## wpiea2024251-print-pdf

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---

### Data and Methodology
- Data sources and sample:
  - Data source: World Economic Outlook (WEO) Spring and Fall releases.
  - Sample period: 1998-2022.
  - Country coverage: 39 Advanced Economies (AEs) and 73 Emerging Markets (EMs), total of 112 countries.
  - Rationale: harmonized release dates across a large number of countries for real-time analysis.
- Variables of interest:
  - Output gap: deviation of current output from the economy’s productive potential; indicator of the economic cycle.
  - Cyclically-adjusted primary balance (CAPB): CAPB as a percentage of potential GDP; interpreted as discretionary fiscal policy (excludes automatic stabilizers and interest expenses on public debt).
- Vintage convention and interpretation:
  - Vintage notation: x_{i,t|τ+v} where v ∈ {S, F}.
  - Spring (S) vintage interpreted as real-time forecasts available to policymakers; Fall (F) vintage as realized values (ex post).
  - Real-time use: if τ = t and v = S, then x_{i,t|t+S} is the real-time estimate used to set fiscal stance.
- Forecast error metrics (definitions preserved):
  - Real-time forecast error (RTFE): RTFE_{i,t} = x_{i,t|t+F} − x_{i,t|t+S}
  - Final forecast error (FFE): FFE_{i,t} = x_{i,t|2022+F} − x_{i,t|t+S}
- Conceptual context:
  - Output gap estimates subject to considerable uncertainty (measurement errors, model quality, parameter changes, ex post data revisions).
  - Information lag for policymakers roughly one and a half year for OECD countries; potential for significant forecast errors and sub-optimal fiscal policy decisions.
  - Real-time data may be generated from a misspecified probabilistic model; output gap estimation depends on chosen model.

### Theoretical and empirical framing; study aim
- Literature draws on work on real-time assessment errors, real-time fiscal reaction functions, data-vintage dependence of fiscal stance, and pro-cyclicality findings in EMs.
- Policy design under uncertainty: references to Brainard (1967) and robustness approaches (Hansen and Sargent (1999); Woodford (2010)).
- Aim: assess implications of output gap uncertainty on fiscal adjustment; bridge real-time data literature in EMs and uncertainty literature.
- Approach:
  - Introduce a New Keynesian DSGE with an uncertainty-averse policymaker facing output gap estimation uncertainty and fiscal implementation uncertainty.
  - Calibrate to an average EM and solve for the optimal fiscal reaction function.
- Key reported findings (text):
  - Empirical: fiscal policy is counter-cyclical in AEs, less strongly so in EMs.
  - Model: uncertainty-averse EM policymaker facing both uncertainties produces a less counter-cyclical reaction function than benchmark with no uncertainty.
  - Welfare and debt trade-offs: placing more weight on output gap stabilization can offset welfare losses from weaker fiscal response; increases public debt short run but yields stronger output recovery and faster debt stabilization long run.

### Summary Statistics — Output Gap (empirical)
- Real-time Spring estimate (GAP i,t|t+S):
  - Mean GAP i,t|t+S: AEs = -1.369; EMs = -1.636
  - SD GAP i,t|t+S: AEs = 2.385; EMs = 3.240
  - p-value H0: GAP i,t|t+S = 0: AEs = 0.000; EMs = 0.000
- Real-time Fall estimate (GAP i,t|t+F):
  - Mean GAP i,t|t+F: AEs = -1.151; EMs = -1.520
  - SD GAP i,t|t+F: AEs = 2.259; EMs = 3.459
  - p-value H0: GAP i,t|t+F = 0: AEs = 0.000; EMs = 0.000
- Final estimate (GAP i,t|2022+F):
  - Mean GAP i,t|2022+F: AEs = -0.567; EMs = -0.470
  - SD GAP i,t|2022+F: AEs = 2.742; EMs = 4.977
  - p-value H0: GAP i,t|2022+F = 0: AEs = 0.000; EMs = 0.035
- Real-time forecast errors (RTFE):
  - Mean RTFE i,t: AEs = 0.219; EMs = 0.117
  - SD RTFE i,t: AEs = 1.157; EMs = 1.558
  - p-value H0: RTFE i,t = 0: AEs = 0.000; EMs = 0.094
  - Variance ratio Var EM / Var AE (RTFE): 1.815
  - p-value H0: Var EM (RTFE) / Var AE (RTFE) = 1: 0.000
- Final forecast errors (FFE):
  - Mean FFE i,t: AEs = 0.803; EMs = 1.166
  - SD FFE i,t: AEs = 2.360; EMs = 3.747
  - p-value H0: FFE i,t = 0: AEs = 0.000; EMs = 0.000
  - Variance ratio Var EM / Var AE (FFE): 2.521
  - p-value H0: Var EM (FFE) / Var AE (FFE) = 1: 0.000
- Key empirical patterns:
  - WEO real-time estimates tend to underestimate economic upswing in real-time for AEs (significantly negative real-time estimates; positive RTFE).
  - For EMs, the null that output gap averages 0 cannot be rejected in the final estimate.
  - Revisions using latest vintage are significantly different from 0 and positive, particularly for EMs; average magnitude of FFE almost twice that of AEs.
  - Revisions more volatile for EMs; EM forecast error distributions exhibit fatter tails and more extreme values.
- Dynamics and stability of revisions:
  - Discrepancy between real-time and later vintages higher at beginning of sample.
  - Final estimates and estimates 5 years after are very close for most countries; 5 years often enough for revisions to stabilize.

### Forecast-error predictability (tests and implications)
- Predictability regressions:
  - RTFE i,t = α + x′ i,t|t+S γ + ν i,t
  - FFE i,t = α + x′ i,t|t+S γ + η i,t
  - x includes GAP i,t|t+S, surplus-to-GDP, deficit-to-GDP (from CAPB-to-GDP), lagged forecast error, log(GDP deflator i,t|t+S).
- Table 2 regression highlights (significance: ∗ p<0.1; ∗∗ p<0.05; ∗∗∗ p<0.01):
  - RTFE i,t (AEs Col 1): Output Gap i,t|t+S coefficient = -0.320 ∗∗∗ (0.045); Observations = 165; R2 = 0.403
  - RTFE i,t (EMs Col 2): Output Gap i,t|t+S coefficient = -0.019 (0.059); Observations = 186; R2 = 0.010
  - FFE i,t (AEs Col 3): Output Gap i,t|t+S coefficient = -0.501 ∗∗∗ (0.046); FFE i,t−1 = 0.188 ∗∗ (0.074); Observations = 165; R2 = 0.572
  - FFE i,t (EMs Col 4): Output Gap i,t|t+S coefficient = -0.370 ∗∗∗ (0.086); FFE i,t−1 = 0.295 ∗∗∗ (0.096); Observations = 186; R2 = 0.149
- Interpretations:
  - RTFE for EMs not predictable given available information; other regressions show systematic, predictable errors.
  - Negative output gap coefficient: lower real-time output gap estimate → larger final forecast error (actual output gap likely larger than forecast when RTFE positive); risk of Type I error if policymakers react strongly to a negative forecast.
  - Fiscal variables affect final forecast errors but not real-time errors:
    - Surplus (AEs) positively predicts FFE (AEs).
    - Deficit (EMs) negatively predicts FFE (EMs).
  - Final forecast errors are serially autocorrelated (FFE i,t−1 significant), stronger for EMs.

### CAPB — empirical summary
- RTFE (percent of GDP):
  - Mean RTFE: AEs = -0.304; EMs = -0.283
  - SD RTFE: AEs = 1.482; EMs = 1.367
  - p-value H0: RTFE = 0: AEs = 0.000; EMs = 0.000
  - Variance ratio Var EM / Var AE (RTFE) = 0.851
  - p-value Var EM (RTFE) / Var AE (RTFE) = 1: 0.123
- FFE (percent of GDP):
  - Mean FFE: AEs = 0.652; EMs = 0.006
  - SD FFE: AEs = 1.508; EMs = 1.637
  - p-value H0: FFE = 0: AEs = 0.000; EMs = 0.943
  - Variance ratio Var EM / Var AE (FFE) = 1.179
  - p-value Var EM (FFE) / Var AE (FFE) = 1: 0.118
- Key CAPB findings:
  - CAPB forecast errors for AEs significantly different from 0 in both vintages.
  - EMs CAPB forecast errors biased in real-time but not in long run.
  - Variability of CAPB forecast errors similar across AEs and EMs.
  - Distributional evidence: RTFE and FFE distributions for CAPB similar across income groups.
  - Interpretation: CAPB is a policy variable subject to implementation discrepancies and frictions; similar uncertainty across groups plausible.

### Real-time fiscal reaction functions — methodology
- Estimated regression (real-time):
  - CAPB i,t|t+S = β0 GAP i,t|t+S + β1 IEM × GAP i,t|t+S + γ′ x i,t|t+S + ε i,t
  - β0: sensitivity for average AE; β0 + β1 for average EM.
  - Counter-cyclicality: β0 > 0 and β0 + β1 > 0; β1 < 0 indicates EMs less counter-cyclical.
- Covariates x include:
  - Lagged change in debt-to-GDP (∆DEBT t−1).
  - Election year dummy (lead; Election Year t+1).
  - Institutional quality: negative of Corruption Perception Index (higher = lower institutional quality).
  - Interaction: GAP × ∆DEBT t−1.
  - Asymmetry tests using indicators I[GAP t < 0].
  - All regressors lagged as appropriate.
- Instrumental variable:
  - Regional weighted average GAP as instrument:
    - ̄GAP IV g,t|t+S = Σ i∈g GDP i,t|t+S GAP i,t|t+S / Σ i∈g GDP i,t|t+S
  - Regions g ∈ {AFR, APD, EUR, MCD, WHD}

### Real-time fiscal reaction functions — fixed effects results (Table 4)
- Column (1) — symmetric specification (selected coefficients, standard errors in parentheses):
  - CAPB t−1 = 0.236 ∗∗∗ (0.067)
  - GAP t = 0.708 ∗∗∗ (0.073)
  - GAP t−1 = 0.334 ∗∗∗ (0.080)
  - I EM = 1.826 (1.707)
  - I EM × GAP t = -0.283 ∗∗ (0.119)
  - Corruption Perception = -0.186 ∗∗∗ (0.053)
  - Observations = 402; R2 = 0.382; Adjusted R2 = 0.255
- Column (2) — lagged GAP response:
  - CAPB t−1 = 0.205 ∗∗∗ (0.076)
  - GAP t = 0.680 ∗∗∗ (0.075)
  - GAP t−1 = 0.499 ∗∗∗ (0.102)
  - I EM × GAP t = -0.191 (0.122)
  - I EM × GAP t−1 = -0.314 ∗∗ (0.126)
  - Corruption Perception = -0.172 ∗∗∗ (0.053)
  - Observations = 402; R2 = 0.384; Adjusted R2 = 0.253
- Column (3) — robustness / asymmetry:
  - CAPB t−1 = 0.115 (0.089)
  - GAP t = -0.337 (1.308)
  - GAP t−1 = 0.637 ∗∗∗ (0.117)
  - Corruption Perception = -0.150 ∗∗∗ (0.056)
  - I EM × GAP t = -0.202 (0.136)
  - Observations = 402; R2 = 0.355; Adjusted R2 = 0.214
- Empirical summary and interpretation:
  - AEs: significant counter-cyclical response in real-time; CAPB-to-GDP increases by 0.708 percentage points for a 1 percentage point increase in GAP (Column 1).
  - EMs: lower magnitude counter-cyclicality; on-impact response = 0.708 + (−0.283) = 0.425 percentage points per 1 percentage point GAP increase (Column 1).
  - Dynamic responses (Column 2): contemporaneous responses similar across groups (~0.68 on impact); EMs show lower persistence (one-year-after response EMs = 0.185; AEs = 0.499 implied).
  - No systematic effect of ∆DEBT t−1 or Election Year t+1 in these specifications.
  - Institutional quality: higher corruption perception predicts lower CAPB-to-GDP consistently.
  - No robust evidence of asymmetry in fiscal reaction to positive vs negative GAP in reported specifications.
  - Overall: AEs more responsive to output-gap shocks on impact and dynamically; lower sensitivity in EMs may reflect higher uncertainty in output-gap measurement and policy reluctance.

### Robustness checks (Appendix summary)
- Random effects and Hausman tests:
  - Random effects Table 6 shows alternative coefficients; Hausman tests (Table 7) reject random effects in symmetric specifications but not in asymmetric specification including potential asymmetries.
  - Authors prefer fixed effects (Table 4) to capture country-specific differences.
- Alternative specifications (Equation (51), ∆CAPB):
  - Evidence that fiscal policy appears more history-dependent for EMs; lagged CAPB coefficient about twice for EMs vs AEs in some specifications.
- Results robust across multiple specifications (Tables 6–9), with variations in persistence and some interaction effects with debt.

### Model — Households, Firms, Government, and Equilibrium
- Households:
  - Fraction φ_C financially constrained (consume current income only); intratemporal FOC: (C^C_t)^{1/σ} (N^C_t)^φ = (1−τ) W_t / P_t.
  - Unconstrained households maximize lifetime utility; Euler equation: β E_t [ (C^U_{t+1}/C^U_t)^{−1/σ} (1+i_t)/Π_{t+1} ] = 1; labor supply (C^U_t)^{1/σ} (N^U_t)^φ = (1−τ) W_t / P_t.
  - Only unconstrained households enter intertemporal decisions and price-setting discounting via Λ_{t,t+s}.
- Firms:
  - Final good: perfect competition, CES aggregation with elasticity ε; demand Y^d_t(i) = [P_t(i)/P_t]^{−ε} Y_t.
  - Intermediate goods: Y_t(i) = A_t N_t(i); Calvo pricing with probability 1−θ to reset price; reset price solves discounted profit maximization with stochastic discount factor Λ_{t,t+s}.
- Government and monetary policy:
  - Government budget constraint: P_{t+1} B_t = (1+i_t)(P_t B_{t−1} + P_t G_t − τ P_t Y_t).
  - Monetary Taylor rule: (1+i_t) = β^{−1} (Π_t)^{φ_π} (Y_t/Y^p_t)^{φ_y}; Π_t = P_t / P_{t−1}.
- Equilibrium and log-linearization:
  - Goods market: φ_C C^C_t + (1−φ_C) C^U_t + G_t = Y_t; equivalent (1−φ_C) C^U_t + G_t = (1−(1−τ) φ_C) Y_t.
  - Labor market and bond market clearing specified.
  - Log-linearized relations:
    - Euler: c^U_t = E_t c^U_{t+1} − σ (i_t − E_t π_{t+1})
    - Dynamic IS: ˆy_t = E_t ˆy_{t+1} + μ_g (ˆg_t − E_t ˆg_{t+1}) − χ (i_t − E_t π_{t+1} − r^n_t), with μ_g = γ / (1−(1−τ) φ_C) and χ = σ (1−φ_C)(1−γ) / (1−(1−τ) φ_C)
    - Phillips curve: π_t = κ ˆy_t + β E_t π_{t+1}
    - Taylor rule log-linear: i_t = r^n_t + φ_y ˆy_t + φ_π π_t
    - Debt dynamics: β ˆb_t = ˆb_{t−1} + δ γ ˆg_t − δ τ ˆy_t + β (i_t − E_t π_{t+1} − r^n_t), δ = Y / B

### Benchmark optimal fiscal policy (control problem)
- Shocks and objective:
  - ˆy_t = E_t ˆy_{t+1} + μ_g (ˆg_t − E_t ˆg_{t+1}) − χ (i_t − E_t π_{t+1} − r^n_t) + x_t; x_t = ρ x_{t−1} + ε_t, ε ∼ N(0,1)
  - Policymaker objective: max −1/2 E_0 Σ_{t≥0} β^t [ ˆy_t^2 + λ (ˆb_t − b)^2 ]
- Constraints: IS, debt accumulation (BC), Phillips curve (PC), monetary policy (MP), shock law.
- Lagrangian FOCs yield a linear dynamic system pinning endogenous dynamics and multipliers.

### Optimal fiscal policy under uncertainty — modeling and qualitative effects
- Real-time perceived misspecification introduced via measurement errors:
  - ˆy_{t|t} and ˆb_{t|t} include additive errors σ_w w_t and σ_v v_t respectively.
- Robust policymaker solves max-min selecting g_t against worst-case disturbances (w_t, v_t) subject to relative entropy bound:
  - 1/2 Σ_{t≥0} β^t (w_t^2 + v_t^2) ≤ η / (1−β)
  - Lagrangian penalty θ^{−1}(η − w_t^2 − v_t^2); θ^{−1} inverse represents sensitivity to measurement errors (higher θ^{−1} ⇒ more pessimistic/cautious).
- First-order minimization conditions:
  - θ^{−1} w_t + μ^{IS}_t = 0
  - θ^{−1} v_t + μ^{BC}_t = 0
- Qualitative effect:
  - Uncertainty aversion changes quantitative responses and policy paths (less counter-cyclical responses when both uncertainties present) but not the fundamental output gap vs debt stabilization tradeoff.

### Calibration and simulation (EMs focus)
- Calibration approach:
  - β: inverse of average gross long term real rate across EM sample.
  - δ and τ: average debt-to-GDP ratio and revenues-to-GDP ratio.
  - γ: τ + average CAPB-to-GDP ratio.
  - φ_C from Bhattacharya and Patnaik (2013).
  - σ_w and σ_v: standard deviations of final forecast errors for output gap and CAPB.
  - λ set to match fiscal spending response coefficient from estimated fiscal rule.
  - θ calibrated by matching policy function coefficients.
- Selected calibrated parameters (Table 5):
  - Discount factor (annual) β = 0.8235
  - Target/SS Debt-to-GDP (inverse δ) b = 0.4892
  - SS tax-to-GDP τ = 0.28
  - SS government spending-to-GDP γ = 0.285
  - CRRA σ = 1
  - Relative weight on debt λ = 0.51
  - Persistence of output gap shock ρ = 0.5
  - Fraction of constrained HH φ_C = 0.786
  - Slope of the Phillips curve κ = 0.2
  - SD of IS error σ_w = 1.558
  - SD of debt equation error σ_v = 1.352
  - Taylor rule output gap coefficient φ_y = 0.7
  - Taylor rule inflation coefficient φ_π = 1.3
  - Uncertainty aversion θ = 0.3226

### Simulation experiments: 1% negative output gap shock (scenarios and quantitative responses)
- Scenarios:
  - Benchmark: no uncertainty.
  - Both output gap and fiscal implementation uncertainty (w and v non-zero), θ = 0.3226.
  - Only output gap uncertainty (v_t = 0).
- Main quantitative responses (impact and dynamics, reported):
  - Benchmark:
    - Government spending increases by 0.63% on impact, then sharply falls.
    - Debt increases to 0.75%.
    - Output gap decreases by 0.44%.
  - Both uncertainties:
    - Government spending increases by 0.56% on impact, more persistent and hump-shaped.
    - Output gap falls by 0.57% on impact (larger fall than benchmark) but recovers faster.
    - Public debt surges on impact to 0.6%.
  - Only output gap uncertainty:
    - Government spending increases by 0.77% on impact, more persistent than benchmark.
    - Output gap drops by 0.48% on impact, then reverts faster.
    - Public debt jumps to 0.79%, then reverts faster due to stronger growth and higher tax revenues.
- Interpretation:
  - Concern only about output gap uncertainty → stronger fiscal response, larger immediate debt accumulation, faster recovery.
  - Concern about both uncertainties → more timid fiscal response to control debt dynamics; weaker counter-cyclicality of spending and larger output fall on impact relative to benchmark.
  - Welfare: both uncertainties induce welfare losses relative to uncertainty-neutral policymaker due to less effective output gap stabilization.
- Sensitivity to λ:
  - Varying λ in [0.01, 0.51] finds λ = 0.21 minimizes distance between output gap response and benchmark (no uncertainty).
  - Lowering λ (more weight on output gap stabilization) can replicate benchmark fiscal response without necessarily generating explosive public debt dynamics because faster output recovery raises tax revenues.

### Conclusions and policy implications
- Policymakers aim to stabilize output while limiting debt accumulation; real-time uncertainty in output gap estimates and fiscal implementation can cause Type I and II errors leading to excessive debt or recessionary risks.
- Empirically, real-time dispersion for output gap and fiscal stance is significant; dispersion is higher for EMs than AEs.
- Model results:
  - Uncertainty about output gap and fiscal implementation reduces counter-cyclicality of fiscal policy relative to benchmark (no uncertainty) and reduces efficiency in output stabilization.
  - If only output gap uncertainty is relevant, fiscal responses can be more counter-cyclical and sometimes overshoot benchmark, leading to faster recoveries but higher immediate debt.
- Policy implication:
  - Given high output gap uncertainty in EMs, fiscal responses should be more counter-cyclical, and policymakers could lower the relative weight on debt stabilization (λ) to better stabilize output gaps without necessarily provoking unsustainable debt dynamics due to faster output recoveries.
- Caution: reliance on real-time output gap estimates in EMs for fiscal policy design and implementation is risky.

*Source: wpiea2024251-print-pdf — Sections 2.1–3.8 (Data and Methodology; Summary Statistics; Real-time fiscal reaction functions; Model; Calibration and Simulation) and Conclusions.*

### 2.1  Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   4

### 2.1  Data and Methodology

### Data sources and sample
- Data source: World Economic Outlook (WEO) Spring and Fall releases.
- Sample period: 1998-2022.
- Country coverage: 39 Advanced Economies (AEs) and 73 Emerging Markets (EMs), total of 112 countries.
- Rationale for WEO: harmonized release dates across a large number of countries, crucial for real-time analysis.

### Variables of interest
- Output gap: measure of deviation of current output from the economy’s productive potential; used as the indicator of the economic cycle.
- Cyclically-adjusted primary balance (CAPB): CAPB as a percentage of potential GDP used as a measure of fiscal stance and interpreted as the discretionary component of fiscal policy (excludes automatic stabilizers and interest expenses on public debt).

### Vintage convention and interpretation
- Vintage notation: x_{i,t|τ+v} denotes the estimate of variable x for country i in year t from the vintage τ+v, where v ∈ {S, F}.
- Interpretation:
  - Spring (S) vintage of a given year is interpreted as the forecasts for that year (real-time estimates available to policymakers).
  - Fall (F) vintage is considered the realized values (ex post).
- Real-time use: if τ = t and v = S, then x_{i,t|t+S} is the real-time estimate policymakers use to set fiscal stance.

### Forecast error metrics (definitions preserved)
- Real-time forecast error (RTFE):
  - RTFE_{i,t} = x_{i,t|t+F} − x_{i,t|t+S}
- Final forecast error (FFE):
  - FFE_{i,t} = x_{i,t|2022+F} − x_{i,t|t+S}

### Conceptual context on uncertainty and policy relevance
- Output gap estimates are subject to considerable uncertainty due to measurement errors, model quality, parameter changes from new data, and ex post data revisions.
- Information available to policymakers at decision time may differ significantly from ex post information because of data vintage issues and long fiscal implementation lags (the overall information lag for policymakers can be roughly quantified to be around one and a half year for OECD countries, implying potential for significant forecast errors and sub-optimal fiscal policy decisions).
- Real-time data can be viewed as generated from a probabilistic model that may be misspecified relative to the true data generating process (DGP); output gap estimation itself depends on a chosen model which can be misspecified.

### Theoretical and empirical framing used in the paper
- Literature spans:
  - Real-time assessment errors in cyclical conditions (Orphanides (2001); Orphanides and van Norden (2002)).
  - Real-time fiscal reaction functions (Forni and Momigliano (2004)).
  - Evidence that fiscal stance assessments depend heavily on data vintage (Golinelli and Momigliano (2006); Hallet, Kattai, and Lewis (2007); Cimadomo (2012)).
  - Pro-cyclicality findings in EMs based on ex post data (Gavin and Perotti (1997); Kaminsky, Reinhart, and Végh (2005); Talvi and Vegh (2005); Alesina, Campante, and Tabellini (2008); Marioli and Vegh (2023)).
- Policy design under uncertainty:
  - Brainard (1967) — uncertainty changes optimal policy in a static framework.
  - Robustness approach in dynamic New Keynesian models (Hansen and Sargent (1999); Woodford (2010)): policy maker uncertainty-averse, considers neighborhood of models in relative entropy, maximizes objective under worst-case scenario.

### Study aim and methodological approach (as stated)
- Aim: assess implications of output gap uncertainty on fiscal adjustment; bridge real-time data literature in EMs and uncertainty literature.
- Approach:
  - Introduce a New Keynesian DSGE model in which the policymaker is uncertainty-averse and faces both output gap uncertainty and fiscal policy implementation uncertainty.
  - Calibrate the DSGE model to an average EM and solve for the optimal fiscal reaction function.
- Key empirical and model-based findings (reported in the text):
  - Empirical: fiscal policy is counter-cyclical in AEs, and less strongly so in EMs.
  - Model: when an EM policymaker is concerned about uncertainty around both output gap estimation and fiscal implementation, the resulting reaction function is less counter-cyclical than the benchmark case with no uncertainty.
  - Welfare and debt trade-offs: welfare losses from weaker fiscal response in EMs can be counteracted by placing more weight on output gap stabilization in the policymaker’s objective; this increases public debt in the short run but leads to a stronger recovery of output gap and faster stabilization of public debt over the long run.

*Source: wpiea2024251-print-pdf — Section 2.1 "Data and Methodology" (WEO vintages, 1998-2022; definitions and RTFE/FFE formulas).*

### 2.2  Summary Statistics

### 2.2  Summary Statistics

### 2.2.1 Output Gap — empirical summary
- Sample disaggregation: Advanced Economies (AEs) and Emerging Markets (EMs).
- Real-time Spring estimate (GAP i,t|t+S):
  - Mean GAP i,t|t+S: AEs = -1.369; EMs = -1.636
  - SD GAP i,t|t+S: AEs = 2.385; EMs = 3.240
  - p-value H0: GAP i,t|t+S = 0: AEs = 0.000; EMs = 0.000
- Real-time Fall estimate (GAP i,t|t+F):
  - Mean GAP i,t|t+F: AEs = -1.151; EMs = -1.520
  - SD GAP i,t|t+F: AEs = 2.259; EMs = 3.459
  - p-value H0: GAP i,t|t+F = 0: AEs = 0.000; EMs = 0.000
- Final estimate (GAP i,t|2022+F):
  - Mean GAP i,t|2022+F: AEs = -0.567; EMs = -0.470
  - SD GAP i,t|2022+F: AEs = 2.742; EMs = 4.977
  - p-value H0: GAP i,t|2022+F = 0: AEs = 0.000; EMs = 0.035
- Real-time forecast errors (RTFE):
  - Mean RTFE i,t: AEs = 0.219; EMs = 0.117
  - SD RTFE i,t: AEs = 1.157; EMs = 1.558
  - p-value H0: RTFE i,t = 0: AEs = 0.000; EMs = 0.094
  - Variance ratio test (Var EM / Var AE (RTFE)): 1.815
  - p-value H0: Var EM (RTFE) / Var AE (RTFE) = 1: 0.000
- Final forecast errors (FFE):
  - Mean FFE i,t: AEs = 0.803; EMs = 1.166
  - SD FFE i,t: AEs = 2.360; EMs = 3.747
  - p-value H0: FFE i,t = 0: AEs = 0.000; EMs = 0.000
  - Variance ratio test (Var EM / Var AE (FFE)): 2.521
  - p-value H0: Var EM (FFE) / Var AE (FFE) = 1: 0.000
- Key empirical patterns:
  - WEO real-time estimates tend to underestimate economic upswing in real-time for AEs (significantly negative real-time estimates; positive real-time forecast errors).
  - For EMs, the null that output gap averages 0 cannot be rejected in the final estimate.
  - Revisions using the latest vintage are significantly different from 0 and positive, particularly for EMs; average magnitude of final forecast error is almost twice that of AEs.
  - Revisions (uncertainty proxy) are more volatile for EMs than for AEs; EM forecast error distributions exhibit fatter tails and more extreme values.

### Output gap dynamics and stability of revisions
- Figure 2 evidence (selected EMs: Russia, South Africa, Turkey, India, Indonesia, Mexico, Argentina, Brazil, China):
  - Discrepancy between real-time estimates and later vintages is higher at the beginning of the sample period.
  - Final estimates and estimates 5 years after are very close for most countries; in some cases they perfectly overlap, implying 5 years is a long enough period for revisions to stabilize.

### Forecast-error predictability — tests and implications
- Forecast-error model and hypotheses:
  - Under full-information measurement, RTFE i,t should be white noise: RTFE i,t = ν i,t ∼ WN(0,σ).
  - Predictability assessed by regressions:
    - RTFE i,t = α + x′ i,t|t+S γ + ν i,t
    - FFE i,t = α + x′ i,t|t+S γ + η i,t
  - x i,t|t+S includes GAP i,t|t+S, surplus-to-GDP, deficit-to-GDP (from CAPB-to-GDP), lagged forecast error, and log(GDP deflator i,t|t+S).
- Table 2 regression results (coefficients with standard errors in parentheses; significance: ∗ p<0.1; ∗∗ p<0.05; ∗∗∗ p<0.01):
  - Dependent variable RTFE i,t:
    - AEs (Col 1): Output Gap i,t|t+S coefficient = -0.320 ∗∗∗ (0.045)
    - EMs (Col 2): Output Gap i,t|t+S coefficient = -0.019 (0.059)
    - Observations: AEs = 165; EMs = 186
    - R2: AEs = 0.403; EMs = 0.010
  - Dependent variable FFE i,t:
    - AEs (Col 3): Output Gap i,t|t+S coefficient = -0.501 ∗∗∗ (0.046)
    - EMs (Col 4): Output Gap i,t|t+S coefficient = -0.370 ∗∗∗ (0.086)
    - FFE i,t−1 significant: AEs 0.188 ∗∗ (0.074); EMs 0.295 ∗∗∗ (0.096)
    - Observations: AEs = 165; EMs = 186
    - R2: AEs = 0.572; EMs = 0.149
- Interpretations and policy implications from predictability tests:
  - Only the RTFE equation for EMs yields jointly not-significant coefficients: real-time forecast errors for EMs are not predictable given available information; other regressions show systematic, predictable errors.
  - The negative sign on the output gap coefficient implies: the lower the real-time output gap estimate, the larger the final forecast error (i.e., actual output gap is likely larger than the forecast when RTFE is positive). Hence, a policymaker reacting strongly to a negative output gap forecast risks a Type I error (overreacting to an underestimated upswing).
  - Fiscal variables affect final forecast errors (long run) but not real-time errors:
    - Surplus (AEs) positive predictor of FFE (AEs): implies higher surplus leads to a higher output gap than forecasters expected (negative impact of tighter fiscal policy is over-estimated).
    - Deficit (EMs) negative predictor of FFE (EMs): implies forecasters over-estimated the positive impact of expansionary fiscal policy in EMs.
  - Final forecast errors are serially autocorrelated (FFE i,t−1 significant), stronger for EMs: forecasters over-rely on priors; shocks are harder to identify in EMs, hindering fiscal response effectiveness.

### 2.2.2 Cyclically Adjusted Primary Balance (CAPB) — empirical summary
- Table 3 summary statistics for CAPB forecast errors (percent of GDP):
  - RTFE:
    - Mean RTFE: AEs = -0.304; EMs = -0.283
    - SD RTFE: AEs = 1.482; EMs = 1.367
    - p-value H0: RTFE = 0: AEs = 0.000; EMs = 0.000
    - Variance ratio (Var EM / Var AE (RTFE)) = 0.851
    - p-value Var EM (RTFE) / Var AE (RTFE) = 1: 0.123
  - FFE:
    - Mean FFE: AEs = 0.652; EMs = 0.006
    - SD FFE: AEs = 1.508; EMs = 1.637
    - p-value H0: FFE = 0: AEs = 0.000; EMs = 0.943
    - Variance ratio (Var EM / Var AE (FFE)) = 1.179
    - p-value Var EM (FFE) / Var AE (FFE) = 1: 0.118
- Key CAPB findings:
  - CAPB forecast errors for AEs are significantly different from 0 in both real-time and final vintages.
  - EMs CAPB forecast errors are biased in real-time but not in the long run.
  - Variability of CAPB forecast errors is similar across AEs and EMs (cannot reject equal uncertainty).
  - Distributional evidence (Figure 3): RTFE and FFE distributions for CAPB are similar across income groups — range and dispersion comparable.
  - Interpretation: CAPB is a policy variable subject to implementation discrepancies (planning vs implementation), administrative and political frictions drive forecast errors; hence similar uncertainty across groups is plausible.

### 2.3 Real-time fiscal reaction functions — methodology
- Estimated regression (real-time):
  - CAPB i,t|t+S = β0 GAP i,t|t+S + β1 IEM × GAP i,t|t+S + γ′ x i,t|t+S + ε i,t
  - β0: sensitivity of CAPB to output gap for an average AE; β0 + β1: same for average EM.
  - Counter-cyclicality implies β0 > 0 (AEs) and β0 + β1 > 0 (EMs). β1 < 0 would indicate EMs are less counter-cyclical.
- Covariates x include:
  - Lagged change in debt-to-GDP (∆DEBT t−1) to capture debt-stabilization motive and fiscal capacity.
  - Election year dummy (lead; Election Year t+1) to capture political incentives (expansion before elections).
  - Institutional quality proxied by negative of Corruption Perception Index (higher value = lower institutional quality).
  - Interaction: GAP × ∆DEBT t−1 to capture initial fiscal capacity effect.
  - Asymmetry test: I[GAP t < 0] and I[GAP t < 0] × GAP t to allow different responses in bad times.
  - All regressors lagged as appropriate.

### 2.3.2 Instrumental variables
- Instrument for output gap: weighted average of output gaps from other countries in same WEO regional department:
  - ̄GAP IV g,t|t+S = Σ i∈g GDP i,t|t+S GAP i,t|t+S / Σ i∈g GDP i,t|t+S
  - Regions g ∈ {AFR, APD, EUR, MCD, WHD}
- Identification assumption: regional weighted GAP is exogenous to fiscal policy in country i but conveys external output-gap shocks.

### 2.3.3 Results — fixed effects estimates (Table 4)
- Dependent variable: CAPB-to-GDP (real-time).
- Key coefficient estimates (standard errors in parentheses; significance as above):
  - Column (1) — symmetric specification:
    - CAPB t−1 = 0.236 ∗∗∗ (0.067)
    - GAP t = 0.708 ∗∗∗ (0.073)
    - GAP t−1 = 0.334 ∗∗∗ (0.080)
    - I EM = 1.826 (1.707)
    - I EM × GAP t = -0.283 ∗∗ (0.119)
    - Corruption Perception = -0.186 ∗∗∗ (0.053)
    - Observations = 402; R2 = 0.382; Adjusted R2 = 0.255
  - Column (2) — allow lagged GAP response:
    - CAPB t−1 = 0.205 ∗∗∗ (0.076)
    - GAP t = 0.680 ∗∗∗ (0.075)
    - GAP t−1 = 0.499 ∗∗∗ (0.102)
    - I EM × GAP t = -0.191 (0.122)
    - I EM × GAP t−1 = -0.314 ∗∗ (0.126)
    - Corruption Perception = -0.172 ∗∗∗ (0.053)
    - Observations = 402; R2 = 0.384; Adjusted R2 = 0.253
  - Column (3) — robustness / asymmetry specification:
    - CAPB t−1 = 0.115 (0.089)
    - GAP t = -0.337 (1.308)
    - GAP t−1 = 0.637 ∗∗∗ (0.117)
    - Corruption Perception = -0.150 ∗∗∗ (0.056)
    - I EM × GAP t = -0.202 (0.136)
    - Observations = 402; R2 = 0.355; Adjusted R2 = 0.214
- Summary of empirical findings:
  - Counter-cyclicality:
    - AEs: significant counter-cyclical response in real-time; CAPB-to-GDP increases by 0.708 percentage points for a 1 percentage point increase in GAP (Column 1).
    - EMs: evidence of counter-cyclicality but with lower magnitude; interaction term I EM × GAP t is significantly negative in Column (1), implying on-impact response for EMs = 0.708 + (−0.283) = 0.425 percentage points per 1 percentage point GAP increase.
  - Dynamics:
    - When allowing for lagged GAP (Column 2), contemporaneous responses are statistically similar across groups (both ~0.68 percentage points on impact), but EMs show lower persistence:
      - EMs: one-year-after response = 0.185 percentage points (implied); AEs: one-year-after response = 0.499 percentage points.
  - No significant evidence that initial fiscal capacity (∆DEBT t−1) or imminent elections (Election Year t+1) systematically affect CAPB response in these specifications.
  - Institutional quality: higher corruption perception (worse institutions) predicts lower CAPB-to-GDP consistently (Corruption Perception coefficients = -0.186 ∗∗∗, -0.172 ∗∗∗, -0.150 ∗∗∗ across columns).
  - No robust evidence of asymmetry in fiscal reaction to positive vs negative GAP in the reported specifications.
- Interpretation:
  - AEs appear more responsive to output-gap shocks both on impact and dynamically.
  - Lower sensitivity in EMs may reflect higher uncertainty in output-gap measurement and greater reluctance by policymakers to respond aggressively when measurement is noisy.

*Source: wpiea2024251-print-pdf - 2.2  Summary Statistics*

### 3.1  Households

### 3.1  Households

### Households: structure and optimization
- A fraction φ_C of households is financially constrained: they cannot save or borrow and must consume all current income. Their intratemporal problem:
  - max_{C^C_t, N^C_t} (C^C_t)^{1−1/σ}/(1−1/σ) − (N^C_t)^{1+φ}/(1+φ)
  - subject to P_t C^C_t = (1−τ)(W_t N^C_t + Θ_t)
- Θ_t represents firm profits rebated lump-sum to households. Both labor income and profits face proportional tax rate τ.
- Labor supply of constrained households:
  - (C^C_t)^{1/σ} (N^C_t)^φ = (1−τ) W_t / P_t

- The remaining 1−φ_C fraction are unconstrained and maximize lifetime expected utility choosing C^U_t, nominal bonds B^n_t, and hours N^U_t:
  - max_{C^U_t,B^n_t,N^U_t} E_0 Σ_{t≥0} β^t [ (C^U_t)^{1−1/σ}/(1−1/σ) − (N^U_t)^{1+φ}/(1+φ) ]
  - subject to B^n_t = (1+i_t)[B^n_{t−1} + (1−τ)(W_t N^U_t + Θ_t) − P_t C^U_t]
- First-order conditions yield:
  - Euler equation: β E_t [ (C^U_{t+1}/C^U_t)^{−1/σ} (1+i_t)/Π_{t+1} ] = 1
  - Labor supply: (C^U_t)^{1/σ} (N^U_t)^φ = (1−τ) W_t / P_t

### Key parameters and roles
- Constrained share: φ_C
- Income tax rate: τ
- Bond demand: B^n_t
- Discounting and preferences governed by β, σ, φ

### Implications
- Only unconstrained households enter intertemporal decisions and price-setting discounting via their marginal utility (Λ_{t,t+s}).

---

### 3.2  Firms

### Final good producer
- Perfect competition; CES aggregation with elasticity ε.
- Cost minimization:
  - min_{Y_t(i)} ∫ P_t(i) Y_t(i) di subject to [∫ Y_t(i)^{(ε−1)/ε} di]^{ε/(ε−1)} = Y_t
- Demand for variety i:
  - Y^d_t(i) = [P_t(i)/P_t]^{−ε} Y_t

### Intermediate goods producers
- Each variety i produced with Y_t(i) = A_t N_t(i).
- Monopolistic producers face isoelastic demand Y^d_t(i) and set prices under Calvo staggered pricing:
  - With prob 1−θ firms can reset price P^*_t(i). Reset price solves:
    - max_{P^*_t(i)} Σ_{s≥0} θ^s Λ_{t,t+s} [ P^*_t(i) Y_{t+s}(i)/P_{t+s} − W_{t+s} Y_{t+s}(i)/A_{t+s} ]
  - Demand constraint: Y_{t+s}(i) = [P^*_t(i)/P_{t+s}]^{−ε} Y_{t+s}
- Λ_{t,t+s} is stochastic discount factor of the unconstrained household because households own firms.

---

### 3.3  Government

### Fiscal and monetary policy rules
- Government budget constraint:
  - P_{t+1} B_t = (1+i_t)(P_t B_{t−1} + P_t G_t − τ P_t Y_t)
- Monetary authority Taylor rule:
  - (1+i_t) = β^{−1} (Π_t)^{φ_π} (Y_t/Y^p_t)^{φ_y}
  - Π_t = P_t / P_{t−1} (gross inflation); Y^p_t is potential output

---

### 3.4  Equilibrium

### Market clearing conditions
- Goods market:
  - φ_C C^C_t + (1−φ_C) C^U_t + G_t = Y_t
  - Equivalent form: (1−φ_C) C^U_t + G_t = (1−(1−τ) φ_C) Y_t
- Labor market:
  - N_t = (1−φ_C) N^U_t + φ_C N^C_t, where N_t = Y_t / A_t
  - Rewritten condition:
    - [ (1−φ_C) (C^U_t)^{−1/σ φ} + φ_C (C^C_t)^{−1/σ φ} ]^{−φ} N^φ_t = (1−τ) W_t / P_t
- Bond market clearing:
  - B^n_t = P_{t+1} B_t

---

### 3.5  Log-linear economy

### Log-linearized key relations
- Euler (log-linear):
  - c^U_t = E_t c^U_{t+1} − σ (i_t − E_t π_{t+1})
- Goods market (log-linear; γ is steady-state government spending ratio):
  - (1−φ_C)(1−γ) c^U_t = (1−(1−τ) φ_C) y_t − γ g_t
- Dynamic IS curve (closed form accounting for fiscal policy):
  - ˆy_t = E_t ˆy_{t+1} + μ_g (ˆg_t − E_t ˆg_{t+1}) − χ (i_t − E_t π_{t+1} − r^n_t)
  - where μ_g = γ / (1−(1−τ) φ_C)
  - χ = σ (1−φ_C)(1−γ) / (1−(1−τ) φ_C)
  - r_t = i_t − E_t π_{t+1}; r^n_t natural level
- New Keynesian Phillips curve with fiscal transfers:
  - π_t = κ ˆy_t + β E_t π_{t+1}
- Taylor rule (log-linear):
  - i_t = r^n_t + φ_y ˆy_t + φ_π π_t
- Government debt dynamics (log-linear):
  - β ˆb_t = ˆb_{t−1} + δ γ ˆg_t − δ τ ˆy_t + β (i_t − E_t π_{t+1} − r^n_t)
  - δ = Y / B is inverse of steady-state debt-to-GDP ratio

---

### 3.6  Benchmark Optimal Fiscal Policy

### Planner's reduced-form problem
- Dynamic IS shocks modeled as:
  - ˆy_t = E_t ˆy_{t+1} + μ_g (ˆg_t − E_t ˆg_{t+1}) − χ (i_t − E_t π_{t+1} − r^n_t) + x_t
  - x_t = ρ x_{t−1} + ε_t, ε ∼ N(0,1)
- Policy maker objective (quadratic loss with debt weight λ):
  - max −1/2 E_0 Σ_{t≥0} β^t [ ˆy_t^2 + λ (ˆb_t − b)^2 ]
- Constraints include IS, debt accumulation (BC), Phillips curve (PC), monetary policy (MP), and shock law (x_t).
- Lagrangian and first-order conditions lead to a linear dynamic system pinning down endogenous dynamics; multipliers labeled by constraint.

---

### 3.7  Optimal Fiscal Policy under Uncertainty

### Modeling real-time misspecification and robustness
- Real-time perceived misspecification: denote real-time estimates by ˆ·_{t|t}. IS and debt perceived with measurement errors:
  - ˆy_{t|t} = E_t ˆy_{t+1|t} + μ_g (ˆg_{t|t} − E_t ˆg_{t+1|t}) − χ (i_{t|t} − E_t π_{t+1|t} − r^n_{t|t}) + x_t + σ_w w_t
  - β ˆb_{t|t} = ˆb_{t−1|t} + δ γ ˆg_{t|t} − δ τ ˆy_{t|t} + β (i_{t|t} − E_t π_{t+1|t} − r^n_{t|t}) + σ_v v_t
  - (w_t, v_t)_{t≥0} are random errors
- Robust (uncertainty-averse) policy maker solves a max-min problem selecting policy g_t against worst-case disturbances (w_t, v_t), subject to a relative entropy bound:
  - 1/2 Σ_{t≥0} β^t (w_t^2 + v_t^2) ≤ η / (1−β)
- Equivalent Lagrangian adds penalty θ^{−1}(η − w_t^2 − v_t^2) and θ^{−1} inverse represents sensitivity to measurement errors (higher θ^{−1} => more pessimistic/cautious).
- First-order conditions include two additional minimization conditions:
  - θ^{−1} w_t + μ^{IS}_t = 0
  - θ^{−1} v_t + μ^{BC}_t = 0

### Qualitative effect of uncertainty aversion
- Uncertainty aversion alters quantitative responses to shocks and the path of policy/endogenous variables without changing the fundamental tradeoff between output gap and debt stabilization.

---

### 3.8  Calibration and Simulation

### Calibration approach (EMs focus)
- Use real-time dataset to calibrate steady-state parameters focusing on EMs.
- Key calibration steps:
  - β: inverse of average gross long term real rate across EM sample
  - δ and τ: average debt-to-GDP ratio and revenues-to-GDP ratio (inverse and level)
  - γ: τ + average CAPB-to-GDP ratio
  - φ_C: share of financially constrained households from Bhattacharya and Patnaik (2013)
  - σ_w and σ_v: standard deviations of final forecast errors for output gap and CAPB
  - λ: set to match fiscal spending response coefficient from estimated fiscal rule in Section 2
  - θ (uncertainty aversion) calibrated by matching policy function coefficients

### Table 5: calibrated parameters (selected)
- Discount factor (annual) β = 0.8235
- Target/SS Debt-to-GDP (inverse δ) b = 0.4892
- SS tax-to-GDP τ = 0.28
- SS government spending-to-GDP γ = 0.285
- CRRA σ = 1
- Relative weight on debt λ = 0.51
- Persistence of output gap shock ρ = 0.5
- Fraction of constrained HH φ_C = 0.786
- Slope of the Phillips curve κ = 0.2
- SD of IS error σ_w = 1.558
- SD of debt equation error σ_v = 1.352
- Taylor rule output gap coefficient φ_y = 0.7
- Taylor rule inflation coefficient φ_π = 1.3
- Uncertainty aversion θ = 0.3226

### Simulation experiments: 1% negative output gap shock
- Scenarios analyzed:
  - Benchmark case: no uncertainty
  - Both output gap and fiscal implementation uncertainty (w and v non-zero), θ = 0.3226
  - Only output gap uncertainty (v_t = 0)
- Main quantitative responses (impact and dynamics):
  - Benchmark:
    - Government spending increases by 0.63% on impact, then sharply falls
    - Debt increases to 0.75%
    - Output gap decreases by 0.44%
  - Both uncertainties:
    - Government spending increases by 0.56% on impact, more persistent and hump-shaped
    - Output gap falls by 0.57% on impact (larger fall than benchmark) but recovers faster
    - Public debt surges on impact to 0.6%
  - Only output gap uncertainty:
    - Government spending increases by 0.77% on impact, more persistent than benchmark
    - Output gap drops by 0.48% on impact, then reverts faster
    - Public debt jumps to 0.79%, then reverts faster due to stronger growth and higher tax revenues

### Interpretation of simulated responses
- Concern only about output gap uncertainty -> stronger fiscal response, larger immediate debt accumulation, faster recovery of output and revenues.
- Concern about both uncertainties -> more timid fiscal response to control debt dynamics, weaker counter-cyclicality of spending, larger output fall on impact relative to benchmark.
- Welfare implication: presence of both uncertainties induces welfare losses relative to uncertainty-neutral policy maker because of less effective output gap stabilization.

### Sensitivity to λ
- Vary λ in [0.01, 0.51]; there exists λ = 0.21 that minimizes distance between output gap response and benchmark (no uncertainty).
- Lowering λ (more weight on output gap stabilization) can replicate benchmark fiscal response without necessarily generating explosive public debt dynamics due to faster output recovery raising tax revenues.

---

### 4  Conclusions

- Policy makers aim to stabilize output while limiting debt accumulation, using output gap estimates as cyclical indicators; real-time uncertainty in output gap estimates and fiscal implementation can cause Type I and II errors leading to excessive debt or recessionary risks.
- Empirically, real-time dispersion for output gap and fiscal stance is significant; dispersion is higher for EMs than AEs.
- Model results:
  - Uncertainty about output gap and fiscal implementation reduces counter-cyclicality of fiscal policy relative to benchmark (no uncertainty) and reduces efficiency in output stabilization.
  - If only output gap uncertainty is relevant, fiscal responses can be more counter-cyclical and sometimes overshoot the benchmark, leading to faster recoveries but higher immediate debt.
- Policy implication:
  - Given high output gap uncertainty in EMs, fiscal responses should be more counter-cyclical, and policymakers could lower the relative weight on debt stabilization (λ) to better stabilize output gaps without necessarily provoking unsustainable debt dynamics due to faster output recoveries.
- Caution advised in relying on real-time output gap estimates in EMs for fiscal policy design and implementation.

*Source: wpiea2024251-print-pdf - 3.1  Households*

### References

### wpiea2024251-print-pdf - References

### References
- Bibliographic list of works cited including empirical and theoretical studies on output gap estimation, real-time fiscal and monetary policy, fiscal procyclicality, robustness and ambiguity in policy design, and New Keynesian frameworks. Notable authors and works appearing in the list include Aastveit and Trovik (2014); Alesina, Campante, and Tabellini (2008); Benigno and Woodford (2003); Borio, Disyatat, and Juselius (2017); Coibion and Gorodnichenko (2015); Galí (2015); Orphanides (2001); Orphanides and van Norden (2002); Woodford (2003, 2010); and multiple IMF and NBER working papers.

### Appendix — Fiscal Reaction Function Robustness Checks
- Methodology and model choice:
  - Random effects models estimated for Equation (9) and modifications; comparison with fixed effects models from Section 2.3.
  - Hausman test applied to decide between fixed and random effects.
  - Alternative specification following Golinelli and Momigliano (2006): Equation (51)
    - ∆CAPB_{i,t|t+S} = α + ρ CAPB_{i,t−1|t+S} + β GAP_{i,t−1|t+S} + x′_{i,t|t+S} γ + ε_{i,t}
- Key empirical findings (from Tables 6–9):
  - Table 6 (Random Effects, dependent variable: Cyclically adjusted primary balance-to-GDP_t):
    - CAPB_{t−1}: 0.820 ∗∗∗ (column 1), 0.829 ∗∗∗ (column 2), 0.164 ∗ (column 3) with standard errors (0.047), (0.050), (0.090).
    - GAP_t: 0.740 ∗∗∗, 0.739 ∗∗∗, −1.349 with standard errors (0.084), (0.089), (1.364).
    - GAP_{t−1}: 0.017, 0.236 ∗∗, 0.632 ∗∗∗ with standard errors (0.081), (0.109), (0.115).
    - ∆DEBT_{t−1}: 0.047 ∗∗, 0.092 ∗∗∗, −0.014 with standard errors (0.024), (0.028), (0.032).
    - I_EM coefficient: −1.079 ∗∗, −1.723 ∗∗∗, 0.895 (std. errors (0.465), (0.517), (1.306)).
    - I_EM × GAP_t: −0.417 ∗∗∗, −0.292 ∗∗, −0.186 (std. errors (0.135), (0.142), (0.138)).
    - Observations: 402 across columns; R^2: 0.511, 0.509, 0.291; Adjusted R^2: 0.501, 0.496, 0.269.
    - F Statistics: 462.851 ∗∗∗, 455.365 ∗∗∗, 228.949 ∗∗∗.
  - Table 7 (Hausman test: Fixed vs Random Effect models for Equation (9)):
    - Statistics: 153.732 (column 1), 92.904 (column 2), 0.216 (column 3).
    - p-values: 0.000, 0.000, 1.000.
    - Interpretation: Hausman test rejects random effects for the two symmetric specifications but not for the specification including potential asymmetries.
  - Table 8 (Equation (51), dependent variable: ∆Cyclically adjusted primary balance-to-GDP):
    - For AEs (columns 1 and 3) and EMs (columns 2 and 4):
      - CAPB_{t−1}: −0.481 ∗∗ (AEs, col 1), −0.928 ∗∗∗ (EMs, col 2); −0.498 ∗ (AEs, col 3), −1.148 ∗∗ (EMs, col 4). Std. errors (0.200), (0.141), (0.269), (0.449).
      - GAP_t: 0.547 ∗∗∗, 0.572 ∗∗∗, 0.289, 0.414 ∗ with std. errors (0.137), (0.219), (0.190), (0.250).
      - GAP_{t−1}: 0.176 ∗∗∗ (AEs col 1), 0.110 (EMs col 2) with std. errors (0.051), (0.069).
      - DEBT_{t−1}: −3.485 ∗ (AEs col 1), −0.120 (EMs col 2) with std. errors (2.012), (0.329).
      - Observations: 135 (AEs col1), 144 (EMs col2), 124 (AEs col3), 63 (EMs col4).
      - R^2: 0.221, 0.269, 0.116, 0.379. Adjusted R^2: −0.076, −0.035, −0.221, 0.104.
      - F Statistics: 28.323 ∗∗∗, 59.467 ∗∗∗, 8.529 ∗, 28.677 ∗∗∗.
    - Interpretation: Fiscal policy appears more history-dependent for EMs; coefficient on lagged CAPB-to-GDP about twice for EMs compared to AEs.
  - Table 9 (Real-time fiscal reaction functions, Debt in Levels; dependent variable: Cyclically adjusted primary balance-to-GDP_t):
    - Selected coefficients across specifications:
      - CAPB_{t−1}: 0.299 ∗∗ (AEs col1), 0.191 (EMs col2), 0.320 ∗∗ (AEs col3), 0.108 (EMs col4), 0.460 (AEs col5), 0.096 (EMs col6). Std. errors shown per cell.
      - GAP_t: 0.618 ∗∗∗, 0.518 ∗∗∗, 0.519 ∗∗∗, 1.258 ∗∗∗, −2.328, 1.636 with std. errors (0.056), (0.094), (0.101), (0.239), (19.290), (2.059).
      - GAP_{t−1}: 0.420 ∗∗∗, 0.237, 0.406 ∗∗∗, 0.219, 0.630, 0.299 ∗ with std. errors provided.
      - DEBT_{t−1} and interactions (GAP_t × DEBT_{t−1}): coefficients include 0.028, 0.009, 0.032, −0.028, 0.118, −0.023 and GAP_t × DEBT_{t−1}: 0.001, −0.015 ∗∗∗, −0.0004, −0.018 ∗∗∗ with std. errors.
      - Observations alternate at 135 and 144; R^2 ranges: 0.740, 0.292, 0.739, 0.417, 0.663, 0.374; Adjusted R^2 ranges: 0.637, −0.012, 0.632, 0.158, 0.514, 0.077.
      - F Statistics: highly significant across columns (e.g., 289.325 ∗∗∗, 54.046 ∗∗∗, 285.071 ∗∗∗, 81.167 ∗∗∗, 206.203 ∗∗∗, 72.384 ∗∗∗).
    - Interpretation: Results are consistent with Section 2.3.3; robust relationships when fiscal capacity proxied by debt-to-GDP in levels.

- Authors’ methodological choice and reporting:
  - Despite Hausman rejection in some specifications, fixed effects models reported in Table 4 are preferred to capture country-specific differences.
  - Random effects specifications display higher persistence (large autoregressive coefficient) relative to fixed effects estimates.
  - Evidence of EMs consolidating after output gap shocks: cyclically adjusted primary balance moves opposite to lagged output gap in some specifications.

### Model Solution — Key equations and relations
- Household problem:
  - Unconstrained household Lagrangian (equations (52)–(53)) and first-order conditions lead to Euler equation and labor supply.
  - Constrained household Lagrangian (equation (54)) and FOCs yield labor supply for constrained households.
- Final good producer:
  - Minimization problem and price index derivation yield price index P_t as in (56).
  - Individual demand: Y^d_t(i) = (P_t(i)/P_t)^{−ε} Y_t (equation (57)).
- Intermediate goods producers:
  - Profit-maximization with optimal reset price P^*_t(i) expression in (60) and aggregate reset price P^*_t in (63).
  - Price index aggregation: P_t = [ (1−θ) P^{* 1−ε}_t + θ P^{1−ε}_{t−1} ]^{1/(1−ε)} and inflation relations Π_t and Π^*_t (equations through (65)).
  - Auxiliary constructs X_aux,1_t and X_aux,2_t defined in (61)–(62); optimal reset price P^*_t = ε/(ε−1) X_aux,1_t / X_aux,2_t (63).
- Equilibrium and market-clearing:
  - Goods market equilibrium: φ_C(1−τ) Y_t + (1−φ_C) C_t + G_t = Y_t leading to (1−φ_C) C_t + G_t = (1−(1−τ)φ_C) Y_t (68).
  - Labor market clearing: N_t expression combining (1−φ_C) and φ_C households (69).
- Log-linearization and linearized equilibrium:
  - Steady state: Π_ss = 1 and Y_ss = Y^p_ss.
  - Linearized supply side: π_t = ψ ˆmc_t + β E_t π_{t+1} (70); ˆmc_t = w_t − a_t (71).
  - Log-linearized equilibrium conditions (72)–(74) and modified Phillips curve (75) and (76).
  - Natural output and flexible-price relations: y^n_t expression in (77) and Phillips-like relation π_t = κ ˆy_t + β E_t π_{t+1} (78).
- Real debt accumulation and log-linearized dynamics:
  - Nominal accumulation: P_{t+1} B_t = (1 + i_t)(P_t B_{t−1} + P_t G_t − τ P_t Y_t) (79).
  - Debt-to-GDP dynamics in log-linearized form: β b_t = b_{t−1} + (G/B) ˆg_t − (Y/B) τ y_t + β (i_t − E_t π_{t+1}) with 1 + i = β^{−1} (equations (80)–(82)). Compact expression: β ˆb_t = ˆb_{t−1} + δ(γ ˆg_t − τ ˆy_t) + β(i_t − E_t π_{t+1} − r^n_t) (82) where δ = Y/B.
- Benchmark optimal control conditions and FOCs:
  - First-order conditions and multipliers lead to system (83) and (84) linking output deviations, debt, and multipliers with parameters δ, γ, τ, μ_g, μ_IS, χ, λ, κ, φ_y, φ_π appearing in corresponding relations.
  - Example result: ˆy_t = β^{−1} δ χ (γ − μ_g τ) / μ_g μ_IS_t (equation (83)).
  - Debt dynamics relation: b_t = b − β μ_g (γ − μ_g τ) λ δ τ ˆy_t + β μ_g (γ − μ_g τ) λ δ τ E_t ˆy_{t+1} (84).

*Source: wpiea2024251-print-pdf - References (Working Paper No. WP/2024/251).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024251-print-pdf.pdf_