## wpiea2023010-print-pdf - Section 3 presents the workhorse macroeconomic model with real rigidities in a dynamic stochastic

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---

### Optimal Price Setting with Quasi-kinked Demand
- Setup and key formulas:
  - Firms maximize π = p*y − mc*y subject to quasi-kinked demand y = a − p^b, with a, b > 0.
  - Demand elasticity: ε ≡ −(dy/dp) * (p/y) = b/a * p^(b−1), which is increasing in p.
  - Marginal revenue is a concave function of price because elasticity rises with p.
- Illustrative parameter example:
  - a = 3, b = 100, mc = 0.9.
- Mechanism and implications:
  - For a given percentage change in marginal costs, firms raise prices by more than they cut them.
  - Low marginal costs → high markups → firms reluctant to cut prices (limited ability to crowd in extra demand under quasi-kinked demand).
  - High marginal costs → low markups → firms have large incentives to raise prices even if demand falls substantially.
  - Concavity of marginal revenue (due to rising elasticity) generates asymmetric pricing responses and a convex optimal price as a function of marginal cost (a “banana-type” schedule), producing a nonlinear Phillips curve in general equilibrium.

### The Workhorse Macroeconomic Model (model overview)
- Model choice and structure:
  - Baseline: Smets and Wouters (2007) model (multi-shock version of Christiano, Eichenbaum and Evans (2005)) with endogenous capital accumulation.
  - Real rigidities via Kimball (1995) aggregation (following Dotsey and King (2005) and Levin, López‑Salido and Yun (2007)).
  - Monopolistic competition in goods and labor markets; nominal frictions via sticky prices and wages; households may index prices/wages to composite of steady-state and lagged inflation.
  - Real rigidities: habit formation in consumption, investment adjustment costs, variable capital utilization, fixed costs in production.
- Shocks and processes:
  - Seven structural shocks drive dynamics.
  - Monetary policy shocks: AR(1).
  - Two inefficient cost-push shocks (wage and price): ARMA(1,1).
  - Four efficient shocks (TFP, risk premium, investment-specific technology, government spending): AR(1).
- Central pricing equations:
  - Linearized Phillips curve (Smets and Wouters style):
    - bπ_t − ι_p bπ_{t−1} = β(E_t bπ_{t+1} − ι_p bπ_t) + κ cmc_t + ê_{p,t}
    - κ = (1 − ξ_p β)(1 − ξ_p) / [ξ_p (1 + (φ_p − 1) ε_p)]
    - 1 − ξ_p is the probability of a firm reoptimizing price each period; ε_p is curvature of SW aggregator; φ_p is steady state gross price markup; ê_{p,t} is cost-push shock rescaled by 1/κ to enter with unit coefficient.
  - Nonlinear recursive pricing system where the markup shock ε_{p,t} multiplies marginal costs mc_t, representing an exogenous shock to desired markup.

### Calibration, estimation, and solution method
- Observables (quarterly US series) used for estimation:
  - Log-differences of real per capita GDP, consumption, investment.
  - Log-differences of compensation per hour and the GDP deflator.
  - Log-deviations of hours worked per capita from average.
  - Federal funds rate.
- Estimation approach and posterior modes:
  - Authors adapt SW parameters estimated on pre-Global Financial Crisis and Covid data, except pricing parameters φ_p, ξ_p, ε_p.
  - Re-estimation for sample 1965Q1−2007Q4 with alternative prior φ_p ∼ N(1.2, 0.05) to obtain lower gross markup than the 61 percent estimated by SW.
  - Calibration/priors and resulting posterior modes:
    - ξ_p = 0.667.
    - Prior ε_p ∼ N(75, 25) → posterior mode ε_p = 64.5.
    - Posterior mode φ_p = 1.34.
    - κ in eq. (1) equals .008 (authors’ value).
    - SW’s κ was .026; with SW priors and sample ending 2007:4, authors obtain κ = .018.
  - Re-estimated model improves marginal likelihood by roughly 5 log points relative to original SW parameterization for the specified sample.
- Solution algorithm and implementation:
  - Fair and Taylor (1983) two-point boundary value (time-stacking) solution algorithm (certainty equivalence imposed on nonlinear model).
  - Equilibrium equations fed into Dynare; use Dynare’s perfect foresight/deterministic simulation algorithm via the ‘simul’ command.
  - Algorithm handles ZLB by writing the Taylor rule with the max operator.

### Results — Phillips Curve and nonlinearities
- Simulation design:
  - Stochastic simulations with monetary policy shocks ε_{r,t} only.
  - Standard deviation of monetary policy shocks (σ_r) sized to imply variations in model-consistent output gap between roughly minus 15 and plus 15 percent in the linearized solution.
  - Long-run simulations: 10,000 periods using identical shock sequence {ε_{r,t}}_{t=1}^{10,000}.
- Main empirical pattern:
  - Linearized solution → constant downward-sloped Phillips curve.
  - Nonlinear solution → “banana-shaped” Phillips curve (nonlinear), reflecting Kimball aggregation: firms reluctant to change prices much when relative demand is low but more willing when relative demand is high.
  - Nonlinear model produces episodes with more elevated price inflation than linearized model in booms.
- State-dependent transmission of markup shocks:
  - Linearized model: transmission of a given-sized markup shock is independent of output gap level; Phillips curve shifts outward in parallel.
  - Nonlinear model: transmission is state-dependent:
    - When inflation is subdued (recessions/low relative demand), a given-sized markup shock shifts out inflation very little.
    - When inflation is high (booms/high relative demand), the same-sized markup shock shifts the Phillips curve outward substantially more than in the linearized model.
  - Implication: supports inflation scares in booms and asymmetric/state-dependent inflation responses to cost-push shocks.

### Key numeric examples and takeaways (from nonlinear vs linearized)
- dy/dx slopes for nonlinear inflation responses by driving shock:
  - TFP + price cost-push shock: dy/dx = 0.26
  - Risk premium + price cost-push shock: dy/dx = 0.24
  - Fiscal + price cost-push shock: dy/dx = 0.25
  - Investment + price cost-push shock: dy/dx = 0.24
  - Monetary + price cost-push shock: dy/dx = 0.24
  - Price cost-push + price cost-push shock: dy/dx = 0.22
  - Wage cost-push + price cost-push shock: dy/dx = 0.26
  - All shocks + price cost-push shock: dy/dx = 0.21
- dy/dx slopes for nonlinear output gap responses by driving shock:
  - TFP + price cost-push shock: dy/dx = -0.38
  - Risk premium + price cost-push shock: dy/dx = -0.34
  - Fiscal + price cost-push shock: dy/dx = -0.33
  - Investment + price cost-push shock: dy/dx = -0.34
  - Monetary + price cost-push shock: dy/dx = -0.35
  - Price cost-push + price cost-push shock: dy/dx = -0.32
  - Wage cost-push + price cost-push shock: dy/dx = -0.38
  - All shocks + price cost-push shock: dy/dx = -0.3
- Example dispersions:
  - Initial inflation = 8 percent: one-year-ahead inflation impulse of the same-sized positive price cost-push shock can be roughly 0.5 or above 2 percent.
  - Initial inflation = 10 percent with all shocks combined: one-year-ahead inflation impact of a positive price-markup shock can vary from about 0 to over 3 percent.

### Implications for policy and risk
- Conditional volatility of inflation increases with the inflation level; inflation risk is higher when inflation is high, whereas linearized model implies constant inflation risk.
- Larger inflation transmission from price-markup shocks lowers real wages (acting as a tax) and worsens activity.
- The Taylor rule implies stronger endogenous policy rate increases when initial inflation is high, yet the nonlinear model may require even tighter policy to keep inflation pass-through to normal levels.

---

### 4.2 Propagation of Cost-Push Shocks — Experimental design and state-dependent findings
- Simulation and identification:
  - Simulate linearized and nonlinear solutions for T = 10,000 periods with only one shock active at a time; use estimated standard deviations to obtain y_nonlin_t and y_lin_t.
  - For each t = 1,...,10,000, add one σ_p positive price cost-push shock and compute impact on one-year-ahead average inflation and on the output gap.
  - Repeat for seven shock types separately and for all shocks combined.
- Filtering for Post-Covid assessment:
  - Use inversion filter for 1965Q1–2022Q1 to obtain one-sided filtered innovations ε_t|t and initialize nonlinear filter using Kalman-smoothed linearized estimates.
  - Impose ELB = 10 (annualized) basis points in nonlinear model when filtering.
- State-dependent inflation effects (Figure 3 summary):
  - Linearized: impact of given price cost-push shock on one-year-ahead inflation independent of initial inflation (horizontal).
  - Nonlinear: impact is upward-sloping in initial inflation for nearly all shock types — larger initial inflation → larger one-year-ahead inflation response.
  - Numeric slopes listed above (dy/dx = 0.21 to 0.26).
- State-dependent output gap effects (Figure 4 summary):
  - Adverse impact on output gap is larger when initial inflation is higher; slopes listed above (dy/dx = -0.30 to -0.38).
- Inflation surge vs descend (Figure 5):
  - When Δinflation_t > 0 (surge) the distribution of inflation responses to the same cost-push shock is substantially wider than when Δinflation_t < 0 (descend).
  - Output is also at greater risk when inflation is rising prior to the shock.

### IRFs for 2022Q1 and policy trade-offs
- IRFs in 2022Q1: price markup shock (Figure 6)
  - Add one σ_p price cost-push shock to filtered 2022Q1 state.
  - Linearized model: average of first four quarters equals 0.55.
  - Nonlinear model: one-year inflation and output-gap effects are more than twice as large over first year compared with linearized model.
  - Policy rate in nonlinear model increases about twofold relative to linearized model; inflation and output-gap effects amplified nonetheless.
- IRFs in 2022Q1: monetary policy shock (Figure 7)
  - Add one σ_r monetary policy shock to filtered 2022Q1 state.
  - Monetary shock has similar effects on output gap and output growth across models.
  - Initial impact on inflation about twice as high in nonlinear model.
  - Trade-off example: to reduce inflation by 0.1 percentage points the policy maker needs to accept a decline of more than 1% in the output gap over one year.
- Required tightening and output cost to fully stabilize one-year-ahead inflation (Figure 8)
  - Linearized model:
    - Required tightening invariant to initial inflation and is a little more than two percent, on average, to stabilize inflation during first year following a price-markup shock.
    - Output cost of such stabilization policy is a little above 6 percent (over one year).
  - Nonlinear model:
    - Required tightening and output cost increase with initial inflation level.
    - Price-markup shocks have much larger absolute effects as initial inflation rises, necessitating increasingly more policy tightening.

### Practical filtering and policy caveats
- ELB set at 10 (annualized) basis points.
- Linearized model average first-four-quarter inflation impulse = 0.55.
- Trade-off example: 0.1 percentage point inflation reduction requires >1% output gap decline over one year.
- Linearized full stabilization tightening ≈ a little more than two percent; output cost ≈ a little above 6 percent. Nonlinear tightening and output costs rise with initial inflation.

---

### 4.4 Conditional Forecast Distributions — Method and main findings
- Methodology:
  - Filtered states obtained for T = {2020Q4, 2021Q2, 2021Q4, 2022Q1}.
  - Construct conditional forecast distributions for T+1,...,T+h with 1,000 dynamic forecasts where shocks hit as surprises during horizon.
  - Projections based on filtered state in linearized model to hold initial state identical across solutions.
  - Nonlinear model imposes ELB on policy rate; linearized model allows negative policy rate (probability of binding ELB low for these cases).
- Main findings:
  - For 2020Q4 initial state:
    - Median forecasts in linearized and nonlinear models differ little; inflation near steady state.
    - Lower uncertainty bands differ: linearized model allows more persistent negative inflation; nonlinear model reduces deflation probability.
    - Both models underpredict subsequent uptick in inflation; imposing a constant interest rate path aligned with data would have narrowed prediction gap.
  - For later initial quarters as initial inflation rises (2021Q2, 2021Q4, 2022Q1):
    - Differences between nonlinear and linearized models increase with initial inflation.
    - Nonlinear model implies notably higher near-term inflation risk and higher median inflation forecast, especially in late 2021 and early 2022.
    - Higher near-term inflation risk translates into higher median policy rate projections and elevated risk of tighter policy in nonlinear model.
    - Nonlinear model implies somewhat more downside risk to output growth in conditional distributions.
- Persistence and horizon:
  - Inflation risk differences between nonlinear and linearized models are mostly near-term; after two years most differences dissipate.
  - Limiting features: ARMA(1,1) for price/wage cost-push shocks and modest intrinsic persistence in estimated wage and price setting curves.
  - Alternative assumptions could generate more persistent differences.

---

### Appendix A — The Smets and Wouters (2007) Model (structural summaries)
- Firms and price setting:
  - Final output Y_t produced from continuum of differentiated goods using Kimball (1995) aggregator: ∫_0^1 G_Y[Y_t(f)/Y_t] df = 1.
  - Aggregator specified with gross markup parameter φ_p ≥ 1 and curvature parameter ε_p; ε_p = 0 → Dixit-Stiglitz.
  - Intermediate goods firms produce Y_t(f) with Cobb-Douglas: Y_t(f) = ε^a_t K_t(f)^α (γ_t L_t(f))^{1−α} − γ_t Φ.
  - TFP: ln ε^a_t = (1 − ρ_a) ln ε^a + ρ_a ln ε^a_{t−1} + η^a_t, η^a_t ∼ N(0, σ_a).
  - Fixed cost Φ related to φ_p so steady-state profits zero.
  - Calvo-Yun staggered price setting: probability to reoptimize = 1 − ξ_p; non-optimizers index to lagged and steady-state inflation with ι_p.
  - Price-markup shock ε^p_t: ln ε^p_t = (1 − ρ_p) ln ε^p + ρ_p ln ε^p_{t−1} + η^p_t − μ_p η^p_{t−1}, η^p_t ∼ N(0, σ_p).
- Households and wage setting:
  - Labor aggregator Kimball form: ∫_0^1 G_L[L_t(h)/L_t] dh = 1, with ε_w and φ_w.
  - Representative utility with external habit κ and parameters β, σ_c, σ_l.
  - Budget constraint includes consumption, investment, nominal government bonds, contingent claims, labor income, capital rental income, utilization cost, profits share, lump-sum tax.
  - Capital accumulation: K^p_{t+1}(h) = (1 − δ) K^p_t(h) + ε^i_t [1 − S(I_t(h)/I_{t−1}(h))] I_t(h); S(x) = (φ/2) (x − γ)^2.
  - Investment-specific shock: ln ε^i_t = ρ_i ln ε^i_{t−1} + η^i_t, η^i_t ∼ N(0, σ_i).
  - Risk shock: ln ε^b_t = ρ_b ln ε^b_{t−1} + η^b_t, η^b_t ∼ N(0, σ_b).
  - Capital utilization adjustment function a(Z_t(h)) defined with parameters r_k, ̃z_1, z_1, ψ.
  - Wage setting Calvo with reoptimization probability 1 − ξ_w, indexation ι_w; wage-markup shock ε^w_t: ln ε^w_t = (1 − ρ_w) ln ε^w + ρ_w ln ε^w_{t−1} + η^w_t − μ_w η^w_{t−1}, η^w_t ∼ N(0, σ_w).
- Monetary policy:
  - Taylor rule with explicit ZLB: R_t = max[ 1 + ̄b, R_{t−1}^ρ_R ̄R^{(1−ρ_R)} ( (π_t / ̄π)^{r_π (1−ρ_R)} (y_t / y^{pot}_t)^{r_y (1−ρ_R)} ( (y_t/y^{pot}_t) / (y_{t−1}/y^{pot}_{t−1}) )^{r_{∆y} (1−ρ_R)} ) + ε_{r,t} ].
  - ̄b = 10 (annualized) basis points to match ELB period behavior.
  - y^{pot}_t is output in flexible price/wage economy absent inefficient policy and markup shocks.
- Market clearing and aggregations:
  - Government purchases G_t exogenous; ln g_t process specified.
  - Final goods allocation: Y_t = C_t + I_t + G_t + a(Z_t) ̄K_t.
- Parameterizations:
  - Two parameterizations: SW baseline and alternative with prior φ_p ∼ N(1.2, 0.05) and ε_p = 64.5.
  - Selected key parameter values preserved exactly (see Panel A/B/C values in source), including δ = 0.025, φ_w = 1.50, g_y = 0.18, φ = 5.48 (SW baseline), ε_p = 64.5 (alternative), φ_p = 1.34 (alternative), ξ_p = 0.667 (alternative), and detailed shock process parameters (e.g., ρ_p = 0.90, μ_p = 0.74, σ_p = 0.14 in SW baseline; alternative ρ_p = 0.83, μ_p = 0.69, σ_p = 0.13).
- Model equations and linearization:
  - Nonlinear model summarized with 28 equations and 28 endogenous variables; flexible price/wage allocations by setting ξ_p = ξ_w = 0.
  - Linearized model equations provided for consumption Euler, investment Euler, price-setting, wage-setting, monetary policy (no ZLB), and market clearing (log-linear forms preserved as in source).
- Observer equations mapping to data:
  - π^{obs}_t = 100 ln π_t
  - ∆w^{obs}_t = 100 ln (w_t / w_{t−1}) + γ
  - R^{obs}_t = 100 (R_t − 1)
  - ∆y^{obs}_t = 100 ln (y_t / y_{t−1}) + γ
  - ∆c^{obs}_t = 100 ln (c_t / c_{t−1}) + γ
  - ∆i^{obs}_t = 100 ln (i_t / i_{t−1}) + γ
  - l^{obs}_t = 100 ln (l_t / l_{t−1})
  - Data: same dataset as Smets and Wouters (2007) updated through 2014Q2.

*Italic: wpiea2023010-print-pdf - Section 3 presents the workhorse macroeconomic model with real rigidities in a dynamic stochastic; source content as provided.*

### Section 3 presents the workhorse macroeconomic model with real rigidities in a dynamic stochastic

### wpiea2023010-print-pdf - Section 3 presents the workhorse macroeconomic model with real rigidities in a dynamic stochastic

### Optimal Price Setting with Quasi-kinked Demand
- Setup:
  - Firms maximize π = p*y − mc*y subject to quasi-kinked demand y = a − p^b, with a, b > 0.
  - Demand elasticity: ε ≡ −(dy/dp) * (p/y) = b/a * p^(b−1), which is increasing in p.
  - Marginal revenue is a concave function of price because elasticity rises with p.
- Illustrative parameter example (broadly in line with Section 3 parameterization):
  - a = 3, b = 100, mc = 0.9.
- Key intuition and mechanism:
  - For a given percentage change in marginal costs, firms raise prices by more than they cut them.
  - When marginal costs are low → markups are high → firms are reluctant to cut prices (limited ability to crowd in extra demand when prices fall under quasi-kinked demand).
  - When marginal costs are high → markups are low → firms have large incentives to raise prices even if demand falls substantially.
  - The concavity of marginal revenue (due to rising elasticity) is key to this asymmetric pricing response.
  - The optimal price as a function of marginal cost is convex (a “banana-type” schedule), which is the key force generating a nonlinear Phillips curve in the general equilibrium model.

### The Workhorse Macroeconomic Model (model overview)
- Model choice and features:
  - Baseline: Smets and Wouters (2007) model (multi-shock version of Christiano, Eichenbaum and Evans (2005)) with endogenous capital accumulation.
  - Introduces real rigidities in price and wage setting using Kimball (1995) aggregation (following Dotsey and King (2005) and Levin, López‑Salido and Yun (2007)).
  - Monopolistic competition in goods and labor markets; nominal frictions via sticky prices and wages.
  - Households may index prices/wages to a composite of steady-state and lagged inflation.
  - Real rigidities: habit formation in consumption, investment adjustment costs, variable capital utilization, fixed costs in production.
- Shocks and stochastic processes:
  - Seven structural shocks drive dynamics.
  - Monetary policy shocks: AR(1).
  - Two inefficient cost-push shocks in wage and price setting: ARMA(1,1).
  - Four efficient shocks (TFP, risk premium, investment-specific technology, government spending): AR(1).
- Central pricing equations:
  - Linearized Phillips curve (Smets and Wouters style):
    - bπ_t − ι_p bπ_{t−1} = β(E_t bπ_{t+1} − ι_p bπ_t) + κ cmc_t + ê_{p,t}
    - κ = (1 − ξ_p β)(1 − ξ_p) / [ξ_p (1 + (φ_p − 1) ε_p)]
    - 1 − ξ_p is the probability of a firm reoptimizing price each period; ε_p is curvature of SW aggregator; φ_p is steady state gross price markup; ê_{p,t} is cost-push shock rescaled by 1/κ to enter with unit coefficient.
  - Nonlinear recursive pricing system (equations (2)–(5) as in the source) where the markup shock ε_{p,t} multiplies marginal costs mc_t, representing an exogenous shock to desired markup.

### Calibration, estimation, and solution method
- Observables used for estimation (quarterly US series):
  - Log-differences of real per capita GDP, consumption, investment.
  - Log-differences of compensation per hour and the GDP deflator.
  - Log-deviations of hours worked per capita from average.
  - Federal funds rate.
- Estimation approach:
  - Smets and Wouters (2007) use full-information Bayesian techniques; authors adapt SW parameters estimated on pre-Global Financial Crisis and Covid data, except pricing parameters φ_p, ξ_p, ε_p.
  - Re-estimation for sample 1965Q1−2007Q4 with alternative prior φ_p ∼ N(1.2, 0.05) to obtain lower gross markup than the 61 percent estimated by SW.
  - Calibration/priors and resulting posterior modes:
    - ξ_p = 0.667 (in line with micro evidence; close to SW posterior mode of .65).
    - Prior ε_p ∼ N(75, 25) → posterior mode ε_p = 64.5.
    - Posterior mode φ_p = 1.34.
    - κ in eq. (1) equals .008 (the authors’ value).
    - SW’s κ was .026; with SW priors and sample ending 2007:4, the authors obtain κ = .018.
  - Note: re-estimated model improves marginal likelihood by roughly 5 log points relative to original SW parameterization for the specified sample.
- Solution algorithm and implementation:
  - Use Fair and Taylor (1983) two-point boundary value (time-stacking) solution algorithm (certainty equivalence imposed on nonlinear model).
  - Equilibrium equations fed into Dynare; use Dynare’s perfect foresight/deterministic simulation algorithm via the ‘simul’ command.
  - Algorithm handles ZLB by writing the Taylor rule with the max operator.

### Results — Phillips Curve and nonlinearities
- Simulation design:
  - Stochastic simulations with monetary policy shocks ε_{r,t} only.
  - Standard deviation of monetary policy shocks (σ_r) sized to imply variations in model-consistent output gap between roughly minus 15 and plus 15 percent in the linearized solution.
  - Long-run simulations: 10,000 periods using identical shock sequence {ε_{r,t}}_{t=1}^{10,000}.
- Main empirical pattern:
  - Linearized solution → constant downward-sloped Phillips curve.
  - Nonlinear solution → “banana-shaped” Phillips curve (nonlinear), reflecting Kimball aggregation: firms reluctant to change prices much when relative demand is low but more willing when relative demand is high.
  - Consequence: nonlinear model produces episodes with more elevated price inflation than linearized model in booms.
- State-dependent transmission of markup shocks:
  - Method: For states with output gap near −15, −10, −5, ..., 15, compute contemporaneous effect of adding same-sized one σ_p positive markup shocks and average resulting inflation.
  - Linearized model: transmission of a given-sized markup shock is independent of output gap level; Phillips curve shifts outward in parallel.
  - Nonlinear model: transmission is state-dependent:
    - When inflation is subdued (recessions/low relative demand), a given-sized markup shock shifts out inflation very little.
    - When inflation is high (booms/high relative demand), the same-sized markup shock shifts the Phillips curve outward substantially more than in the linearized model.
  - Implication: supports the concept of inflation scares in booms and highlights asymmetric/state-dependent inflation responses to cost-push shocks.

*Italic: wpiea2023010-print-pdf - Section 3 presents the workhorse macroeconomic model with real rigidities in a dynamic stochastic; source content as provided.*

### 4.2    Propagation of Cost-Push Shocks

### 4.2    Propagation of Cost-Push Shocks

### Experimental design and method
- Simulate linearized and nonlinear solutions of the estimated Smets and Wouters model for T = 10,000 periods with only one shock active at a time, using the estimated standard deviations for each shock to obtain simulated paths y_nonlin_t and y_lin_t.
- For each simulated state t = 1,...,10,000, add one σ_p positive price cost-push shock and compute its impact on one-year-ahead average inflation and on the output gap.
- Repeat for seven shock types separately (stationary technology (TFP), risk premium, fiscal spending, investment-specific technology, monetary, price-markup, wage-markup) and for all shocks combined.
- Filtering for the Post-Covid assessment: use the inversion filter for 1965Q1–2022Q1 on US data to obtain one-sided filtered innovations ε_t|t and initialize the nonlinear filter using Kalman-smoothed linearized estimates. Impose ELB = 10 (annualized) basis points in the nonlinear model when filtering.

### State-dependent effects on inflation (Figure 3) — key findings
- In the linearized model, the impact of a given price cost-push shock on one-year-ahead inflation is independent of the initial inflation level (horizontal line).
- In the nonlinear model, the impact is an upward-sloping function of the initial inflation level for nearly all shock types: inflation responses are larger when initial inflation is higher.
- Slopes (dy/dx) of the nonlinear inflation response regressions by driving shock:
  - TFP shocks + price cost-push shock: dy/dx = 0.26
  - Risk premium shocks + price cost-push shock: dy/dx = 0.24
  - Fiscal shocks + price cost-push shock: dy/dx = 0.25
  - Investment shocks + price cost-push shock: dy/dx = 0.24
  - Monetary shocks + price cost-push shock: dy/dx = 0.24
  - Price cost-push shocks + price cost-push shock: dy/dx = 0.22
  - Wage cost-push shocks + price cost-push shock: dy/dx = 0.26
  - All shocks + price cost-push shock: dy/dx = 0.21
- Price- and wage-markup shocks generate markedly more variation in initial inflation levels. Examples from simulations:
  - With an initial inflation rate of 8 percent, the one-year-ahead inflation impulse of the same-sized positive price cost-push shock can be roughly 0.5 or above 2 percent.
  - With all shocks combined and an initial inflation level of 10 percent, the impact of a positive price-markup shock on one-year-ahead inflation can vary from about 0 to over 3 percent.
- Implication: conditional volatility of inflation increases with the inflation level; inflation risk is higher when inflation is high, whereas in the linearized model inflation risk is constant.

### State-dependent effects on the output gap (Figure 4) — key findings
- The adverse impact of a given markup shock on the output gap is larger when initial inflation is higher.
- Slopes (dy/dx) of the nonlinear output gap response regressions by driving shock:
  - TFP shocks + price cost-push shock: dy/dx = -0.38
  - Risk premium shocks + price cost-push shock: dy/dx = -0.34
  - Fiscal shocks + price cost-push shock: dy/dx = -0.33
  - Investment shocks + price cost-push shock: dy/dx = -0.34
  - Monetary shocks + price cost-push shock: dy/dx = -0.35
  - Price cost-push shocks + price cost-push shock: dy/dx = -0.32
  - Wage cost-push shocks + price cost-push shock: dy/dx = -0.38
  - All shocks + price cost-push shock: dy/dx = -0.3
- Mechanisms:
  - Larger inflation transmission from price-markup shocks lowers real wages (working as a tax) and worsens economic activity.
  - The Taylor rule implies stronger endogenous policy rate increases when initial inflation is high, as the central bank responds more to increased inflationary pressure despite a larger deterioration in activity.

### Inflation surge versus inflation descend episodes (Figure 5)
- Separate the responses by the sign of the one-period change in inflation prior to the shock (inflation surging when Δinflation_t > 0; inflation descending when Δinflation_t < 0).
- Findings:
  - Inflation risk (distribution of inflation responses to the same cost-push shock) is substantially higher when inflation is increasing before the shock hits.
  - Output is also at greater risk when inflation is on the rise, explaining the trumpet-like dispersion of output gap responses observed in Figure 4.

### Post-Covid period assessment and filtering approach
- Filter shocks up to 2022Q1 and analyze impulse responses starting from the filtered state in 2022Q1.
- Practical filtering notes:
  - ELB imposed at 10 (annualized) basis points in the nonlinear model.
  - When the ELB binds, monetary policy shocks are assumed zero except when a monetary policy shock is needed to match observed variables where the model-implied rate is predicted above the ELB.

### IRFs in 2022Q1: price markup shock (Figure 6)
- Add a one σ_p price cost-push shock to the filtered 2022Q1 state in both models.
- Results:
  - In the linearized model the impulse is invariant to the state; the average of the first four quarters equals 0.55 (black horizontal line referenced).
  - In the nonlinear model the one-year inflation and output-gap effects are more than twice as large over the first year compared with the linearized model.
  - The policy rate in the nonlinear model increases about twofold relative to the linearized model in response to the same shock, yet the inflation and output-gap effects are amplified.

### IRFs in 2022Q1: monetary policy shock (Figure 7)
- Add a one σ_r monetary policy shock to the filtered 2022Q1 state in both models.
- Results:
  - The monetary shock has similar effects on the output gap and output growth across models.
  - The initial impact on inflation is about twice as high in the nonlinear model relative to the linearized model.
  - Trade-off quantified: to reduce inflation by 0.1 percentage points the policy maker needs to accept a decline of more than 1% in the output gap over one year.
  - In the current high inflation risk situation, Figure 6 implies the nonlinear model requires even tighter monetary policy than shown to maintain pass-through of price cost-push shocks to inflation at normal levels.

### Required tightening and output cost to fully stabilize one-year-ahead inflation (Figure 8)
- Compute the additional policy tightening over the first year (beyond the endogenous rule response) required to fully stabilize one-year-ahead inflation impulses from price cost-push shocks, as a function of initial inflation level.
- Linearized model:
  - Required tightening is invariant to initial inflation level and is a little more than two percent, on average, to stabilize inflation during the first year following a price-markup shock.
  - Output cost of such stabilization policy is a little above 6 percent in the linearized model (over one year).
- Nonlinear model:
  - Required tightening and output cost increase with the initial inflation level.
  - Even though monetary policy and price-markup shocks become more potent on average as initial inflation rises, price-markup shocks have much larger absolute effects, necessitating increasingly more policy tightening to keep inflation in check.

### Key quantitative takeaways and examples
- dy/dx slopes for inflation responses in the nonlinear model range from 0.21 to 0.26 depending on the driving shock; the linearized model response is state-invariant.
- dy/dx slopes for output gap responses in the nonlinear model range from -0.32 to -0.38 depending on the driving shock; the linearized model response is state-invariant.
- Example dispersion:
  - Initial inflation = 8 percent: one-year-ahead inflation impulse from a same-sized positive price cost-push shock can be roughly 0.5 or above 2 percent.
  - Initial inflation = 10 percent with all shocks combined: one-year-ahead inflation impact of a positive price-markup shock can vary from about 0 to over 3 percent.
- Filtering and policy caveats:
  - ELB set at 10 (annualized) basis points.
  - Linearized model average first-four-quarter inflation impulse = 0.55.
  - Trade-off example: 0.1 percentage point inflation reduction requires >1% output gap decline over one year.
  - Linearized full stabilization tightening ≈ a little more than two percent; output cost ≈ a little above 6 percent. Nonlinear tightening and output costs rise with initial inflation.

*Source: 4.2 Propagation of Cost-Push Shocks (wpiea2023010-print-pdf).*

### 4.4    Conditional Forecast Distributions

### 4.4    Conditional Forecast Distributions

### Methodology and setup
- Filtered states obtained for T = {2020Q4, 2021Q2, 2021Q4, 2022Q1} using the filtering procedure in Section 4.3.
- Conditional forecast distributions for periods T+1,...,T+h constructed by computing 1,000 dynamic forecasts in which economic shocks hit as surprises during the forecast horizon.
- Results reported for both the nonlinear and linearized solutions; projections are based on the filtered state in the linearized model to hold the initial state identical across solutions.
- Nonlinear model imposes the ELB on the policy rate; the linearized model allows the policy rate to become negative.
- Note: the probability of a binding ELB in the linearized model is low for these cases, so this technical difference plays a minor role for distributional differences.

### Main findings on conditional distributions
- For forecasts conditional on 2020Q4:
  - Median forecasts in the linearized and nonlinear models differ very little, reflecting that inflation is close to steady state at this initial state.
  - Lower uncertainty bands differ notably: in the linearized model inflation can generally become persistently negative (except not for the 5th percentile given this initial state).
  - The nonlinear model reduces deflation probability because shocks have less impact on inflation when inflation is lower.
  - Both models underpredict the subsequent uptick in inflation; this underprediction is conditional on a quick and sharp normalization of the policy rate. Imposing a constant interest rate path (in line with the data) would have narrowed the gap between predicted and actual inflation outcomes.
- For later initial quarters (as initial inflation rises: 2021Q2, 2021Q4, 2022Q1):
  - Differences between nonlinear and linearized models increase with the initial inflation level.
  - The nonlinear model implies notably higher near-term inflation risk and a notably higher median inflation forecast, especially in the second-half of 2021 and the beginning of 2022.
  - Higher near-term inflation risk in the nonlinear model translates into higher median policy rate projections and an elevated risk of a notably tighter policy rate stance in the nonlinear model.
  - As a result, the nonlinear model implies somewhat more downside risk to output growth in the conditional distributions (consistent with Adrian et al. (2019)).

### Persistence and horizon of nonlinear effects
- Inflation risk differences between the nonlinear and linearized models are mostly a near-term phenomenon; after two years most differences in conditional distributions have dissipated.
- Two key model features limiting persistence:
  - The ARMA(1,1) feature of the price- and wage cost-push shocks.
  - The modest role for intrinsic persistence in the estimated wage and price setting curves in the SW model.
- These features imply that price-wage cost-push shocks have transient effects on the economy.
- Alternative assumptions about these parameters could produce more persistent differences in conditional forecast distributions.

*Source: 4.4 Conditional Forecast Distributions (wpiea2023010-print-pdf)*

### Appendix A    The Smets and Wouters (2007) Model

### Appendix A    The Smets and Wouters (2007) Model

### Firms and Price Setting
- Final output Y_t is produced from a continuum of differentiated intermediate goods Y_t(f) using the Kimball (1995) aggregator:
  - ∫_0^1 G_Y[Y_t(f)/Y_t] df = 1. (A.1)
  - G_Y[...] is specified as in equation (A.2) with gross markup parameter φ_p ≥ 1 and curvature parameter ε_p.
  - When ε_p = 0 the aggregator reduces to Dixit-Stiglitz; positive ε_p introduces additional strategic complementarity in price setting.
- Final goods firms are perfectly competitive and solve:
  - max_{Y_t, Y_t(f)} P_t Y_t − ∫_0^1 P_t(f) Y_t(f) df subject to (A.1). (A.3)
- Intermediate goods firms:
  - Produce Y_t(f) with Cobb-Douglas technology:
    - Y_t(f) = ε^a_t K_t(f)^α (γ_t L_t(f))^{1−α} − γ_t Φ. (A.4)
  - TFP follows: ln ε^a_t = (1 − ρ_a) ln ε^a + ρ_a ln ε^a_{t−1} + η^a_t, η^a_t ∼ N(0, σ_a). (A.5)
  - Fixed cost Φ is related to φ_p so steady-state profits are zero.
- Price setting uses Calvo-Yun staggered contracts:
  - Probability to reoptimize price each period is 1 − ξ_p; non-optimizers index to lagged and steady-state inflation with indexation parameter ι_p (0 ≤ ι_p ≤ 1).
  - Optimization problem for reoptimizing firms is given in (A.6).
- Price-markup shock ε^p_t enters first-order conditions as an exogenous ARMA(1,1):
  - ln ε^p_t = (1 − ρ_p) ln ε^p + ρ_p ln ε^p_{t−1} + η^p_t − μ_p η^p_{t−1}, η^p_t ∼ N(0, σ_p). (A.7)
  - In the nonlinear model a scaling factor is used so log-linearization matches SW.

### Households and Wage Setting
- Continuum of monopolistically competitive households supply differentiated labor L_t(h); labor aggregator has Kimball form:
  - ∫_0^1 G_L[L_t(h)/L_t] dh = 1. (A.8)
  - Aggregator characterized by ε_w (convexity) and φ_w (gross wage markup).
- Representative household period utility:
  - E_t ∑_{j=0}^∞ β^j [ (1/(1−σ_c)) (C_{t+j}(h) − κ C_{t+j−1})^{1−σ_c} exp((σ_c − 1)/(1 + σ_l) L_{t+j}(h)^{1+σ_l}) ]. (A.9)
  - External habit parameter 0 ≤ κ ≤ 1; discount factor 0 < β < 1.
- Household budget constraint (A.10) includes consumption, investment, nominal government bonds, contingent claims, labor income W_t(h) L_t(h), capital rental income R^k_t Z_t(h) K^p_t(h), utilization cost a(Z_t(h)) K^p_t(h), profits share Γ_t(h), and lump-sum tax T_t(h).
- Capital accumulation:
  - K^p_{t+1}(h) = (1 − δ) K^p_t(h) + ε^i_t [1 − S(I_t(h)/I_{t−1}(h))] I_t(h). (A.11)
  - Adjustment S(x_t) = (φ/2) (x_t − γ)^2 with S(γ) = 0, S′(γ) = 0, S′′(γ) = φ. (A.12)
  - Investment-specific shock: ln ε^i_t = ρ_i ln ε^i_{t−1} + η^i_t, η^i_t ∼ N(0, σ_i). (A.13)
- Risk shock on bond returns:
  - ln ε^b_t = ρ_b ln ε^b_{t−1} + η^b_t, η^b_t ∼ N(0, σ_b). (A.14)
- Capital utilization adjustment function:
  - a(Z_t(h)) = r_k ̃z_1 [exp(̃z_1 (Z_t(h) − 1)) − 1], where a(1) = 0, a′(1) = r_k, a′′(1) ≡ r_k ̃z_1.
  - Set a′′(1) = z_1 = ψ/(1 − ψ) > 0 with ψ ∈ [0,1), implying ̃z_1 ≡ z_1 / r_k.
- Wage setting follows Calvo staggered contracts with probability 1 − ξ_w to reoptimize and indexation with parameter ι_w:
  - Reoptimization problem in (A.16); wage-markup shock ε^w_t follows ARMA(1,1):
    - ln ε^w_t = (1 − ρ_w) ln ε^w + ρ_w ln ε^w_{t−1} + η^w_t − μ_w η^w_{t−1}, η^w_t ∼ N(0, σ_w). (A.17)
  - Scaling factor applied for log-linear consistency with SW.

### Monetary Policy
- Monetary authority follows a Taylor rule with explicit ZLB via a nonlinear policy rule:
  - R_t = max[ 1 + ̄b, R_{t−1}^ρ_R ̄R^{(1−ρ_R)} ( (π_t / ̄π)^{r_π (1−ρ_R)} (y_t / y^{pot}_t)^{r_y (1−ρ_R)} ( (y_t/y^{pot}_t) / (y_{t−1}/y^{pot}_{t−1}) )^{r_{∆y} (1−ρ_R)} ) + ε_{r,t} ]. (A.18)
  - ̄b > 0 reflects the level of the effective lower bound (ELB); set ̄b = 10 (annualized) basis points to match observed federal funds oscillation around 10 annualized basis points during ELB period.
  - y^{pot}_t is output in the flexible price and wage economy absent inefficient monetary policy and markup shocks.

### Market Clearing Conditions
- Government purchases G_t exogenous; g_t = G_t / (γ_t Y_t) follows:
  - ln g_t = (1 − ρ_g) ln g + ρ_g [ ln g_{t−1} − ρ_{ga} ln ε^a_{t−1} ] + ε^g_t, ε^g_t ∼ N(0, σ_g). (A.19)
  - Government purchases do not affect marginal utility of consumption and government budget balanced via lump-sum taxes.
- Final goods allocation:
  - Y_t = C_t + I_t + G_t + a(Z_t) ̄K_t. (A.20)
- Aggregate production constraint depends on technology, capital, labor, fixed costs, and price/wage dispersion.

### Model Parameterization
- Two parameterizations considered:
  - Baseline: adopt SW posterior mode estimates given in Table A.1.
  - Alternative: re-estimate with prior φ_p ∼ N(1.2, 0.05) (vs SW prior φ_p ∼ N(1.25, 0.125)) and set ε_p = 64.5 (instead of ε_p = 10 in SW). Calibrated parameters kept as in SW.
- Key parameter values (selected from Table A.1, SW baseline):
  - Panel A: Calibrated
    - δ = 0.025
    - φ_w = 1.50
    - g_y = 0.18
  - Panel B: Estimated (SW baseline)
    - φ = 5.48
    - σ_c = 1.39
    - κ = 0.71
    - l = 0.25
    - σ_l = 1.92
    - ξ_w = 0.73
    - ξ_p = 0.65
    - ι_w = 0.59
    - ι_p = 0.22
    - α = 0.19
    - γ = 1.0043
    - π = 0.0081
    - β = 0.9984
    - ρ_R = 0.81
    - r_{∆y} = 0.22
    - r_y = 0.08
    - r_π = 2.03
    - ψ = 0.54
    - φ_p = 1.61
  - Panel C: Shock Processes (persistence, MA(1), and standard deviations)
    - Neutral technology: ρ_a = 0.95 − σ_a = 0.45
    - Risk premium: ρ_b = 0.18 − σ_b = 0.24
    - Gov’t spending: ρ_g = 0.97, ρ_{ga} = 0.52, σ_g = 0.52
    - Investment-specific: ρ_i = 0.71, σ_i = 0.45
    - Price markup: ρ_p = 0.90, μ_p = 0.74, σ_p = 0.14
    - Wage markup: ρ_w = 0.97, μ_w = 0.88, σ_w = 0.24
    - Monetary policy: ρ_r = 0.2 − σ_r = 0.24
- Key parameter values (alternative in Table A.2 with prior φ_p ∼ N(1.2, 0.05) and ε_p = 64.5):
  - Panel A: Calibrated
    - δ = 0.025
    - φ_w = 1.50
    - g_y = 0.18
    - ξ_p = 0.67
  - Panel B: Estimated (alternative)
    - φ = 5.58
    - σ_c = 1.41
    - κ = 0.68
    - ξ_w = 0.80
    - σ_l = 2.20
    - ε_p = 64.5
    - φ_p = 1.34
    - γ = 1.0044
    - π = 0.0087
    - β = 0.9987
    - ρ_R = 0.82
    - r_{∆y} = 0.25
    - r_y = 0.097
    - r_π = 1.93
    - ψ = 0.49
  - Panel C: Shock Processes (alternative)
    - ρ_a = 0.95 − σ_a = 0.48
    - ρ_b = 0.22 − σ_b = 0.23
    - ρ_g = 0.97, ρ_{ga} = 0.53, σ_g = 0.47
    - ρ_i = 0.70, σ_i = 0.40
    - ρ_p = 0.83, μ_p = 0.69, σ_p = 0.13
    - ρ_w = 0.97, μ_w = 0.93, σ_w = 0.28
    - ρ_r = 0.11 − σ_r = 0.23
- Note: price- and wage-markup shock processes are adapted and rescaled so log-linearization matches original SW model.

### Summary of Nonlinear Model Equations
- Variables detrended by deterministic trend γ; nominal variables replaced by real counterparts. Definitions include:
  - k_t = K_t / γ_t; w_t = W_t / P_t γ_t; r^k_t = R^k_t / P_t; mc_t = MC_t / P_t; and others.
- Market clearing:
  - c_t + i_t + g_t y + a(U_t) k^h_{t−1} / γ = y_t. (A.1)
  - y_t ε_p + s^p_t / (1 + ε_p) = ... (full expression in (A.2)–(A.3))
- Capital and investment dynamics:
  - k^s_t = (1/γ) U_t k^h_{t−1}. (A.3)
  - k^h_t = (1 − δ)/γ k^h_{t−1} + ε^i_t [1 − S(i_t / γ i_{t−1})] i_t. (A.4)
- Price and wage dispersion and auxiliary variables s^p_t, s^w_t, δ^p_t, δ^w_t, s^{pl}_t, s^{wl}_t follow recursive equations (A.5)–(A.12).
- Firms:
  - k^s_t L_t = (α/(1 − α)) w_t / r^k_t. (A.13)
  - mc_t = [ (α/(1 − α)) w_t / r^k_t ]^{−α} w_t^{1−α} ε^a_t. (A.14)
- Households:
  - ξ_t ≡ Ξ_t / γ^{σ_c} c_t = ε^d_t (c_t − κ/γ c_{t−1})^{−σ_c} exp((σ_c − 1)/(1 + σ_l) L^h_t^{1+σ_l}). (A.15)
  - First-order conditions for Q_t, R_t, and capital price Q_t in (A.16)–(A.19).
- Wage setting summarized in (A.20)–(A.23); price setting in (A.24)–(A.27).
- Monetary policy in nonlinear form restated in (A.28).
- System size:
  - 28 equations and 28 endogenous variables listed (c_t, y_t, L_t, i_t, k^s_t, k^h_t, U_t, p^*_t, w^*_t, π_t, s^p_t, s^w_t, δ^p_t, δ^w_t, s^{pl}_t, s^{wl}_t, r^k_t, w_t, mc_t, ξ_t, R_t, Q_t, γ^p_{1,t}, γ^p_{2,t}, γ^p_{3,t}, γ^w_{1,t}, γ^w_{2,t}, γ^w_{3,t}).
  - Flexible price and wage allocations obtained by setting ξ_p = ξ_w = 0, defining y^{pot}_t. Only four shocks affect flex allocations: ε^a_t, ε^b_t, ε^i_t, and g_t.

### Summary of Linearized Model Equations
- Log-linear notation: ˆx_t = dx_t / x; shocks denoted ˆε_t = ln ε_t.
- Consumption Euler (log-linear):
  - b c_t = 1/(1 + κ/γ) E_t b c_{t+1} + (κ/γ)/(1 + κ/γ) b c_{t−1} − (1 − κ/γ)/(σ_c (1 + κ/γ)) (ˆR_t − E_t b π_{t+1} + ˆε^b_t) − ((σ_c − 1)(w^h_* L / c_*) /(σ_c (1 + κ/γ))) (E_t b L_{t+1} − b L_t). (A.29)
- Investment Euler:
  - b i_t = 1/(1 + βγ) [ b i_{t−1} + βγ E_t b i_{t+1} + (1/γ^2) φ_b Q^k_t ] + b ε^q_t. (A.30)
- Price of capital:
  - b Q^k_t = −(b ̃R_t − E_t b π_{t+1} + b ε^b_t) + q_1 E_t r^k_{t+1} + (1 − q_1) E_t Q^k_{t+1}. (A.31)
- Capital utilization:
  - b u_t = (1 − ψ)/ψ b r^k_t. (A.32)
  - b k_t = b u_t + b ̄k_{t−1}. (A.33)
- Capital accumulation (linearized):
  - b ̄k_t = κ_1 b ̄k_{t−1} + (1 − κ_1) b i_t + κ_2 b ε^q_t. (A.34)
- Optimal capital–labor input:
  - b k_t = b w_t − b r^k_t + b L_t. (A.35)
- Production function (log-linear):
  - b y_t = φ_p (α b k_t + (1 − α) b L_t + b ε^a_t). (A.36)
- Aggregate demand equals supply:
  - b y_t = (c^*/y^*) b c_t + (i^*/y^*) b i_t + g_t + (r^k^* k^*/y^*) b u_t. (A.37)
- Log-linearized price-setting (with indexation ι_p):
  - b π_t − ι_p b π_{t−1} = π_1 (E_t b π_{t+1} − ι_p b π_t) − π_2 b μ^p_t + ˆε^p_t, with π_1 = β and π_2 = (1 − ξ_p β)(1 − ξ_p)/[ξ_p (1 + (φ_p − 1) ε_p)]. (A.38)
  - Price markup b μ^p_t = − c m c_t and c m c_t = (1 − α) b w^{real}_t + α b r^k_t − b ε^a_t. (A.39)
- Wage-setting (log-linear) with indexation ι_w:
  - (1 + β γ) b w^{real}_t − b w^{real}_{t−1} − β γ E_t b w^{real}_{t+1} = (1 − ξ_w β γ)(1 − ξ_w)/[ξ_w (1 + (φ_w − 1) ε_w)] [ 1/(1 − κ/γ) b c_t − (κ/γ)/(1 − κ/γ) b c_{t−1} + σ_l b L_t − b w_t ] − (1 + β γ ι_w) b π_t + ι_w b π_{t−1} + β γ E_t b π_{t+1} + ˆε^w_t. (A.40)
- Linearized monetary policy (no ZLB):
  - b R_t = ρ_R b R_{t−1} + (1 − ρ_R) (r_π b π_t + r_y ˆy^{gap}_t) + r_{∆y} ∆ˆy^{gap}_t + ˆε^r_t. (A.41)
  - ˆy^{gap}_t = b y_t − b y^{pot}_t. The flex price-wage solution for b y^{pot}_t obtained by setting ξ_p = ξ_w = 0 and removing ˆε^w_t and ˆε^p_t.
- In linearization the ZLB is not taken into account; policy rate equals shadow rate from (A.41) at all times.

### Observer Equations and Data
- Nonlinear observer equations mapping model variables to data (percent and annualized where noted):
  - π^{obs}_t = 100 ln π_t
  - ∆w^{obs}_t = 100 ln (w_t / w_{t−1}) + γ
  - R^{obs}_t = 100 (R_t − 1)
  - ∆y^{obs}_t = 100 ln (y_t / y_{t−1}) + γ
  - ∆c^{obs}_t = 100 ln (c_t / c_{t−1}) + γ
  - ∆i^{obs}_t = 100 ln (i_t / i_{t−1}) + γ
  - l^{obs}_t = 100 ln (l_t / l_{t−1})
- Data: same dataset as Smets and Wouters (2007) updated through 2014Q2.

*Source: Appendix A, The Smets and Wouters (2007) Model — Understanding Post-COVID Inflation Dynamics Working Paper No. WP/23/10*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023010-print-pdf.pdf_