## wpiea2025051-print-pdf - 1. The literature is a very large one, but Christiano, Eichenbaum and Evans

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---

### Background and motivation
- Seminal references: Christiano, Eichenbaum and Evans (2005) and Smets and Wouters (2003, 2007).
- Motivation:
  - Different practices for historical shock decomposition generate divergent interpretations of the same estimated shocks.
  - Need methods that isolate the impact of exogenous shocks for particular periods.
- Practical constraint: Dynare 4.5.7’s current decomposition corresponds to DC2 and “thus far there is no Dynare facility for” implementing DC1; authors program a general Matlab routine to compute DC1 for Dynare 4.5.7 estimates (available upon request).

### Shock decomposition methods (formal definitions and implications)
- State-space reduced-form (log-linear, first order around balanced growth path):
  - s_t = A s_{t−1} + B ε_t
  - x_t = C s_t
  - Iterated form: x_t = C A^t s_0 + sum_{j=1}^t C A^{t−j} B ε_j
- Three decomposition approaches:
  - DC1 (isolate shocks after t):
    - x_{t+h} = C A^h s_t + sum_{j=t+1}^{t+h} C A^{t+h−j} B ε_j, h∈{1,...,k}.
    - First term captures initial condition at start of episode (distance from balanced growth path).
  - DC2 (common in literature and Dynare 4.5.7):
    - x_{t+h} = C A^{t+h} s_0 + sum_{j=1}^t C A^{t+h−j} B ε_j + sum_{j=t+1}^{t+h} C A^{t+h−j} B ε_j, h∈{1,...,k}.
    - Uses smoothed shocks for entire sample; places significant weight on impacts of shocks before the event because persistent pre-event shocks remain in the decomposition.
  - DC3 (differencing approach used by DU):
    - x_{t+h} − x_t = C(A^{t+h} − A^t) s_0 + sum_{j=1}^t C A^{t−j} (A^h − I) B ε_j + sum_{j=t+1}^{t+h} C A^{t+h−j} B ε_j, h∈{1,...,k}.
    - Nets out initial value at start of sample but does not remove influence of prior shocks accumulated from the initial period onward.
- Practical note: DC3 can mitigate DC2’s sensitivity to earlier shocks but still includes prior-shock influence via its second term.

### Model used to illustrate differences (specification and extensions)
- Base: extended Drautzburg and Uhlig (DU, 2015) built on Smets and Wouters (2007); extensions include fiscal policy, financial frictions, and increasing returns from public capital externality.
- Data and shocks:
  - Twelve data series and twelve exogenous shocks (expanded from DU’s ten series and ten shocks).
  - Four observed interest rates (determined by data): policy rate r_f, intermediate-term government rate r_b, long-term government rate r_l, private/corporate rate r_k.
  - Four corresponding wedges (determined by data): ω_f, ω_b, ω_l, and ω_k (sum of the three).
- Preference shock added (multiplicative to discount factor β):
  - log ς_{p,t} = ρ_P log ς_{p,t−1} + ε_{p,t}, ε_{p,t} ∼ N(0, σ^2_{εp}).
  - Affects only the Euler equation; positive shock increases discount factor, raising savings and lowering consumption for unconstrained agents.
- Production and public capital externality:
  - Micro firm: Y_t(i) = ẽ^a_t ( (K^g_t)^Λ (∫_0^1 Y_t(j) dj + Φ μ_t) )^{ζ/(1−ζ)} (u_t(i) K^p_t(i))^{α} (μ_t n_t(i))^{1−α} − Φ μ_t
  - Aggregate: Y_t = ẽ^a_t (K^g_t)^ζΛ (u_t K^p_t)^{α(1−ζ)} (μ_t n_t)^{(1−ζ)(1−α)} − Φ μ_t
  - Λ is estimated, allowing government capital to generate nonconstant returns to scale.
- Marginal costs and frictions:
  - MC_t = α^{−α} (1−α)^{−(1−α)} W_t^{1−α} (R^k_t)^α μ_t^{−(1−α)} ( (K^g_{t−1})^Λ / (Y_t + μ_t Φ) )^{ζ/(1−ζ)} ε_t^{α}
  - Calvo pricing in goods and labor markets; unions set wages to maximize representative household welfare.

### Historical decomposition results and episode-specific findings
- Estimation sample for parameters: data from 1948Q2 to 2020Q1. Model used to analyze outcomes up to 2023Q1.
- Long-run detrended output pattern (1948Q2–2023Q1):
  - US economy experienced an eight month recession at the end of World War II; analysis begins the quarter before the second postwar recession starting end-1948 lasting nearly a year.
  - From mid 1950 (start of Korean war) to start of the Great Recession in 2008, detrended output is relatively elevated in nearly every quarter.
  - From the Great Recession onward, detrended output declines and remains permanently below detrended steady state value from mid 2015 onward.
- Episode comparisons (differences across DC1, DC2, DC3):
  - 1964Q2–1966Q1:
    - DC2/DC3 commonly highlight fiscal policy (Johnson tax cuts, 1964 Economic Opportunity Act).
    - DC1 (preferred) emphasizes momentum from legacy shocks from previous quarters as explaining much of performance.
  - 2006Q4–2009Q3 (financial crisis):
    - DC2: year preceding crisis appears relatively stable; expansionary monetary policy and positive investment shocks partly offset by downward pressure from intermediate-term interest rates and shocks to TFP and preferences.
    - DC3: crisis driven largely by negative preference shocks plus intensification of prior negative private investment shocks; corporate bond term spreads muted or mild pro-cyclical impacts; monetary policy had little ameliorating effect.
    - DC1 (preferred): assigns far more important role to interest rate spreads alongside preference shock; identifies extraordinary countercyclical monetary and fiscal policies during crisis as shocks to government expenditure and the estimated Taylor rule.
  - 1979–1987 (Volcker era): all three methods consistently highlight profound negative impact of Volcker’s contractionary monetary policy and positive role for private investment in mid-1980s.
  - 2016Q2–2020Q1: methods diverge on roles of private investment, price markups, preference shocks, fiscal and monetary policy; DC1 finds ongoing negative contributions from private investment and markups, modest expansionary roles for tax and monetary policy.
  - 2020Q2–2023Q1 (Covid period, model using parameters estimated to 2020Q1):
    - All methods assign similarly large contractionary effect to the intertemporal preference shock followed by drop in private investment.
    - DC1 largely agrees with DC3 on positive countervailing impact of fiscal policy and TFP.
    - DC1 suggests economy was already below long-run balanced growth path at pandemic start; accumulated legacy shocks mildly exacerbated crisis.

### Role of the added preference shock and interest-rate observables
- Preference shocks:
  - Large negative preference shocks identified at:
    - 2001Q3 (coinciding with 9/11),
    - last two quarters of 2008 (collapse of Lehman Brothers),
    - 2020Q2 (start of covid pandemic).
- Observed interest rates and shadow rate:
  - Extended model includes shadow rate (Wu and Xia (2016)) when policy rate hits effective zero bound (financial crisis and aftermath).
  - Inclusion of four observed interest rates avoids counterfactual responses from omitting rates that often do not move together during episodes.
- Contribution to output variance:
  - Once two yields on government bonds are added as observable variables, spread ω_f accounts for 22.27% of fluctuations in output for the entire sample (extended model result).
  - Monetary policy contribution to output variance remains nearly unchanged between models at roughly 18%–19%.

### Monetary policy, interest-rate spreads, and fiscal history
- Monetary policy long-run pattern:
  - Very expansive monetary policy during the 1950s; more muted expansion in 1960s and 1970s.
  - Sharp contraction during the 1980s associated with Volcker (starting 1979) and continued under Greenspan till 1992.
  - From 1992 on, monetary shocks largely expansionary, except a brief interlude at pandemic start.
  - Period 2009Q3 to 2015Q4: shadow policy rate descended below zero; monetary policy shocks mirror data.
- Interest-rate spreads:
  - Four interest rates highly correlated, but gaps fluctuate; widening spreads appear to depress output, particularly the 5-year government bond interest.
- Fiscal (Debt/GDP milestones):
  - 1948Q2 to 1950Q2 (start of Korean War): total U.S. net debt burden declined from 61.2% of GDP to 43.8%.
  - 1960Q4 to 1979Q4: debt burden dropped from 45.1% to 25.3%.
  - After Economic Recovery Tax Act first phase (October 1 1981; Congress passed Aug 4, 1981; signed Aug 13, 1981) debt burden began to rise and accelerated thereafter.
  - Stabilised at about 49% in early/mid 1990s; dropped to 30.7% in 2001Q2.
  - From 2001Q2 onward debt grew rapidly (debt/GDP ratio fluctuates substantially due to denominator fluctuations during GFC and covid).
- Fiscal transmission:
  - Output affected most directly through the labor tax, to a lesser extent through government expenditure, and indirectly through government bonds.
  - Sample partitioning:
    - Immediate post–World War II: labor tax impact negative while debt falls steeply and output below trend.
    - 1960s recovery: as debt declines, labor tax shocks nearly always positive for output.
    - 1980s onward: despite rising debt, labor tax impact remains mostly positive until 2012Q2.
    - From 2012Q2, when debt burden crosses 75% of GDP, labor tax shocks become uniformly negative except for 2020Q2 and 2020Q3.

### Individual shocks’ effects on output (selected patterns)
- Private investment:
  - Negative effect starting mid 1951, intensifying through 1950s, damping in mid 1970s.
  - Strongly positive 1981Q1 to recession starting 1989Q4; positive thereafter before declining precipitously during late 2008 GFC.
  - From 2016Q3 to 2023Q1 impact always negative, with a particularly large effect in 2020Q2.
- Price markup:
  - Positive 1953Q2 to 1974Q3 and 1984Q2 to 2005Q1.
  - From 2005Q1 onward effect negative and increasingly so toward sample end.
- Technology shock:
  - Increasingly positive from 1950Q4 till late 1970s; declines thereafter.
  - Negative from 1987Q4 to 2009Q2.
- Government investment:
  - Positive effect from 1951Q2 reflecting strong external effect of public capital.
- Preference shocks:
  - Typically positive through much of late twentieth century.
  - Turn decidedly negative in 2001Q3 and remain negative thereafter, intensifying in 2003, dissipating by end-2005.
  - Large negative shocks in 2008Q4 and 2020Q2.

### Estimation, variance decomposition, and parameter results
- Solution and estimation:
  - Solved as log-linear first order approximation around balanced growth path and estimated using Dynare (Dynare 4.5.7 referenced).
- Variance decomposition differences (DU vs this paper, 1948Q2–2008Q4):
  - Technology: Drautzburg 21.35 ; This paper 16.00
  - Price Markup: Drautzburg 8.58 ; This paper 6.91
  - Wage Markup: Drautzburg 9.17 ; This paper 1.47
  - Lab. Tax: Drautzburg 8.26 ; This paper 6.94
  - Gov. Spending: Drautzburg 3.39 ; This paper 3.29
  - Priv. Inv.: Drautzburg 18.77 ; This paper 16.80
  - Gov. Inv.: Drautzburg 4.75 ; This paper 6.16
  - Policy Rate: Drautzburg 18.77 ; This paper 18.40
  - 5 y. Gov. Bonds: This paper 7.22
  - 10 y. Gov. Bonds: This paper 6.36
  - 20 y. Gov. Bonds: This paper 2.62
  - 20 y. Priv. Bonds: Drautzburg 1.20 ; This paper 1.78
  - Pref. Shock: Drautzburg 12.41
- Selected posterior modes and intervals (Table 2 highlights):
  - Returns to scale Λ: Posterior Mode 1.2826 ; Mean Estimate 1.276 ; HPD inf 0.7995 ; HPD sup 1.8565
  - Capital share α: Posterior Mode 0.2213 ; Mean Estimate 0.224 ; HPD inf 0.2072 ; HPD sup 0.2412
  - Risk aversion σ: Posterior Mode 1.0965 ; Mean Estimate 1.115 ; HPD inf 0.9729 ; HPD sup 1.2547
  - Habit h: Posterior Mode 0.7958 ; Mean Estimate 0.798 ; HPD inf 0.7580 ; HPD sup 0.8400
  - Disc. factor 100 × 1 − β: Posterior Mode 0.0692 ; Mean Estimate 0.084 ; HPD inf 0.0319 ; HPD sup 0.1336
  - Taylor smoothing ρ_R: Posterior Mode 0.8762 ; Mean Estimate 0.880 ; HPD inf 0.8542 ; HPD sup 0.9060
  - Mean spread ω_f: Posterior Mode 0.4004 ; Mean Estimate 0.400 ; HPD inf 0.3362 ; HPD sup 0.4666
  - AR(1), technology ρ_a: Posterior Mode 0.9645 ; Mean Estimate 0.965 ; HPD inf 0.9487 ; HPD sup 0.9824
  - AR(1), gov. bond spread ω_f: Posterior Mode 0.8887 ; Mean Estimate 0.892 ; HPD inf 0.8510 ; HPD sup 0.9332
- Selected shock standard deviations (Table 3 highlights):
  - s.d. tech. invg: Posterior Mode 0.4521 ; Mean Estimate 0.454 ; HPD inf 0.4188 ; HPD sup 0.4871
  - s.d. bond invg: Posterior Mode 0.1495 ; Mean Estimate 0.151 ; HPD inf 0.1398 ; HPD sup 0.1610
  - s.d. pref. invg: Posterior Mode 12.8913 ; Mean Estimate 13.483 ; HPD inf 10.1839 ; HPD sup 16.7559
  - s.d. gov. invg: Posterior Mode 0.3516 ; Mean Estimate 0.355 ; HPD inf 0.3285 ; HPD sup 0.3799
  - s.d. mon. pol. invg: Posterior Mode 0.1862 ; Mean Estimate 0.189 ; HPD inf 0.1744 ; HPD sup 0.2025
  - s.d. inv. price invg: Posterior Mode 0.8968 ; Mean Estimate 0.904 ; HPD inf 0.7693 ; HPD sup 1.0358
  - s.d. gov. inv. price invg: Posterior Mode 0.5565 ; Mean Estimate 0.558 ; HPD inf 0.4640 ; HPD sup 0.6490
- Diagnostics: Figures 13–18 present priors and posteriors for parameters and shock s.d.’s; authors report using posterior modes for variance calculations.

### Methodological implications and recommendations
- Choice of shock-decomposition method materially affects historical interpretation:
  - Formula (4) / DC1 preferred when isolating in-episode shocks from earlier legacy effects.
  - Formula (5) / DC2 aggregates cumulative effects and can overemphasize persistent pre-episode shocks.
  - Formula (6) / DC3 (differencing) contrasts cumulative pre-episode impacts with in-episode behavior but removes level information relative to steady state.
- Combining decompositions (e.g., solid bars from (4) and striped bars for difference between (4) and (5) as in Figure 12) can illuminate contemporaneous policy changes versus carried-forward cumulative effects.
- Practical suggestion: analysts using Dynare should be aware that Dynare 4.5.7 implements DC2 by default; authors supply Matlab code to compute DC1 for Dynare-estimated models.

*Source: wpiea2025051-print-pdf — Evaluating Historical Episodes using Shock Decompositions in the DSGE Model — Working Paper No. WP/2025/051 — https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025051-print-pdf.pdf*

### 1. The literature is a very large one, but Christiano, Eichenbaum and Evans (2005) and Smets and Wouters

### wpiea2025051-print-pdf - 1. The literature is a very large one, but Christiano, Eichenbaum and Evans

### Background and motivation
- Christiano, Eichenbaum and Evans (2005) and Smets and Wouters (2003, 2007) are widely regarded as seminal contributions.
- Motivation: different practices for historical shock decomposition generate divergent interpretations of the same estimated shocks; need methods that isolate the impact of exogenous shocks for particular periods.
- Practical constraint: "thus far there is no Dynare facility for" implementing one preferred approach (DC1) in common use.

### Shock decomposition methods (formal definitions and implications)
- State-space reduced-form of DSGE (log-linear, first order around balanced growth path):
  - s_t = A s_{t−1} + B ε_t
  - x_t = C s_t
  - Iterated form: x_t = C A^t s_0 + sum_{j=1}^t C A^{t−j} B ε_j
- Three decomposition approaches described:
  - DC1 (isolate shocks after t): x_{t+h} = C A^h s_t + sum_{j=t+1}^{t+h} C A^{t+h−j} B ε_j, h∈{1,...,k}.
    - First term captures initial condition at start of episode (distance from balanced growth path).
  - DC2 (common in literature and Dynare 4.5.7): use smoothed shocks for entire sample and extract decomposition for period analyzed; places significant weight on impacts of shocks before the event because persistent pre-event shocks remain in the decomposition.
    - Expressed as x_{t+h} = C A^{t+h} s_0 + sum_{j=1}^t C A^{t+h−j} B ε_j + sum_{j=t+1}^{t+h} C A^{t+h−j} B ε_j, h∈{1,...,k}.
  - DC3 (differencing approach used by DU): examine differences x_{t+h} − x_t = C(A^{t+h} − A^t) s_0 + sum_{j=1}^t C A^{t−j} (A^h − I) B ε_j + sum_{j=t+1}^{t+h} C A^{t+h−j} B ε_j, h∈{1,...,k}.
    - Nets out initial value at start of sample but does not remove influence of prior shocks accumulated from the initial period onward.
- Practical note: one can correct for DC2's sensitivity to earlier shocks by considering DC3 (differencing), but DC3 still includes influence of prior shocks via second term.

### Model used to illustrate differences (specification and extensions)
- Base: extended version of Drautzburg and Uhlig (DU, 2015) built on Smets and Wouters (2007), expanded to include fiscal policy, financial frictions, and increasing returns from public capital externality.
- Key features and extensions:
  - Twelve data series and twelve exogenous shocks (expanded from DU’s ten series and ten shocks).
  - Four observed interest rates (all determined by data): policy rate r_f, intermediate-term government rate r_b, long-term government rate r_l, and private/corporate rate r_k.
  - Four corresponding wedges determined by data: ω_f (policy rater_f vs five-year government rate), ω_b (five-year vs twenty-year government yields), ω_l (twenty-year government vs twenty-year corporate yields), and ω_k (sum of these three).
  - Added preference shock (multiplicative to discount factor β): log ς_{p,t} = ρ_P log ς_{p,t−1} + ε_{p,t}, ε_{p,t} ∼ N(0, σ^2_{εp}).
    - Preference shock affects only the Euler equation; a positive shock increases the discount factor, raising savings and lowering consumption for unconstrained agents.
  - Production: intermediate goods use Cobb-Douglas combining effective private capital (K^p_t times utilization u_t), effective labor (μ_t n_t), fixed cost Φ, and external effect from contemporaneous government capital K^g_t:
    - Micro firm: Y_t(i) = ẽ^a_t ( (K^g_t)^Λ (∫_0^1 Y_t(j) dj + Φ μ_t) )^{ζ/(1−ζ)} (u_t(i) K^p_t(i))^{α} (μ_t n_t(i))^{1−α} − Φ μ_t
    - Aggregate: Y_t = ẽ^a_t (K^g_t)^ζΛ (u_t K^p_t)^{α(1−ζ)} (μ_t n_t)^{(1−ζ)(1−α)} − Φ μ_t
    - Inclusion of exponent Λ as an estimated parameter allows government capital to generate nonconstant returns to scale.
  - Marginal costs: MC_t = α^{−α} (1−α)^{−(1−α)} W_t^{1−α} (R^k_t)^α μ_t^{−(1−α)} ( (K^g_{t−1})^Λ / (Y_t + μ_t Φ) )^{ζ/(1−ζ)} ε_t^{α}
  - Market and frictions: Calvo pricing in goods and labor markets; unions set wages to maximize representative household welfare; households supply differentiated labor.

### Historical decomposition results and empirical findings (sample periods and key episodes)
- Estimation sample for parameters: data from 1948Q2 to 2020Q1. Model used to analyze outcomes up to 2023Q1.
- Main long-run patterns:
  - Detrended output across sample 1948Q2–2023Q1:
    - US economy experienced an eight month recession at the end of World War II; analysis begins the quarter before the second postwar recession starting end-1948 lasting nearly a year.
    - From start of the Korean war in mid 1950 to start of the Great Recession in 2008, detrended output is relatively elevated relative to steady state in nearly every quarter.
    - From the Great Recession onward, detrended output declines and remains permanently below detrended steady state value from mid 2015 onward.
- Episode-specific comparative interpretations (differences across DC1, DC2, DC3):
  - 1964Q2 to 1966Q1 (period of relatively high growth):
    - Two decompositions suggest fiscal policy played important role (Johnson’s tax cuts and 1964 Economic Opportunity Act in 1964Q4).
    - Preferred decomposition (DC1) suggests momentum from legacy shocks from previous quarters explains much of the performance.
  - 2006Q4 to 2009Q3 (period surrounding financial crisis):
    - Common decomposition: year preceding crisis appears relatively stable; expansionary monetary policy and positive investment shocks partly offset by downward pressure from intermediate-term interest rates and shocks to TFP and preferences. From 2007Q4 a growing imbalance tips economy into recession.
    - Alternative decomposition: crisis largely driven by negative preference shocks plus intensification of prior negative shocks to private investment; corporate bond term spreads had muted or mild pro-cyclical impacts; monetary policy (conventional and unconventional) had little ameliorating effect.
    - Preferred decomposition: assigns a far more important role to interest rate spreads in explaining crisis evolution alongside preference shock; identifies extraordinary countercyclical monetary and fiscal policies during the crisis as shocks to government expenditure and the estimated Taylor rule.
  - 1979 to 1987 (Volcker era):
    - All three methods consistently highlight profound negative impact of Volcker's contractionary monetary policy and positive impact of private investment shocks in mid-1980s.
  - 2016Q2 to 2020Q1 (four years prior to Covid downturn):
    - Preferred decomposition: private investment and price markups increasingly negative for output; ongoing impact from previous negative preference shocks and shocks to government spending and monetary policy; modest expansionary roles for tax and monetary policy.
    - Most commonly used decomposition: monetary policy initially plays significant positive but diminishing role.
    - Another decomposition: monetary policy exerts ever-increasing downward pressure on output.
    - Different methods assign important but opposing roles to intermediate-term bond spread over the policy rate.
  - 2020Q2 to 2023Q1 (Covid period, model experiments using parameters estimated to 2020Q1):
    - All methods assign a similarly large contractionary effect to the intertemporal preference shock followed by drop in private investment.
    - Preferred method largely agrees with differencing approach on positive countervailing impact of fiscal policy and TFP.
    - Preferred method suggests economy was already performing below long-run balanced growth path at pandemic start; accumulated legacy shocks mildly exacerbated the crisis.
    - Section 6 (not reproduced here) separates impacts accruing from start of episode from continuing legacy shocks; resulting figure combines traditional approach with preferred method to represent shock effects best.
- Variance and historical decompositions (model diagnostics and role of added shocks):
  - Inclusion of preference shock plays a significant role:
    - Large negative preference shocks identified at:
      - third quarter of 2001 (coinciding with 9/11),
      - last two quarters of 2008 (collapse of Lehman Brothers),
      - 2020Q2 (start of covid pandemic).
  - Extended model includes shadow rate (Wu and Xia (2016)) when policy rate hits effective zero bound in financial crisis and aftermath.
  - Improvements to public debt measure: use net federal debt including state and local debt (instead of gross federal debt).
  - Extended sample used for analysis up to first quarter of 2023.
  - Inclusion of four observed interest rates avoids counterfactual responses from omitting rates that often do not move together during episodes of interest.

### Estimation, software, and data notes
- Model solution and estimation:
  - Solved as log-linear first order approximation around balanced growth path and estimated using Dynare.
  - Dynare version referenced: Dynare 4.5.7 (current shock decomposition in Dynare generates DC2).
  - Authors have programmed a general Matlab code to compute DC1 for models estimated with Dynare 4.5.7 (available upon request).
- Data and sample choices:
  - Parameters estimated using data from 1948Q2 to 2020Q1; model applied to analyze 2020Q2–2023Q1 without adding special pandemic-specific shocks.
  - Extended model adds two data series (long-term government yields and a preference shock) relative to DU to provide degrees of freedom for estimation.
  - Table references and full prior/posterior details are in Appendix (not reproduced here).

*Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025051-print-pdf.pdf*

### 7. This matches the code for the estimated model developed by DU though the article itself asserts that unions

### 7. This matches the code for the estimated model developed by DU though the article itself asserts that unions maximise the welfare of rational optimizing households only.

### Model replication and correction
- The authors estimate DU’s model using the code DU supply (see https://ideas.repec.org/c/red/ccodes/14-44.html).
- They are able to replicate the estimated model for the whole sample period, aside from the variance decomposition.
- A small error was found in the original code for the linearization of the model’s profit equation; the authors provide their own estimates for a corrected version of the model reported in Table 1.

### Shock decomposition methodology and visualization
- Shock decompositions are calculated using the decomposition formula (6); the shock decompositions in each time period sum to the log of output relative to trend (procedure corresponds to Dynare).
- Figure 2: Shock decomposition 1948Q2–2023Q1, with the black line representing output.
- To isolate policy influences, the authors consider:
  - Monetary policy broadly defined to include: the policy rate, corporate spread, and two term spreads in the model.
  - Fiscal policy defined as: labour taxes, government consumption expenditure, and government investment.
- Figures used to illustrate relationships: Figure 2 (historical shock decompositions), Figure 3 (four interest rates data and their historic shock decompositions), Figure 5 (Debt/GDP series and fiscal decompositions).

### Monetary policy, interest rates, and spreads (findings from Figures 2–4)
- Long-run pattern:
  - Very expansive monetary policy during the 1950s, continuing in a more muted fashion during the 1960s and 1970s.
  - Sharp contraction during the 1980s associated with anti-inflationary policies instituted by Fed Chairman Paul Volcker in 1979 and continued under Chairman Alan Greenspan till 1992.
  - From 1992 on, monetary shocks are largely expansionary, except for a brief interlude at the start of the pandemic.
- Period 2009Q3 to 2015Q4: shadow policy rate descended below zero; the pattern in monetary policy shocks mirrored the data.
- The four interest rates are highly correlated, but gaps (spreads) fluctuate; widening spreads appear to depress output, particularly the 5-year government bond interest.
- Figure 3 details data sources and timing for the four interest rates:
  - Policy Rate: March 1948 to June 1954 3-Month Treasury Bill: Secondary Market Rate; from the start of the subprime mortgage crisis in April 2007 through March 2023 shadow rate calculated by Wu and Xia (2016); for all other months the Federal Funds Rate, converted to quarterly rates.
  - 5-year U.S. Treasury coupon note yield: 1953Q2 to 2023Q1, supplemented by Ibbotson (2016) for 1948Q2 to 1953Q1.
  - 20-year U.S. Treasury coupon note yields: supplemented by Ibbotson (2016) for 1948Q2 to 1953Q1 and 1987Q1 to 1993Q3.
  - Corporate bond yield: Moody’s Baa index at quarterly rates.
- Figure 4: plots policy rate r_f_t and spreads ω_f_t, ω_b_t, and ω_l_t against their effect on output from historical shock decomposition for the policy rate, 5-year government bond, 20-year government bond, and 20-year corporate bond.

### Fiscal history and debt dynamics (key numeric milestones)
- Debt/GDP history and key values:
  - From 1948Q2 (ten quarters after the end of World War II) to the start of the Korean War in the second quarter of 1950, total U.S. net debt burden declined from 61.2% of GDP to 43.8%.
  - From 1960Q4 to 1979Q4 the debt burden dropped from 45.1% to 25.3%.
  - Following enactment of the first phase of the Economic Recovery Tax Act on October 1 1981, the debt burden began to rise and accelerated thereafter.
    - Note: Congress passed the legislation on August 4, 1981, and President Ronald Reagan signed it into law on August 13, 1981.
  - It stabilised at about 49% in the early and mid 1990’s and then began to drop again, reaching 30.7% in 2001Q2.
  - From 2001Q2 onward, following the events of 9/11, the debt grew rapidly (debt/GDP ratio fluctuates substantially due to large denominator fluctuations during the great financial crisis and the covid pandemic).
- Fiscal transmission to output:
  - The history affects output most directly through the labor tax, to a lesser extent through government expenditure, and indirectly through government bonds.
  - Sample partitioning shows:
    - Immediate post–World War II period: labor tax impact is negative while debt is falling steeply and output is below trend.
    - 1960s recovery: as debt declines, labor tax shocks have nearly always positive impact on output.
    - 1980s onward: despite rising debt, labor tax impact remains mostly positive until 2012Q2.
    - From 2012Q2, when the debt burden crosses the threshold of 75% of GDP, labor tax shocks become uniformly negative except for 2020Q2 and 2020Q3.

### Individual shock effects on output (findings from Figures 2 and 5)
- Private investment (shadow price on private investment):
  - Exerts a negative effect starting from mid 1951 that intensifies through the rest of the 1950s, beginning to dampen in the mid 1970s.
  - From 1981Q1 to the start of the recession lasting from 1989Q4 to 1991Q1 the effect is strongly positive, and positive thereafter before declining precipitously during the great financial crisis in late 2008.
  - From 2016Q3 to 2023Q1 the impact is always negative, with a particularly large effect in 2020Q2 coinciding with the start of the covid pandemic.
- Price markup:
  - Positive between 1953Q2 to 1974Q3 and again from 1984Q2 to 2005Q1.
  - From 2005Q1 onward its effect is negative, increasingly so toward the end of the sample.
- Technology shock:
  - Exerts an increasingly positive effect on output from 1950Q4 till the end of the 1970s, thereafter it begins to decline.
  - From 1987Q4 to 2009Q2 it is negative, though declining for much of the period preceding the pandemic.
- Government investment:
  - Exerts a positive effect on output from 1951Q2, reflecting a strong point estimate for the external effect generated by public capital.
- Preference shocks:
  - Typically positive throughout much of the last half of the twentieth century.
  - Turn decidedly negative in 2001Q3 and remain negative thereafter, suggesting a long-lasting decline in the value of ς_p,t beginning in 2001Q3 (coinciding with the events of 9/11), intensifying during most of 2003 (the U.S. invasion of Iraq), and gradually dissipating by the end of 2005.
  - 2004Q1: a very large negative shock offsets a positive shock from the labour tax, possibly linked to passage of the Jobs and Growth Tax Relief Reconciliation Act signed into law in May 2003.
  - Late 2008: a new sharp decline appears in the last quarter of 2008, after bailouts of Fannie Mae and Freddie Mac (July and September 2008) and the collapse of Lehman Brothers (September 2008).
  - 2020Q2: one more large negative shock coinciding with the start of the covid pandemic in the US.

### Combined effects and structural interpretation
- Combined effect of 5-year and 20-year government bonds:
  - Fluctuates around zero through the late 1940s to early 1960s (5-year mostly negative, 20-year mostly positive), and largely positive till the early 1980s.
  - From the early 1980s on, as the debt burden rises, the combined effect is nearly always negative.
- The model highlights:
  - A prominent role for the Federal Reserve’s tight monetary policy during the 1980s and early 1990s.
  - A more important role for bond spreads, particularly intermediate-term yield curve components, compared to DU — an effect pronounced from the late 1980s and intensifying during the financial crisis.
- Methods comparison (decompositions (4)–(6)):
  - The three methods can generate profoundly different interpretations of historical episodes.
  - Figures 6a (1948Q2–1985Q4) and 6b (1986Q1–2023Q1) isolate instantaneous impacts of the shock process term CBe_j in (3) across two three-decade subsamples.
  - Preference shocks, labor tax shocks, and private investment shocks dominate both subsamples, but patterns differ across periods:
    - Raw monetary policy shocks in Figure 6a were more amplified (mostly negative) in the late 1970s and early 1980s; muted in prior and subsequent periods.
    - Shocks to private investment in Figure 6b become notably more muted from the early twenty-first century.
  - The authors focus on decompositions (4) and (5) for three episodes and then discuss decomposition (6) for additional insight.

_Italic: Source — wpiea2025051-print-pdf (section 7) — IMF working paper content provided in the prompt._

### 10. We  find  the  preference  shocks,  but  not  the  other  shocks,  are  negatively  correlated  with  the  cyclicall

### wpiea2025051-print-pdf - 10. We  find  the  preference  shocks,  but  not  the  other  shocks,  are  negatively  correlated  with  the  cyclicall

### Key findings on shocks and asset prices
- The preference shocks, but not the other shocks, are negatively correlated with the cyclically adjusted price earnings ratio calculated by Robert Shiller.
- This correlation suggests that investors bid up the price of equities, relative to earnings, in response to a rise in the willingness to defer consumption and save.

### Comparison of shock-decomposition methods
- Decomposition formula (4) — the authors' preferred state-space decomposition:
  - Isolates the impact of each type of shock from those that preceded it, starting at the initial period within the episode under consideration.
  - Shows that much of the divergence of output above its steady state value can be the cumulative effect of previous shocks to that date rather than shocks occurring during the episode.
- Standard decomposition formula (5):
  - Answers the cumulative effect across time of a particular type of shock on the variable of interest in each period.
  - Conflates the impact of shocks occurring in the time period with those occurring beforehand, so legacy of initial conditions from the beginning of the sample (1948) can persist into later sub-period interpretations.
  - Example: For 1964Q2–1966Q1 (Figure 7b) it attributes the largest contribution to the expansion starting from 1964Q4 to the reduction in price markups, rather than to the pattern of underlying contemporaneous shocks.
- Differencing decomposition formula (6):
  - Attempts to isolate shocks within the time period of interest by differencing, but remains predicated on previous shock behavior because it contrasts cumulative impacts up to the episode with subsequent behavior during the episode.
  - Does not convey where the economy is in the initial period relative to steady state, because differencing removes that information.

### Historical episode analyses and method-dependent interpretations
- 1964Q2–1966Q1 (Figures 6a, 7a–7c):
  - State-space decomposition (4) indicates preference shocks and labor tax shocks have large impacts across the ten-year horizon, but their fluctuating signs prior to 1964 cause cancellation in (5).
  - Standard decomposition (5) inflates the role of government investment and 5-year government bond shocks because of consistent sign and persistence prior to 1964Q2.
- 2006Q4–2009Q3 (Figures 6b, 8a–8c):
  - State-space decomposition (4) (Figure 8a):
    - Output in 2006Q4 was well above its steady-state level; much of the subsequent decline over the next four quarters can be attributed to convergence.
    - By 2008Q1, small negative shocks offset expansionary monetary policy, labor tax shocks, government investment, and preference shock.
    - From 2008Q3 onward, accumulating impacts of preference shocks and, from 2008Q4, private investment shocks contributed to the decline.
    - The largest effect came from shocks to three of the model’s four interest rates: corporate bonds (r_k), and two government bonds (r_b and r_l), which counteracted positive shocks to the policy rate (r_f).
    - These changes imply a large increase in ω_k observed in the data, comprised of increases in ω_b, ω_l and ω_f.
  - Standard decomposition (5) (Figure 8b):
    - Suggests little happened from 2006Q4 to 2009Q3 aside from a large negative preference shock in 2008Q4 and small negative role of price markup.
    - Attributes expansionary roles to monetary policy and government investment well before the crisis.
    - Tends to emphasize effects of shocks occurring before 2006Q4 due to persistence (e.g., positive government investment shocks prior to the episode).
  - Differencing decomposition (6) (Figure 8c):
    - Produces different patterns by contrasting in-episode behavior with prior cumulative impacts; may understate roles of persistent pre-episode shocks and cannot indicate distance from steady state at the episode start.
- 2016Q2–2020Q1 (Figures 6b, 9a–9c):
  - State-space decomposition (9a):
    - Detrended output is flat and unusually constant; private investment, price markups, preference shocks, government spending and technology exert downward pressure.
    - Output is bolstered by shocks to labor taxes and both private and public 20-year bonds; monetary policy modestly supports output from second half of 2016 but dissipates by 2019.
    - Seven years after the end of the great recession the prior accumulation of negative shocks has left output well below steady-state; convergence implies output grows at trend despite prevailing negative shocks.
  - Standard decomposition (9b) and differencing (9c) yield different interpretations because of carry-forward effects and differencing sign reversals respectively (e.g., government investment and price markup influences differ across methods).
  - The three methods can on occasion deliver similar interpretations, but often diverge, indicating method choice is decisive.
- 1979Q3–1987Q3 (Figure 10):
  - All three decompositions capture the initial negative impact on output associated with Paul Volcker’s deflationary policies and suggest persistence of negative effects through much of the 1980s.
  - Recovery thereafter is largely driven by positive shocks to private investment and declines in the price markup across decompositions.
- 2020Q2–2023Q1 (Figures 11 and 12):
  - All three methods produce very similar patterns for the covid pandemic period.
  - The largest initial effects are negative preference shock and private investment.
  - Preferred decomposition (11a) shows monetary policy and government spending also contributing to the initial sharp downturn, partly ameliorated by positive shocks from the labour tax and technology; differencing (11c) yields a largely similar pattern.
  - Combining decompositions (Figure 12) — solid bars from (4), striped bars represent difference between (4) and (5):
    - Reveals that the shock to the labour tax in 2020Q2 played an important role in counteracting the large negative preference shock at the start of the pandemic.
    - Shows previous shocks to the labour tax during the period were entirely negative, highlighting the importance of the policy change in 2020Q2.

### Implications for interpretation and modeling
- The choice of shock-decomposition method materially affects the interpretation of historical episodes when using DSGE models.
- Formula (4) is preferred when the goal is to isolate the impact of shocks within a defined historical episode from the influence of earlier shocks.
- Formulae (5) and (6) include the legacy of earlier shocks in different ways:
  - (5) aggregates cumulative effects across time, potentially overemphasizing persistent pre-episode shocks.
  - (6) contrasts pre-episode cumulative impacts with in-episode behavior via differencing but removes information about the level relative to steady state.
- Combining information from multiple decompositions (as in Figure 12) can illuminate policy changes during episodes by separating contemporaneous shocks from carried-forward cumulative effects.

### Conclusion
- Shock decompositions provide useful insights into DSGE model mechanics and how model-implied shocks may explain historical periods.
- Though sometimes the three methods yield similar interpretations (e.g., Volcker period, covid onset), in general different decomposition methods often generate very different results.
- The authors prefer the decomposition that isolates shocks within the episode analyzed (formula (4)) because it separates episode-specific shocks from prior legacy effects.

*Source: wpiea2025051-print-pdf - 10. We  find  the  preference  shocks... — https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025051-print-pdf.pdf*

### References

### References and Appendix (wpiea2025051-print-pdf)

### References
- Blanchard, Olivier. “Public debt and low interest rates.” American Economic Review 109, no. 4 (2019): 1197-1229.
- Cardani, Roberta, Olga Croitorov, Massimo Giovannini, Philipp Pfeiffer, Marco Ratto, and Lukas Vogel. “The euro area’s pandemic recession: A DSGE-based interpretation.” Journal of Economic Dynamics and Control 143 (2022): 104512.
- Christiano, Lawrence J., Martin Eichenbaum, and Charles L. Evans. “Nominal rigidities and the dynamic effects of a shock to monetary policy.” Journal of Political Economy 113, no. 1 (2005): 1-45.
- Corrado, Luisa, Stefano Grassi, and Aldo Paolillo. Modelling and estimating large macroeconomic shocks during the pandemic. National Institute of Economic and Social Research Discussion Paper No. 530, 2021.
- Del Negro, Marco, Stefano Eusepi, Marc P. Giannoni, Argia M. Sbordone, Andrea Tambalotti, Matthew Cocci, Raiden Hasegawa, and M. Henry Linder. “The FRBNY DSGE model.” FRB of New York Staff Report 647 (2013).
- Drautzburg, Thorsten, and Harald Uhlig. “Fiscal stimulus and distortionary taxation.” Review of Economic Dynamics 18, no. 4 (2015): 894-920.
- Greenwood, Robin, Samuel G. Hanson, Joshua S. Rudolph, and Lawrence Summers. “The optimal maturity of government debt.” The $13 Trillion Question: How America Manages Its Debt. (2015): 1-41. Brookings Institution Press.
- Ibbotson, Roger, Roger J. Grabowski, James P. Harrington, and Carla Nunes. 2016 SBBI Yearbook: Stocks, Bonds, Bills, and Inflation. John Wiley and Sons, Incorporated, 2016.
- Justiniano, Alejandro, Primiceri, Giorgio and Andrea Tambalotti “Investment shocks and the relative price of investment.” Review of Economic Dynamics 14 (2011): 102:121.
- Lindé, Jesper and Mathias Trabandt, “Should We Use Linearized Models to Calculate Fiscal Multipliers?” Journal of Applied Econometrics, (2018): 1-29.
- Meltzer, Allan H. A history of the Federal Reserve, Volume 1: 1913-1951. University of Chicago Press, 2010.
- Mian, A and Sufi, A. “Finance and Business Cycles: The Credit-Driven Household Demand Channel.” The Journal of Economic Perspectives 32, no. 3 (2018): 31-58.
- Smets, Frank, and Raf Wouters. “An estimated dynamic stochastic general equilibrium model of the euro area.” Journal of the European Economic Association 1, no. 5 (2003): 1123-1175.
- Smets, Frank, and Rafael Wouters. “Shocks and frictions in US business cycles: A Bayesian DSGE approach.” American Economic Review 97, no. 3 (2007): 586-606.
- Wu, Jing Cynthia, and Fan Dora Xia. “Measuring the macroeconomic impact of monetary policy at the zero lower bound.” Journal of Money, Credit and Banking 48, no. 2-3 (2016): 253-291.

### Appendix — An Extended Drautzburg-Uhlig Model of the U.S. Economy

#### A1: Modifications to the Data Compared With DU
- Sample and coverage:
  - Of the twelve data series used in estimation, eight are the same as those found in DU, starting in 1948Q2, but are updated and extended to cover the additional forty one quarters from 2009Q1 to 2020Q1.
- The twelve series (as described):
  1. per-capita output: chained 2005 real GDP, growth rates;
  2. per-capita consumption: private consumption expenditure, growth rates;
  3. per-capita investment: private fixed investment, growth rates;
  4. per-capita government investment, growth rates;
  5. per-capita hours worked: civilian employment index×average nonfarm business weekly hours worked index, demeaned log;
  6. inflation: GDP deflator, quarterly growth rates;
  7. wages: nonfarm business, hourly compensation index, growth rates;
  8. policy interest rate: from March 1948 to June 1954 the 3-Month Treasury Bill: Secondary Market Rate, from April 2007 through March 2023 the shadow rate calculated by Wu and Xia (2016), and for all other months the Federal Funds Rate, all coverted to quarterly rates;
  9. corporate bond yield: Moody’s Baa index at quarterly rates;
  10. per-capita net public debt (described below), demeaned log;
  11. 5-year U.S. Treasury coupon note yield supplemented by data calculated by Ibbotson (2016) on intermediate-term U.S. Treasury bond yields for the period between 1948Q2 and 1953Q1;
  12. 20-year U.S. Treasury coupon note yields supplemented by data calculated by Ibbotson (2016) on long-term U.S. Treasury bond yields for the period from 1948Q2 to 1953Q1.
- Per-capita construction:
  - Use the non-institutionalized population in the US (BLS series LNU00000000Q) to derive per-capita variables.
- Rationale and data augmentations:
  - Including government bond yields directly highlights that in DU the implied interest rate on government bonds causes counterfactual volatility and descends below the zero lower bound on several occasions in the sample.
  - Choice of 5-year Treasury bonds (instead of 10-year) better matches the maturity structure of U.S. government debt (citing Greenwood et al. (2015) and Blanchard (2019)).
  - Yields on 5-year Treasury coupon notes are unavailable till April 1954; the series is augmented with intermediate-term (5-year) rates calculated by Ibbotson (2016) for missing periods.
- Alternative public debt measure:
  - Derived as the sum of federal government debt securities and the total liabilities of state and local governments, excluding employee retirement funds (series FL314122005.Q and FL214190005.Q in the Financial Accounts of the United States).
  - This measure excludes intergovernmental federal debt (nonmarketable debt held by the Social Security Trust Fund) but includes total public liabilities of states and localities (excluding employee retirement funds).
  - Compared with DU’s gross federal debt series at par value (Dallas Fed), the constructed series begins the sample approximately 30% of GDP above DU’s series, converges until 1983, then diverges thereafter as the constructed series grows more slowly.
  - “Note the big changes in the debt series in 2020 and after reflect changes in both the debt and sharp fluctations in GDP.”

#### A2: Resulting Differences in Variance and Shock Decomposition
- High-level findings:
  - Addition of the eleventh and twelfth shocks reduces the relative importance of most original ten shocks in DU.
  - Of DU’s three large shocks—monetary policy, technology, and private investment—which together account for nearly 60% of the variance, only the contribution of monetary policy remains nearly undiminished.
  - Price markup, labor tax, government spending, and government investment recede in importance; the wage markup nearly disappears.
  - Once two yields on government bonds are added as observable variables, the shock associated with the spread between policy rate and the 5-year bond associated with government borrowing—ω_f in Figure 1(b)—accounts for 22.27% of the fluctuations in output for the entire sample.
  - The variance decomposition assigns nearly identical values to monetary policy between 18% and 19% of the total variation in output across the two models.
  - In decompositions for the extended model, initial conditions in 1948 imply output is below balanced growth values, and across the enlarged sample output fluctuates closer to and more often below the horizontal axis, producing a somewhat more intuitive measure of trend growth.
- Table 1: Variance decomposition (Drautzburg and Uhlig’s (2015) model 1948Q2 to 2008Q4, with small correction to code)
  - Technology: Drautzburg 21.35 ; This paper 16.00
  - Price Markup: Drautzburg 8.58 ; This paper 6.91
  - Wage Markup: Drautzburg 9.17 ; This paper 1.47
  - Lab. Tax: Drautzburg 8.26 ; This paper 6.94
  - Gov. Spending: Drautzburg 3.39 ; This paper 3.29
  - Priv. Inv.: Drautzburg 18.77 ; This paper 16.80
  - Gov. Inv.: Drautzburg 4.75 ; This paper 6.16
  - Policy Rate: Drautzburg 18.77 ; This paper 18.40
  - 5 y. Gov. Bonds: This paper 7.22
  - 10 y. Gov. Bonds: This paper 6.36
  - 20 y. Gov. Bonds: This paper 2.62
  - 20 y. Priv. Bonds: Drautzburg 1.20 ; This paper 1.78
  - Pref. Shock: Drautzburg 12.41
  - Note: “We are using the posterior modes in Table 2 for these calculations.”
- Estimation outputs (selected results from Table 2: posterior modes and intervals)
  - Returns to scale Λ: Posterior Mode 1.2826 ; Mean Estimate 1.276 ; HPD inf 0.7995 ; HPD sup 1.8565
  - Capital share α: Posterior Mode 0.2213 ; Mean Estimate 0.224 ; HPD inf 0.2072 ; HPD sup 0.2412
  - Risk aversion σ: Posterior Mode 1.0965 ; Mean Estimate 1.115 ; HPD inf 0.9729 ; HPD sup 1.2547
  - Habit h: Posterior Mode 0.7958 ; Mean Estimate 0.798 ; HPD inf 0.7580 ; HPD sup 0.8400
  - Disc. factor 100 × 1 − β: Posterior Mode 0.0692 ; Mean Estimate 0.084 ; HPD inf 0.0319 ; HPD sup 0.1336
  - Taylor smoothing ρ_R: Posterior Mode 0.8762 ; Mean Estimate 0.880 ; HPD inf 0.8542 ; HPD sup 0.9060
  - Mean spread ω_f: Posterior Mode 0.4004 ; Mean Estimate 0.400 ; HPD inf 0.3362 ; HPD sup 0.4666
  - AR(1), technology ρ_a: Posterior Mode 0.9645 ; Mean Estimate 0.965 ; HPD inf 0.9487 ; HPD sup 0.9824
  - AR(1), gov. bond spread ω_f: Posterior Mode 0.8887 ; Mean Estimate 0.892 ; HPD inf 0.8510 ; HPD sup 0.9332
- Structural shock standard deviations (selected entries from Table 3: posterior modes and intervals)
  - s.d. tech. invg: Posterior Mode 0.4521 ; Mean Estimate 0.454 ; HPD inf 0.4188 ; HPD sup 0.4871
  - s.d. bond invg: Posterior Mode 0.1495 ; Mean Estimate 0.151 ; HPD inf 0.1398 ; HPD sup 0.1610
  - s.d. pref. invg: Posterior Mode 12.8913 ; Mean Estimate 13.483 ; HPD inf 10.1839 ; HPD sup 16.7559
  - s.d. gov. invg: Posterior Mode 0.3516 ; Mean Estimate 0.355 ; HPD inf 0.3285 ; HPD sup 0.3799
  - s.d. mon. pol. invg: Posterior Mode 0.1862 ; Mean Estimate 0.189 ; HPD inf 0.1744 ; HPD sup 0.2025
  - s.d. inv. price invg: Posterior Mode 0.8968 ; Mean Estimate 0.904 ; HPD inf 0.7693 ; HPD sup 1.0358
  - s.d. gov. inv. price invg: Posterior Mode 0.5565 ; Mean Estimate 0.558 ; HPD inf 0.4640 ; HPD sup 0.6490
- Diagnostics and visualization:
  - Figures 13–18 present priors and posteriors (density and histogram diagnostics) for model parameters and shock standard deviations across the estimation.

*Evaluating Historical Episodes using Shock Decompositions in the DSGE Model — Working Paper No. WP/2025/051*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025051-print-pdf.pdf_