motivating Example
- Core message: for almost all models there exists some term structure of monetary policy news that can rationalize any given EMPS; without empirical discipline on the term structure, EMPS cannot be used to falsify theory.
- Textbook New Keynesian model (as presented):
  - New Keynesian Phillips curve: π_t = β E[π_{t+1}] + κ y_t
  - Euler equation: i_t = E_t[γ(y_{t+1} − y_t) + π_{t+1}]
  - Taylor rule: i_t = φ_y y_t + φ_π π_t + ν_t
  - ν_t is exogenous and white noise, with news decomposition:
    - ν_t = ν_{0,t} + ν_{1,t−1} + ν_{2,t−2} + ...
    - ν_{h,t−h} are i.i.d. over time and independent of one another; ν_{h,t} ∼ N(0, σ_h^2)
  - ν_{0,t} is the surprise at time t; ν_{h,t−h} represents news at time t−h about policy h periods ahead.
- Empirical comparison:
  - Gertler and Karadi (2015) EMPS produces a gradual deflation over 18 months.
  - The standard New Keynesian surprise ν_{0,t} causes an immediate deflation then rapid stabilization.
  - News shocks ν_{h,t} at different horizons produce different, linearly independent price IRFs; anticipated future tightenings cause smooth deflations.
  - Linear combinations of surprise and news shocks can approximate the empirical IRF:
    - Using the first n news horizons to approximate the Gertler-Karadi IRF improves as n increases.
    - When 24 shocks are used, the Gertler-Karadi IRF is reproduced perfectly.
    - The weights on news shocks that exactly reproduce the empirical IRF define a rationalizing term structure of monetary policy news.
- Implication: without restrictions on the term structure, any EMPS can be rationalized by many models; empirical discipline on the term structure is required.

### 3 Methodology
- Purpose: estimate the term structure of monetary policy news; outline framework, estimation strategy, and theoretical properties.

3.1 Monetary Policy Framework
- Policy rule representation:
  - y_t = x_t φ + r_t  (equation (1))
    - y_t: policy instrument (typically a short-term rate)
    - r_t: exogenous monetary policy residual
    - x_t: row vector of endogenous inputs
    - φ: vector of coefficients
- Residual dynamics allowing smoothing and delayed responses:
  - r_t = Σ_{ℓ=1}^L (ρ_{y,ℓ} y_{t−ℓ} + x_{t−ℓ} φ_ℓ) + ν_t  (equation (2))
- Decomposition of monetary policy innovation into news at H_ν horizons:
  - ν_t = ν_{0,t} + ν_{1,t−1} + ν_{2,t−2} + ... + ν_{H_ν,t−H_ν}  (equation (3))
  - ν_{h,t} ∼ N(0, σ_h^2); Gaussianity assumed to write linear projections as expectations.
- EMPS data-generating process (for EMPS indexed by j ∈ J):
  - w^j_t = Σ_{h=0}^{H_w} β^j_h ν_{h,t} + ξ^j_t  (equation (4))
    - ξ_t orthogonal to ν_{t+h} for all h (measurement error or other factors)
- Term structure definition:
  - γ^j_h ≡ dE[ν_{t+h} | w^j_t] / dw^j_t
  - Linear representation between EMPS and ν_t:
    - ν_t = Σ_{h=0}^{H_w} γ^j_h w^j_{t−h} + u_t  (equation (5))
  - Relationship between β^j_h and γ^j_h:
    - γ^j_h = β^j_h Var(ν_{h,t}) / Var(w^j_t)  (equation (6))
- Note: equation (5) encodes the term structure but ν_t is unobserved, so instrumental variables are required.

3.2 Estimation Strategy
- Main challenges: ν_t unobserved; unknown relation between ν_t and r_t; r_t not orthogonal to x_t.
- Approach: 4-stage estimation relying on availability of a battery of well-identified non-monetary macro shocks z_t (instruments).
- Four stages:
  1. Instrument for endogenous variables x_t in the policy rule using exogenous shocks z_t orthogonal to u_t and w^j_t.
  2. Estimate policy rule coefficients φ̂ by 2SLS.
  3. Compute implied residuals r̂_t = y_t − x_t φ̂ (equation (7)); whiten residuals by projecting r̂_t onto lagged y_t and x_t:
     - r̂_t = Σ_{ℓ=1}^L (y_{t−ℓ} ρ_{y,ℓ} + x_{t−ℓ} ρ_{x,ℓ}) + ν_t (equation (8))
     - write whitening regression as r̂_t = x_t ρ + ν_t (equation (9))
  4. Use estimated ν̂_t innovations to estimate term structure γ by ν̂_t = w_t γ + u_t (equation (10)), where w_t stacks lags of EMPS.
- Closed-form one-step expression (Proposition 1):
  - γ̂ = (W′W)^{−1} W′ M_X (I − X(X′ P_Z X)^{−1} X′ P_Z) y
  - Definitions:
    - x_t stacked lags x_t ≡ [ y_{t−1} x_{t−1} ... y_{t−L} x_{t−L} ]
    - w_t ≡ [ w^j_t w^j_{t−1} ... w^j_{t−H} ]
    - M_X ≡ I − X(X′X)^{−1} X′
    - P_Z ≡ Z(Z′Z)^{−1} Z′
- Estimator for policy coefficients:
  - φ̂ = (X′ P_Z X)^{−1} X′ P_Z y  (equation (11))
- Requirements for validity:
  - Correct specification of the policy rule and adequate lag length in whitening step.
  - Valid instruments z_t that satisfy orthogonality restrictions.
- Monte Carlo validation and robustness checks discussed in appendices.

3.3 Theoretical Properties
- Unbiasedness conditions (Proposition 2):
  - If Z′W = 0 and E[Z′u] = 0, then the 4-stage estimator is unbiased.
- Conditional variance (Proposition 3):
  - If Z′W = 0 and E[Z′u] = 0, then
    - Var(γ̂ | W, X, Z) = (W′W)^{−1} W′ M_X [ I − X(X′ P_Z X)^{−1} X′ P_Z ] Ω [ I − X(X′ P_Z X)^{−1} X′ P_Z ]′ M_X W (W′W)^{−1}
    - where Ω = E[uu′]
  - Standard errors require a consistent estimate of Ω using residuals û_t from equation (10).

3.4 Generalization with Smoothing
- Local projections (Jordà, 2005) used in stage 4 are unbiased but have large variance.
- Penalized (smoothed) local projections reduce variance at the cost of some bias; represented as ridge regression with penalty matrix P_B and penalty parameter λ (selected by cross-validation).
- Smoothed 4-stage estimator (Proposition 4):
  - γ̂_λ = [ W′W + λ P_B ]^{−1} W′ M_X (I − X(X′ P_Z X)^{−1} X′ P_Z) y
  - Conditional variance:
    - Var(γ̂_j | W_j, X, Z, y) = [ (W_j)′ W_j + λ P_B ]^{−1} (W_j)′ M_X [ I − X(X′ P_Z X)^{−1} X′ P_Z ] Ω [ I − X(X′ P_Z X)^{−1} X′ P_Z ]′ M_X W_j [ (W_j)′ W_j + λ P_B ]^{−1} (W_j)′
- The smoothed estimator nests the original estimator when λ = 0.
- The smoothed 4-stage estimator is used in the empirical analysis.

### 4 Estimated Term Structures
- Scope: estimate term structures for a variety of well-known EMPS; provide a summary statistic for the relative importance of news in each EMPS; report a simulation validation in Appendix E.

4.1 Data
- Two data types required: monetary policy shocks (EMPS series) and macro instruments/series to estimate the policy rule.

4.1.1 Monetary Policy Shock Data
- EMPS considered (summary in Table 1; date ranges preserved in source):
  - Gertler and Karadi (2015): HFI, 30 min. window around FOMC decisions, 1990:M1-2007:M12
  - Jarociński and Karadi (2020): HFI, pure monetary shock purged of Fed information, 1990:M1-2016:M12
  - Bundick and Smith (2020): HFI, 2 shocks to term structure uncertainty, 1994:M2-2019:M06
  - Miranda-Agrippino and Ricco (2021): HFI, Orthogonalized w.r.t. Greenbook forecasts, 1991:M1-2009:M12
  - Bu et al. (2021): HFI, Alternative without intraday data, 1994:M2-2024:M12
  - Bauer and Swanson (2023): HFI, Includes Fed minutes and speeches, 1988:M2-2023:M12
  - Swanson (2024): HFI, Decomposed into 3 types of EMPS, 1988:M2-2023:M12
  - Jarociński (2024): HFI, Decomposed into 4 types of EMPS, 1990:M2-2024:M09
  - Romer and Romer (2004): Narrative, Orthogonalized w.r.t. Greenbook forecasts, 1983:M1-2007:M12
  - Aruoba and Drechsel (2024): Narrative, Natural language processing of Fed docs, 1982:M10-2008:M10
- Notes:
  - Many series use intra-day high-frequency identification (e.g., 3-month-ahead federal funds futures in Gertler and Karadi (2015)).
  - Some EMPS attempt to isolate or purge Fed information effects (Jarociński and Karadi (2020); Miranda-Agrippino and Ricco (2021)).
  - Swanson (2024) and Jarociński (2024) decompose EMPS into components (e.g., target rate, forward guidance, LSAP).

4.1.2 Data for Estimating the Monetary Policy Rule
- Baseline policy rule: Effective Federal Funds rate as policy variable; unemployment and PCE inflation as regressors on the right-hand side.
- Instruments (monthly series; summarized in Table 2):
  - Government spending shocks:
    - Romer and Romer (2016): Narrative, Social Security expansions, 1951:M1-1991:M12
    - Fieldhouse et al. (2018): Narrative, Government housing purchases, 1952:M11-2014:M12
  - Oil shocks:
    - Känzig (2021): HFI, Oil supply news, 1975:M1-2023:M6
    - Baumeister and Hamilton (2019): SVAR, Oil supply, consumption/inventory demand, 1975:M2-2024:M3
  - Other shocks:
    - Kim et al. (2025): External ACI severe weather shocks, 1964:M4-2019:M5
    - Adams and Barrett (2024): SVAR, Shocks to inflation expectations, 1979:M1-2024:M5
- Instrument rationale and construction:
  - Romer and Romer (2016) and Fieldhouse et al. (2018) provide narrative, non-cyclically motivated government spending variation.
  - Känzig (2021) and Baumeister and Hamilton (2019) provide oil market shocks via high-frequency oil futures around OPEC announcements and structural VAR identification, respectively.
  - Kim et al. (2025) supply severe weather shocks via the Actuaries Climate Index innovations.
  - Adams and Barrett (2024) provide inflation expectation shocks identified from a structural VAR isolating deviations of survey forecasts from rational expectations.

*Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025128-print-pdf.pdf*

### 4.2    Estimation Results

### 4.2    Estimation Results

### 4.2.1    Estimated Taylor Rules
- Method and identification:
  - First 2 stages of the 4-stage estimator: estimate the Taylor Rule using structural shocks as instrumental variables (IVs).
  - Term structure estimated up to two years after the shock, H_w = 24.
  - When whitening monetary policy residuals, orthogonalize with respect to six lags, L = 6.
  - Include six lags of the IVs because instruments can have persistent effects.
- Main findings on coefficients:
  - Inflation coefficient is roughly 1.5 across multiple specifications and largely satisfies the Taylor principle (φ_π > 1).
  - Baseline specification uses: 12-month growth rate in the PCE index (inflation) and detrended private non-farm employment (activity).
  - Inclusion of excess bond premium has little effect on inflation coefficient.
  - Introduction of inflation expectations (Michigan Survey or Cleveland Fed) substantially changes the size of the inflation coefficient; expectations are highly correlated with current inflation.
- Robustness and comparison to OLS:
  - Results robust to alternative lag lengths, dropping various IVs, alternative sample start (baseline uses data beginning in January 1975 and omits the ZLB and Covid periods), and alternative real activity measures (CF-filtered GDP, GDP growth, HP-filtered GDP, industrial production, unemployment).
  - Exception: inclusion of the ZLB period undermines robustness because policy rates are pinned to zero.
  - Structural IV estimates are stable across specifications and consistent with theory; OLS estimates are highly sensitive to specification choice and produce inconsistent estimates across small changes.
- Selected estimated parameter values (from Table 3 and Table 4; standard errors in parentheses):
  - 12-month PCE Inflation, demeaned (baseline): 1.517*** (0.035)
  - Alternative baseline inflation coefficients reported: 1.536*** (0.035); 1.541*** (0.034); 1.528*** (0.034); 1.512*** (0.037); 1.187*** (0.033); 1.512*** (0.033); -0.249*** (0.057); 2.294*** (0.106)
  - Private nonfarm payroll employment, departures from quadratic log trend, demeaned: 0.370*** (0.010); alternate values: 0.377*** (0.010); -0.033** (0.014); 0.408*** (0.013)
  - GDP, CF-low-pass: 0.697*** (0.076)
  - GDP growth, annualized, demeaned: 0.031*** (0.010)
  - GDP, HP-filtered: 0.273*** (0.068)
  - Industrial Production, CF low-pass filter: 0.058** (0.026)
  - Unemployment rate: 0.265*** (0.005)
  - Excess bond premium, demeaned: 0.440*** (0.090)
  - Cleveland Fed inflation expectations, 1 year, demeaned: 2.180*** (0.052)
  - Michigan inflation expectations, 1 year, demeaned: -1.170*** (0.141)
- Goodness-of-fit and diagnostics (selected entries from Tables 3 and 4):
  - Baseline residual autocorrelation: 0.95; R^2: 0.58; Observations: 569
  - Alternative residual autocorrelation and R^2 examples:
    - CF-filtered GDP: residual autocorrelation 0.95; R^2 0.38; Observations 569
    - Add π^e_1, Clev.: residual autocorrelation 0.89; R^2 0.79; Observations 485
    - 1-month PCE Inflation, demeaned coefficient: 0.598*** (0.076) with residual autocorrelation 0.96 and R^2 0.46; Observations 569
    - 1-month Core PCE Inflation, demeaned: 1.511*** (0.109) with residual autocorrelation 0.89 and R^2 0.34; Observations 569
    - 12-month Core PCE Inflation, demeaned: 2.031*** (0.037)
    - 12-month CPI inflation, demeaned: 1.125*** (0.029)
    - CPI inflation, demeaned: 0.372*** (0.047)

### 4.2.2    Estimated Term Structures
- General description:
  - Figure 2 plots estimated term structures γ_jh: impact of each identified monetary policy shock h periods later on the monetary policy innovation ν̂_{t+h}.
  - Dashed lines in figure show 95 percent confidence intervals.
  - Heterogeneity across EMPS: some shocks spike at low horizons; others concentrate in middle horizons; most—but not all—decay to zero at long horizons.
  - All shocks have a non-trivial term structure at horizons longer than h = 0: even well-identified shocks include information about both monetary surprises and forward guidance.
- Summary statistic R^2_k (Proposition 5):
  - R^2_k measures the share of variation in monetary policy innovation ν_t explained by an EMPS at horizon k:
    - R^2_k ≡ Var(ν_t | w^j_{t−k}) / Var(ν_t | {w^j_{t−h}}_{h=0}^{H_w}) = (γ^j_k)^2 / Σ_{h=0}^{H_w} (γ^j_h)^2
  - Use R^2_{0:1} to interpret immediate change in monetary policy innovation (announcements can happen any day during a month).
- Decomposition of term structures by horizon (Table 5: R^2 sums over monthly horizons):
  - Columns report R^2_{0:1}, R^2_{2:6} (short-run news), R^2_{7:12} (medium-run news), R^2_{13:24} (long-run news).
  - Selected ordered entries (shocks ordered by R^2_{0:1}):
    - Swanson FG: R^2_{0:1} 0.01 ; R^2_{2:6} 0.22 ; R^2_{7:12} 0.59 ; R^2_{13:24} 0.18
    - Bu−Rogers−Wu: 0.02 ; 0.11 ; 0.16 ; 0.71
    - Jarocinski FG: 0.04 ; 0.35 ; 0.36 ; 0.25
    - Jarocinski Info: 0.18 ; 0.11 ; 0.32 ; 0.40
    - Bundick−Herriford−Smith Slope: 0.21 ; 0.17 ; 0.43 ; 0.19
    - Swanson LSAP: 0.23 ; 0.50 ; 0.22 ; 0.05
    - Bundick−Herriford−Smith Level: 0.24 ; 0.22 ; 0.25 ; 0.29
    - Bauer−Swanson: 0.27 ; 0.36 ; 0.26 ; 0.11
    - Jarocinski LSAP: 0.28 ; 0.45 ; 0.07 ; 0.20
    - Jarocinski−Karadi MPS: 0.28 ; 0.25 ; 0.28 ; 0.19
    - Miranda−Agrippino−Ricco: 0.34 ; 0.38 ; 0.16 ; 0.12
    - Jarocinski FFR: 0.43 ; 0.27 ; 0.11 ; 0.18
    - Gertler−Karadi: 0.44 ; 0.35 ; 0.04 ; 0.17
    - Aruoba−Drechsel: 0.47 ; 0.48 ; 0.02 ; 0.04
    - Swanson FFR: 0.49 ; 0.40 ; 0.02 ; 0.09
    - Romer−Romer MPS: 0.52 ; 0.36 ; 0.05 ; 0.07
- Interpretation highlights:
  - Shocks explicitly aiming to identify surprises have large shares at short horizons (e.g., Swanson FFR, Jarocinski FFR, Aruoba−Drechsel, Romer−Romer, Gertler−Karadi).
  - No shock perfectly isolates true policy surprises; even shocks targeting short-term surprises show substantial information at longer horizons.
  - Swanson FG and Jarocinski FG (Forward Guidance shocks) are heavily driven by forward guidance and contain almost no short-run surprise information; Swanson FG concentrates in medium-run while Jarocinski FG spreads over the two-year window.
  - Bu−Rogers−Wu shock overwhelmingly contains news about long-run policy: R^2_{13:24} = 0.71 (no other shock has more than 50% of term structure appear more than a year in the future).

### 4.3    Validation
- Monte Carlo findings (summary):
  - Method delivers unbiased estimates of both the Taylor rule coefficients and the term structure of EMPSs in small samples.
  - Confidence intervals for the term structure are accurate even with weak instruments for macroeconomic shocks.
  - OLS applied to simulated data performs poorly in small samples: biases in Taylor rule estimation are economically small but confidence intervals are spuriously tight, leading to unreliable inference for term structures when using OLS.
- Comparison to related literature:
  - Carvalho et al. (2021) find OLS bias small enough to be economically meaningless and sometimes preferable to lagged-variable GMM in small samples. The authors interpret methods as trading off bias against using informative contemporaneous variation.
  - The structural IV method here isolates variation due to non-monetary shocks to correct endogeneity bias while exploiting contemporaneous variation, delivering unbiased estimates and accurate inference even in small samples and with weak instruments.

### 5    Synthetic Monetary Policy Shocks (start of section included)
- Purpose:
  - Use estimated EMPS term structures to construct synthetic monetary policy shocks (MPS) with desired term structures, enabling study of shock types not directly estimated in the data (e.g., pure surprises, 1-period-ahead news).
- Key definitions and propositions:
  - Let vector γ⃗_j denote normalized term structure coefficients for EMPS j where Var(w^j_t) = 1.
  - Proposition 6: For w^c_t = λ_a w^a_t + λ_b w^b_t, the resulting term structure γ⃗_c is proportional to λ_a γ⃗_a + λ_b γ⃗_b.
  - Proposition 7: EMPS with normalized term structures in set J = {γ⃗_j} can be used to construct any synthetic MPS with term structure γ⃗_s ∈ span{γ⃗_j}_{j∈J}.
  - Corollary 1: If J contains H_w + 1 EMPS with linearly independent term structures, a synthetic MPS can be constructed with any term structure of horizon length up to H_w.
  - Proposition 8: Given Γ_J (matrix of normalized term structures for a linearly independent set J) and a target term structure γ⃗_i, the closest attainable synthetic term structure in Euclidean norm is γ⃗_s = Γ_J (Γ_J′ Γ_J)^{−1} Γ_J′ γ⃗_i.
- Practical implications and applications:
  - Can construct synthetic MPS matching basis vectors (e.g., γ⃗_0 = [1 0 0 ...]′ for a true surprise; γ⃗_1 = [0 1 0 ...]′ for a pure 1-period-ahead news shock).
  - If the desired term structure is not in the span of observed EMPS, approximate it using Proposition 8.
  - Applications include: studying impulse responses to synthetic shocks, extending comparable shock series over periods when a target EMPS is unavailable, and applying methods to other macro shocks with partially anticipated components (fiscal, technological).

*Source: wpiea2025128-print-pdf - 4.2    Estimation Results*

### 5.2    Synthetic Surprise and News

### 5.2 Synthetic Surprise and News

### Construction of synthetic monetary policy shocks (MPS)
- Selected a subset of five empirical MPS (EMPS) to improve parsimony, chosen by average Euclidean distance to other vectors⃗γj: Aruoba-Drechsel, Miranda-Agrippino and Ricco, and the Swanson FFR, FG, and LSAP shocks.
- The five shocks are orthogonalized to be purged of information effects.
- Constructed three synthetic MPS as equally-weighted collections of news shocks at similar horizons:
  - Immediate interest rate tightening: news about the current month and 1 month ahead.
  - Short-term forward guidance: news in the 2-6-month-ahead window.
  - Long-term forward guidance: news about the remaining year and a half (months 7−24).
- Target term structures are vectors⃗γi (Proposition 8); with only five empirical shocks the targets cannot be matched exactly, but the actual term structures (vector⃗γs) approximately match the targets.
- Appendix D repeats the exercise with alternative subsets and additional shocks to get closer to the targets.

### Estimation method for effects on the macroeconomy
- Used a proxy VAR similar to Gertler and Karadi (2015) with variables: 1-year Treasury yields, log CPI, log industrial production, and the excess bond premium (EBP).
- Constructed responses to synthetic shocks by projecting each synthetic shock onto the reduced-form residuals to obtain appropriate reduced-form shock weightings consistent with a synthetic term structure shock.
- Applied the same projection method to EMPS shock series to compare impacts while holding VAR autocovariances fixed.
- VAR lag length is 11, chosen by AIC.
- Confidence intervals: shaded region indicates the 90% confidence interval conditional on impact values; dotted lines indicate the 90% confidence intervals accounting for impact uncertainty.

### Impulse response findings (IRFs)
- Immediate synthetic shock:
  - Characterized as a clear monetary policy tightening: interest rates rise immediately and cause a contraction in real activity.
  - Price level response is perhaps deflationary but not statistically different from zero.
  - Driven largely by Swanson FFR and Aruoba-Drechsel EMPS, which are closest to true policy surprises and cause large declines in real activity with near-zero price effects.
- Short-term news shock (2−6 months):
  - Also increases rates but is expansionary in the initial half year (short-run expansion allows rates to rise without a short-run monetary policy residual).
  - In the longer run this shock becomes contractionary.
  - No large effect on prices at any horizon.
  - Produces an “output puzzle” (short-run expansion despite rate increases) that is hard to reconcile with a textbook New Keynesian model and unlikely to be explained by central bank information effects because component EMPS are orthogonalized.
- Long-run forward guidance shock (months 7−24):
  - The only shock that is deflationary in the results, matching textbook New Keynesian prediction.
  - Counterintuitively decreases rates in the short run (consistent with endogenous short-term monetary policy responses through the Taylor rule because the long-run shock is contractionary).
- Decomposition insight:
  - Empirical MPS appear relatively homogeneous (they mostly predict higher rates, reduced real activity, and zero or modest deflation, except Swanson FG).
  - Synthetic combinations of linearly independent EMPS can produce IRFs that differ markedly from individual component IRFs because they mix news from different horizons.
  - Standard rate increases are driven by immediate and short-run news; standard deflation is driven almost entirely by long-run news; rapid contractions are driven by immediate news rather than forward guidance.

### Tensions and interpretation relative to theory
- Some results align with standard New Keynesian expectations (surprise tightening is contractive; future contractions can cause anticipatory rate declines).
- Puzzles:
  - Short-run expansionary effect of a short-run news shock that raises rates is difficult to explain with the textbook model.
  - Sharp contraction from the immediate policy shock with effectively zero effect on prices is hard to reconcile with New Keynesian mechanisms—this tension echoes known issues in the literature (Ramey, 2016).
- The analysis clarifies that large deflationary effects observed in EMPS are driven by long-run forward guidance components rather than by surprises.

### Robustness checks
- Re-estimated term structures under several alternative specifications and estimation choices:
  - Figure 7: term structure variance decomposition robust to different variables x_t included in Taylor rule estimation; bars show variance share by horizon: impact (R0:1^2), short (R2:6^2), medium (R7:12^2), long (R13:24^2).
  - Figure 8: varied sample and estimation methods (examples include Pre−2020, Post−1985, Inc. ZLB, more instrument lags, fewer instrument lags, No smoothing, Low smoothing, High smoothing).
- Smoothing sensitivity:
  - Tested smoothing parameter values λ = 10 (“low smoothing”) and λ = 0 (“no smoothing”).
  - Even small values of λ do not meaningfully change results; eliminating smoothing entirely causes several shocks to lose most immediate news content.
  - Conclusion: smoothing, if anything, reallocates effects from longer horizons to shorter, making EMPS seem more surprise-like, not less.
- Other sensitivities:
  - Changes in instrument lags have little effect.
  - Changes in sample period matter: including periods when the Federal Reserve was constrained by the zero lower bound (ZLB) yields quite different results, plausibly because the Taylor rule breaks down when the ZLB binds.

### Key takeaways and implications
- Empirical framework: identified monetary policy shocks can be interpreted as a common exogenous disturbance that varies in anticipated timing; decomposed EMPS into surprise and multi-horizon news components and found most shocks have large news components.
- Mapping to theory: by projecting fixed h-period ahead impulses onto imperfectly correlated EMPS, responses to news shocks at multiple horizons can be constructed as linear combinations of estimated IRFs.
- Empirical benchmarks:
  - Positive monetary surprises are contractionary and deflationary.
  - News at longer horizons tends to increase output, employment, and prices; at very long horizons the effect is negligible.
- Methodological guidance:
  - There remains substantial work to better capture monetary policy surprises distinct from news about the future.
  - The synthetic shock approach provides a framework to evaluate and compare future monetary policy shocks and to set empirical benchmarks for theoretical models.

*Source: 5.2 Synthetic Surprise and News, wpiea2025128-print-pdf*

### References

### wpiea2025128-print-pdf - References

### Appendix A — Proofs: key analytical results and formulas
- Proposition 1: OLS and 2SLS estimators for the third and fourth stage regressions
  - OLS third-stage estimator: ˆρ = (X′X)−1 X′ˆR = (X′X)−1 X′(y − X ˆφ)
  - Residual used in fourth stage: ˆν = ˆR − Xˆρ = M_X (y − X ˆφ)
  - OLS fourth-stage estimator: ˆγ = (W′W)−1 W′ˆν = (W′W)−1 W′ M_X (y − X ˆφ)
  - 2SLS estimator: ˆβ = (X′P_Z X)−1 X′P_Z y, and ˆγ = (W′W)−1 W′ M_X (y − X(X′P_Z X)−1 X′P_Z y)
- Proposition 2: Conditional expectation of ˆγ (conditional on the data)
  - E[ˆγ] = E[(W′W)−1 W′ M_X (y − X ˆφ)] = γ + E[(W′W)−1 W′ M_X X(φ − ˆφ)]
  - Using orthogonality of W and u and substitution for φ − ˆφ yields E[ˆγ] = γ when Z is orthogonal to both W and u.
- Proposition 3: Conditional variance of ˆγ (conditional on W, X, Z)
  - V ar(ˆγ|W,X,Z) = (W′W)−1 W′ M_X [I − X(X′P_Z X)−1 X′P_Z] Ω [I − X(X′P_Z X)−1 X′P_Z]′ M_X W (W′W)−1
  - Uses Ω = E[uu′] and orthogonality of u and W to separate terms.
- Additional propositions (5, 6, 8) provide algebraic results used in term-structure construction:
  - Proposition 5: Ratio of conditional variances when EMPS is homoskedastic white noise:
    - V ar(ν_t | w_{j,t−k}) = (γ_{j,k})^2 V ar(w_{j,t})
    - V ar(ν_t | {w_{j,t−h}}_{h=0}^{H_w}) = Σ_{h=0}^{H_w} (γ_{j,h})^2 V ar(w_{j,t})
    - Ratio: V ar(ν_t | w_{j,t−k}) / V ar(ν_t | {w_{j,t−h}}) = (γ_{j,k})^2 / Σ_{h=0}^{H_w} (γ_{j,h})^2
  - Proposition 6: Linear combination of EMPS yields combined term structure coefficients:
    - γ_{c,h} = λ_a γ_{a,h} V ar(w_{a,t})/V ar(w_{c,t}) + λ_b γ_{b,h} V ar(w_{b,t})/V ar(w_{c,t}); when V ar(w_{a,t}) and V ar(w_{b,t}) normalized to 1, ⃗γ_c ∝ λ_a ⃗γ_a + λ_b ⃗γ_b
  - Proposition 8: Synthetic MPS (⃗γ_s) minimizing ∥⃗γ_i − ⃗γ∥_2 in the span of observed EMPS columns Γ_J:
    - ⃗γ_s = Γ_J (Γ′_J Γ_J)−1 Γ′_J ⃗γ_i

### Appendix B — Smooth term structures: methodology and estimators
- Smooth local projection framework (for h = 0,1,...,H):
  - Local projection: y_{t+h} = w_t γ_h + ε_{h,t+h}
  - Equivalent stacked regression (if w_t is white noise): y_t = Σ_{h=0}^H γ_h w_{t−h} + ε_t  (equation (14))
- B-spline approximation of coefficients:
  - γ_h = Σ_{k=0}^K α_k B_k(h)
  - Smoothed shock: v_{t−h} = Σ_{h=0}^H B_k(h) w_{t−h}  (equation (15))
  - Matrix relation: V = W B, γ = B α, and y = V α + ε
- Ridge regression estimator for smoothed projections:
  - ˆα = arg min_α (y − V α)′(y − V α) + λ α′ P α = (V′V + λ P)−1 V′ y
  - Canonical penalty: P = D′_r D_r (rth-difference penalty); λ chosen by cross-validation
- Short-cut via penalty transformation (Proposition 9):
  - Smooth local projection coefficient vector ˆγ found directly by ridge on original regressors with transformed penalty:
    - P_B = (B^{−1})′ P B^{−1}
    - ˆγ = (W′W + λ P_B)−1 W′ y
- Conditional variances (closed form):
  - V ar(ˆα|V) = σ^2 (V′V + λ P)−1 V′V (V′V + λ P)−1
  - V ar(ˆγ|W) = σ^2 (W′W + λ (B^{−1})′ P B^{−1})−1 W′W (W′W + λ (B^{−1})′ P B^{−1})−1
- Application to term structures (smoothed term-structure estimators):
  - ˆγ_j = B (V_j′ V_j + λ P)−1 V_j′ ˆν, with ˆν = M_X (I − X(X′P_Z X)−1 X′P_Z) y
  - Conditional variance (equation (16)):
    - V ar(ˆγ_j | V_j, X, Z, y) = B (V_j′ V_j + λ P)−1 V_j′ V ar(ˆν | X, Z, y) V_j (V_j′ V_j + λ P)−1 B′
  - Equivalent expression in untransformed shocks (equation (17)):
    - ˆγ_j = (W′W + λ P_B)−1 W′ M_X (I − X(X′P_Z X)−1 X′P_Z) y
    - V ar(ˆγ_j | W_j, X, Z, y) = (W_j′ W_j + λ P_B)−1 W_j′ M_X [I − X(X′P_Z X)−1 X′P_Z] Ω [I − X(X′P_Z X)−1 X′P_Z]′ M_X W_j (W_j′ W_j + λ P_B)−1

### Appendix C — Further Taylor rule specification alternatives: robustness checks (selected numeric results)
- Sensitivity to lag length (Table 6): estimated Taylor rule parameters under alternative lag settings
  - Baseline estimates (reported in table):
    - 12-month PCE Inflation, demeaned: 1.517*** (standard error (0.035))
    - Private nonfarm payroll employment, departures from quadratic log trend, demeaned: 0.370*** (standard error (0.010))
    - Residual autocorrelation: 0.95
    - R^2: 0.58
    - Observations: 569
  - Results shown for L = 1, 12, 24 and other variant columns; baseline values above repeat across many columns as shown in Table 6.
- Sensitivity to sample period (Table 7): baseline and alternative samples
  - Baseline: inflation 1.517*** (0.035); payrolls 0.370*** (0.010); residual autocorrelation 0.95; R^2 0.58; observations 569
  - Inc. ZLB: inflation 1.138*** (0.031); payrolls 0.186*** (0.017); residual autocorrelation 0.97; R^2 0.59; observations 569
  - Post-1985: inflation 0.454*** (0.015); payrolls 0.237*** (0.004); residual autocorrelation 0.98; R^2 0.58; observations 449
  - Post-1985, Inc. ZLB: inflation 0.091*** (0.018); payrolls 0.373*** (0.018); residual autocorrelation 0.99; R^2 0.61; observations 449
  - Pre-2020: inflation 1.501*** (0.031); payrolls 0.372*** (0.009); residual autocorrelation 0.96; R^2 0.68; observations 521
- Sensitivity to instruments (Table 8)
  - Baseline: inflation 1.517*** (0.035); payrolls 0.370*** (0.010); residual autocorrelation 0.95; R^2 0.58; observations 569
  - Omit government shock: inflation 1.652*** (0.042); payrolls 0.118*** (0.014); residual autocorrelation 0.96; R^2 0.50; observations 569
  - Omit oil: inflation 1.111*** (0.037); payrolls 0.514*** (0.009); residual autocorrelation 0.97; R^2 0.57; observations 569
- OLS comparisons (Table 9): OLS estimates are highly attenuated and sensitive
  - Baseline IV (repeated): inflation 1.517*** (0.035); payrolls 0.370*** (0.010); residual autocorrelation 0.95; R^2 0.58; observations 569
  - OLS variants show inflation coefficients as low as 0.038 (standard error (0.089)) and other configurations with large changes; OLS residual autocorrelations and R^2 vary widely (examples in Table 9).

### Appendix D — Alternative synthetic shocks: empirical VAR impulse-response findings (qualitative summary)
- Using alternative sets of EMPS to construct synthetic MPS produces broadly similar IRF patterns to baseline:
  - Immediate shock: contractionary on impact with little effect on prices in several alternatives
  - Short-run shock: sometimes expansionary at some horizons; alternative specifications can remove temporary production expansion or delay contraction
  - Long-run shock: deflationary and reduces rates on impact in baseline; variations affect precision and timing
- Figures referenced:
  - Figure 9: IRFs from a smaller EMPS set (drop Swanson LSAP) — short-run news shock lacks temporary production expansion; VAR lag length = 11 (chosen by AIC); 90% CI shading and dotted lines for impact uncertainty
  - Figure 10: IRFs from a larger EMPS set (add Jarocinski and Karadi (2020)) — IRFs more precisely estimated; short-run news has more moderate effect on rates; instantaneous price effect of immediate shock disappears
  - Figures 11–12: IRFs with the largest EMPS set (including Bundick and Smith (2020) level shock) — seven EMPS series used; short-run news effects reduced or altered; immediate shock often has no price effect

### Appendix E — Monte Carlo validation: design, parameters, and performance
- Simulation setup:
  - Data simulated from a three-equation New Keynesian model (Galí, 2008) extended with additive demand and supply shocks and simulated instruments
  - Two assessment exercises: long-sample (25,000 periods) and short-sample repeated experiments (subsamples of 250, 500, 750)
- Table 10 — Parameters of the Monte Carlo Simulation (exact values)
  - 1 − α Returns to scale: 0.67
  - β Utility discount factor: 0.997
  - ρ_a Technology shock persistence: 0.98
  - ρ_r Interest rate smoothing in Taylor rule: 0.49
  - ρ_z Demand shock persistence: 0.56
  - ρ_zpc Supply shock persistence: 0.49
  - σ Risk aversion: 1.0
  - φ Labor supply elasticity: 1.0
  - φ_π Taylor rule inflation coefficient: 1.5
  - φ_y Taylor rule output gap coefficient: 0.125
  - θ Calvo parameter: 0.75
  - Demand elasticity: 6.0
  - {β_i}_{i=0}^{L−1} EMPS lag structure: 1
  - L_EMPS lag length: 4
  - var ξ_MPS measurement error variance: 0.003^2
  - var ν_0 EMPS surprise variance: 0.003^2
  - var ν_1 EMPS 1-period news variance: 0.0008^2
  - var ν_3 EMPS 1-period news variance: 0.0001^2
  - var ν_4 EMPS 1-period news variance: 0.00003^2
- Large-sample properties (Table 11, exact reported values for one set)
  - Four-stage IV (column (1) — strong instruments):
    - Taylor rule φ_π: 1.500 (standard error (0.003))
    - Taylor rule φ_y: 0.125 (0.002)
    - Term-structure γ_0: 0.435 (0.004)
    - γ_1: 0.109 (0.003)
    - γ_2: 0.017 (0.004)
    - γ_3: 0.004 (0.004)
    - Residual lag length L = 1; Demand instrument = 2; Supply instruments = 2; Instrument lags = 0
    - F-test first stage π_t: 374.2; F-test first stage y_t: 102.2
  - Four-stage IV columns (2)–(4) and OLS columns (5)–(8) show degraded performance with weaker instruments and OLS bias; OLS provides misleadingly narrow confidence intervals even in large samples.
  - Key summary: with large sample and strong instruments, four-stage IV recovers correct parameters almost perfectly; instrument weakness degrades performance but adding lags of instruments improves estimation.
- Short-sample properties (repeated-subsample experiments):
  - Distributional results (Figures 13–16):
    - Four-stage IV with strong instruments: unbiased and relatively powerful in samples of size 250–750, with distributions concentrated near true values
    - Four-stage IV with weak instruments: less powerful and more dispersed estimates
    - OLS: tight distribution but biased; biased OLS leads to poor term-structure estimates due to mapping from ˆφ to ˆγ (concavity effects)
  - Coverage and p-value distributions:
    - Four-stage IV yields p-value distributions closer to uniform (better coverage), improving with instrument strength and larger samples
    - OLS yields poor coverage for key parameters (φ_π, φ_y, γ_0) at sample sizes of practical interest
- Practical implications summarized from simulations:
  - The four-stage IV method works well in large samples and with strong instruments.
  - Including lags of instruments in the first stage helps mitigate instrument weakness.
  - Even with weaker instruments, four-stage IV often outperforms OLS for term-structure estimation, particularly for γ_0.
  - OLS estimates of the Taylor rule are inconsistent and can be misleading even with many lags or controls; IV approach corrects endogeneity by isolating non-monetary variation.

*What Are Empirical Monetary Policy Shocks? Estimating the Term Structure of Policy News — Working Paper No. WP/2025/128*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025128-print-pdf.pdf_