## 16.    Transitory shock: U.S. versus Morocco

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---

### Introduction
- Purpose: characterize whether fiscal policy is pro-cyclical or counter-cyclical and show that fiscal policy reacts differently to temporary and permanent income shocks because of the tradeoff between short-run stabilization and debt sustainability.
- Method: Beveridge-Nelson type decomposition of income that distinguishes stochastic trend (permanent) and temporary components, with structural break detection.
- Previewed empirical claim:
  - Government spending tends to be counter-cyclical conditional on temporary shocks and pro-cyclical conditional on permanent shocks.
  - Substantial cross-country dispersion in magnitudes and signs.
  - No clear pattern by income per capita.

### Empirical analysis

- Data and trend-cycle decomposition
  - Data: logarithm of real GDP, annual frequency, real GDP series from IMF World Economic Outlook database (as of August 2014) covering the period 1970-2011.
  - Model: yt = τt + ct with
    - τt = μ + d1(t>Tb) + τt−1 + ηt
    - ct = ρ c t−1 + et
  - Structural break Tb detected using Perron (2005) procedure; model estimated in state-space via Kalman filter.
  - Rationale: HP filter can mis-measure cycles in presence of stochastic trends and structural breaks; Beveridge-Nelson style decomposition with structural break detection yields materially different trend/cycle splits.

- Assessing cyclicality of fiscal policy
  - Estimation equation: gt = β0 + β1 gt−1 + β2 Δτt + β3 Δct + εt
    - Interpretation: β3 negative → counter-cyclical response to temporary shocks; β2 positive → pro-cyclical response to permanent shocks.

- Main empirical results (panel regressions)
  - Mean Group (MG) regressions (Table 1) — three samples (1980-2009; 1985-2009; 1990-2009):
    - gt−1: 0.863*** (0.0239); 0.806*** (0.0307); 0.719*** (0.0542)
    - Δτt: 3.608** (1.579); 6.637*** (2.183); 6.671** (3.057)
    - Δct: -0.969* (0.511); -1.824*** (0.638); -2.629*** (1.007)
    - Constant: 0.826*** (0.166); 1.043*** (0.204); 1.260*** (0.277)
    - Observations: 1,330; 1,092; 838
    - Number of countries: 64; 61; 60
    - Significance legend: *** p<0.01, ** p<0.05, * p<0.1
    - Interpretation: β2>0 and significant; β3<0 and significant across samples.
  - CCEMG/CCEG robustness (Table 2) — 1985-2009 CCEMG example:
    - gt−1: 0.817*** (0.0370)
    - Δτt: 5.937*** (1.790)
    - Δct: -0.815* (0.447)
  - Regressions including debt (Table 3)
    - Debt-to-GDP included (debtt−1); Δτt > 0 and significant; Δct < 0 and significant in MG and CCEG specifications.
    - Coefficient on debt examples: -0.000106 (0.000495); 0.000908 (0.000720); -0.000114 (0.000851) — not consistently significant.
    - Observations and countries vary (examples: Observations 1,130; 1,097; Number of countries 53; 52).
    - Interpretation: simple debt-to-GDP proxy may be inadequate once permanent vs temporary decomposition is accounted for.
  - Cross-country dispersion and income per capita:
    - No clear pattern linking β2 or β3 to log GDP per capita in 1995; poorer countries are not systematically more counter-cyclical.

### Model

- Standard RBC setup (agents, technology, and key conditions preserved)
  - Households: discount factor β ∈ (0,1); inverse intertemporal elasticity σ > 0; weight on leisure η > 0; inverse Frisch elasticity μ > 0.
  - Capital accumulation: Kt+1 = (1−δ)Kt + It.
  - Euler conditions: λt = C−σt; λt = η L μ t (1−τt) wt; λt = β Et λt+1 [(1−τt+1) rk t+1 + 1−δ]; λt = β Et λt+1 rt.
  - Firms: Cobb-Douglas Yt = At Kt α Lt 1−α with 0 < α < 1.
  - Technology decomposition: At = Att (A p t)1−α with
    - log(A p t) = log(A p t−1) + ε p t
    - log(Att) = (1−ρA) log(A) + ρA log(Att−1) + ε t t
    - Shocks ε p t and ε t t iid normal with s.d. σp and σt.

- Government fiscal reaction (spending) rule
  - Δ log(Gt) = ρg Δ log(Gt−1) − (1−ρg)[ρg,y Δ log(Yt) + ρg,b log(ζb t / ̄ζb)] + ε g t
  - Debt-sustainability term: ζb t = Et ∑_{i=0}^∞ ωt+i b t+i where b t+i ≡ Bt+i / Yt+i
  - Two weighting schemes:
    - ω1,t = βb t (declining weights; βb is government subjective discount factor)
    - ω2,t = t κ1−1 exp(−t / κ2) / (κ κ1 2 Γ(κ1)) (asymmetric bell-shaped)
  - Government budget constraint: Bt κt Rt = Bt−1 − (τt Yt − Gt)
  - Tax feedback rule: log(τt / ̄τ) = γτ,g log(gt / ̄g) + γτ,b log(bt / ̄b)

### Theoretical results

- Calibration (baseline)
  - Period = 1 year.
  - Parameters: β = 0.95; σ = 1; Frisch elasticity of labor supply = 0.75; μ chosen so steady-state hours = 0.25; δ = 0.1; α = 0.36; steady-state government spending share = 0.2; steady-state debt share = 0.4.

- Impulse-response experiments
  - Shock: one percentage point decline in productivity; persistence scenarios ρA = 0.25, ρA = 0.75, permanent; fiscal rule variants with and without debt stabilization and with ω1,t or ω2,t.
  - Key dynamics without debt-sensitive rule:
    - Transitory shocks produce short-lived output blips; negative technology shocks yield opposite signs for output and government spending.
    - As shock becomes less persistent, conditional correlation between government spending and output approaches -1.
  - Key dynamics with debt-sensitive rule:
    - For mild persistence and with debt reaction, output response similar but debt ratio increases sharply on impact (example: output declines by about 1.4 percent and debt overshoots by 2.5 percent on impact in one scenario).
    - For high persistence, government spending can switch from positive to negative response (counter- to pro-cyclical) because persistent output decline raises debt ratio and future debt expectations dominate stabilization motives.
    - Discounting scheme matters: ω1,t (declining weights) produces lower counter-cyclicality than ω2,t (bell-shaped) because near-future debt aversion induces sharper spending reversal when shocks persist.

- Conditional correlations from calibrated model (Table 4)
  - Corr(Δgt, Δyt | ε t t ) and Corr(Δgt, Δyt | εp t ):
    - Without debt: -0.983 (ρA=0.25) ; -0.980 (ρA=0.75) ; -0.979 (permanent)
    - With debt, ω1,t: -0.655 (ρA=0.25) ; -0.157 (ρA=0.75) ; 0.229 (permanent)
    - With debt, ω2,t: -0.837 (ρA=0.25) ; -0.350 (ρA=0.75) ; 0.242 (permanent)
  - Interpretation: model generates counter-cyclicality for temporary shocks when government reacts only to output; including debt expectations reduces counter-cyclicality and can yield pro-cyclicality for persistent/permanent shocks.

- Sensitivity analysis
  - Varying ρg,y and ρg,b changes conditional correlations; introducing debt in the rule dampens negative correlation for temporary shocks and can shift sign for persistent/permanent shocks depending on elasticities.

- Filters and mismeasurement
  - Simulated exercise applying HP filter (λ = 100) and growth-rate correlations to model-generated series under varying σp/σt and ρg,b.
  - Finding: HP-filter and growth-rate unconditional correlations can deviate substantially from model-based conditional correlations in regions where trend shocks are more volatile (higher σp/σt). Filters may produce pro-cyclical co-movement between government spending and output even though true conditional relation for temporary shocks is counter-cyclical.
  - Implication: in countries with more volatile stochastic trends (e.g., many developing countries), HP-filter-based assessments may mistakenly classify fiscal policy as pro-cyclical.

### Appendix: mechanisms for cross-country heterogeneity (selected findings)

- A.1. Learning
  - Setup: agents observe composite productivity Zt with signal noise σz; Kalman filter used.
  - Results:
    - For σz → 0, perfect information recovered.
    - Learning "adds little" to change degree of cyclicality; impulse responses to transitory shocks qualitatively similar under perfect and imperfect information.
    - Learning reduces long-run responses to trend shocks (agents partially interpret permanent shock as temporary).
    - Conclusion: learning alone "does not seem to be a sufficient candidate to interpret the heterogeneous counter-cyclicality across countries."

- A.2. Government degree of impatience
  - Vary κ1 (degree of government patience) with values 5 and 20; discount profiles in Figure 12.
  - Findings:
    - κ1 = 20 (more patient) leads to overshooting of government spending after a negative persistent productivity shock; adjustment path of debt delayed.
    - Higher κ1 ⇒ higher degree of counter-cyclical behavior.
    - Short-term impulse-responses to permanent shocks "not very sensitive to κ1"; higher κ1 increases persistence.

- A.3. Endogenous persistence (real rigidities)
  - Extensions: habit formation ψ and investment adjustment cost χ (ψ = 0.8, χ = 10 in illustration).
  - Mechanism: real rigidities amplify sluggishness; costly investment leads households to consume more leisure short term producing reversed hump-shaped output response.
  - Quantitative result: with "reasonably high" ψ and χ, model yields positive conditional correlation between output and government spending—contrasting baseline.
  - Conclusion: endogenous real rigidities are a plausible source of observed heterogeneity.

- A.4. Endogenous tax rules
  - Compare γτ,g = γτ,b = 0 (exogenous taxes) versus γτ,g = γτ,b = 0.5 (reactive taxes).
  - Findings:
    - Reactive taxes exacerbate negative transitory shock impact on output because taxes increase in reaction to higher government spending and debt.
    - Government spending becomes more persistent; conditional correlation between spending and output rises as tax rates become reactive.
    - Conclusion: stronger tax response to debt helps produce negative comovement of government spending and output.

- A.5. Model-based country comparison (United States versus Morocco)
  - Estimation: Bayesian (Metropolis-Hastings) with Kalman filter; annual U.S. data 1970–2008; debt data from Abbas and others (2010).
  - Estimation details: 200,000 draws, first 40,000 discarded; acceptance rate ~25 percent.
  - Log-likelihoods:
    - US benchmark 208.733; US exogenous spending 205.287.
    - Morocco benchmark 120.008; Morocco exogenous spending 116.187.
  - Bayes factor: marginal data density of benchmark model is 3.45 and 3.82 higher on a log-scale for the US and Morocco, respectively — posterior odds favor model with endogenous government spending rule.

  - Selected posterior estimates (posterior mode and 90 percent intervals)
    - Real rigidity parameters (United States, benchmark model)
      - ψ: posterior mean 0.2656 [0.111, 0.413]
      - χ: posterior mean 13.870 [5.326, 22.155]
    - Fiscal policy parameters (posterior means)
      - ρg,y: United States 1.251 [0.784, 1.732]; Morocco 0.820 [0.081, 1.731]
      - ρg,b: United States 0.636 [0.035, 1.314]; Morocco 1.251 [0.206, 2.327]
      - κ1: United States 14.462 [6.051, 23.844]; Morocco 11.435 [5.230, 17.704]
      - γτ,g: United States 0.559 [0.317, 0.797]; Morocco 0.220 [-0.090, 0.511]
      - γτ,b: United States 0.245 [0.060, 0.423]; Morocco 0.362 [0.167, 0.543]
    - Exogenous process parameters (selected)
      - ρAt: United States 0.299 [0.165, 0.431]; Morocco 0.219 [0.108, 0.334]
      - ρg: United States 0.395 [0.222, 0.562]; Morocco 0.339 [0.155, 0.514]
      - σp: United States 0.059 [0.044, 0.073]; Morocco 0.134 [0.103, 0.166]
      - σt: United States 0.020 [0.015, 0.025]; Morocco 0.042 [0.032, 0.052]
      - σg: United States 0.030 [0.023, 0.036]; Morocco 0.106 [0.080, 0.129]
      - σs: United States 0.006 [0.001, 0.013]; Morocco 0.005 [0.001, 0.010]

  - Comparative interpretation and implications
    - US government estimated to be more reactive to output (ρg,y ≈ 1.28) and less reactive to debt (ρg,b ≈ 0.64) than Morocco (ρg,y ≈ 0.82; ρg,b ≈ 1.25).
    - US κ1 ≈ 14.462 versus Morocco κ1 ≈ 11.435 ("government highly cares about debt changes after more than 14 years" for US; "only 11 years for Morocco").
    - Model-implied posterior mode correlation between government spending and output: −0.6 (US) versus −0.4 (Morocco).
    - Interpretation: Morocco’s greater concern with bringing debt back to pre-shock steady state produces less counter-cyclical spending responses.

### Conclusions and policy implications
- Empirical decomposition distinguishing permanent and temporary GDP components yields:
  - Government spending reacts positively to permanent shocks (β2 > 0) and negatively to temporary shocks (β3 < 0).
  - Considerable dispersion across countries in magnitudes and signs; no simple relation with income per capita.
- Theory: an RBC model with an endogenous fiscal rule responding to output gaps and to the expected discounted sum of future debt explains counter-cyclicality to temporary shocks and pro-cyclicality to permanent shocks.
- Mechanisms affecting cross-country dispersion:
  - Degree of aversion to future debt levels (ρg,b)
  - Endogenous persistence (habit formation ψ, investment adjustment cost χ)
  - Tax-rule responsiveness (γτ,g, γτ,b)
  - Learning about shock source explains little of the dispersion.
- Measurement implication: accounting for stochastic trends and structural breaks is crucial — standard HP-filter approaches can mismeasure cyclicality, particularly for countries with volatile permanent shocks.

*Source PDF filename: _wp1659 - REFERENCES*

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

### _wp1659 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### Appendix and thematic breakdown
- Appendix title: A.Sources of Heterogeneity in the Degree of Counter-cyclicality
- Appendix page reference: 31
- Appendix subsections and their page references:
  - A.1. Learning — 31
  - A.2. Government degree of impatience — 35
  - A.3. Endogenous persistence — 36
  - A.4. Endogenous tax rules — 38
  - A.5. Model-based country comparison — 39

### Tables listed (with exact numbering and headings)
- 1.MG Regressions: Dependent Variableg
t
. 10
- 2.CCEMG Regressions: Dependent Variableg
t
1985-2009. 11
- 3.Regressions with debt: Dependent Variableg
t
1985-2009. 12
- 4.Conditional correlations. 23
- 5.Parameter Estimation Results. 42

### Figures listed (with exact numbering and headings)
- 1.GDP Trend-Cycle Decomposition in Selected Countries. 9
- 2.β
2
,β
3
and GDP per capita. 13
- 3.Future debt ratios weighting schemes. 17
- 4.Dynamic responses to -1 percent technology shock: Without debt. 20
- 5.Dynamic responses to -1 percent technology shock: With debt (assumingω
1,t
). 21
- 6.Dynamic responses to -1 percent technology shock: With debt (assumingω
2,t
). 21
- 7.Dynamic responses of the debt ratio. 22
- 8.Conditional correlations: Debt versus output stabilization. 24
- 9.Output and government spending unconditional correlations. 26
- 10. The effect of learning on the impulse-response functions. 33
- 11. The effect of learning on beliefs about transitory and permanent components. 34
- 12. The profiles of the discount function. 35
- 13. The effect of government aversion on the impulse-response functions. 36
- 14. Real rigidities and government spending cyclicality. 37
- 15. The effect of learning on the impulse-response functions. 39

*Source: _wp1659 - References (PDF chapter/section) — canonical PDF listing of appendices, tables, and figures.*

### 16.    Transitory shock: U.S. versus Morocco  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    

### 16.    Transitory shock: U.S. versus Morocco

### Introduction
- Purpose: characterize whether fiscal policy is pro-cyclical or counter-cyclical and show that fiscal policy reacts differently to temporary and permanent income shocks because of the tradeoff between short-run stabilization and debt sustainability.
- Methodological premise: use a Beveridge-Nelson type decomposition of income that distinguishes stochastic trend (permanent) and temporary components, and account for structural breaks.
- Main empirical claim previewed: government spending tends to be counter-cyclical conditional on temporary shocks and pro-cyclical conditional on permanent shocks; substantial cross-country dispersion in magnitudes and signs; no clear pattern by income per capita.

### Empirical analysis

#### Trend-cycle decomposition
- Data: logarithm of real GDP, annual frequency, real GDP series from IMF World Economic Outlook database (as of August 2014) covering the period 1970-2011.
- Model for yt: yt = τt + ct with
  - τt = μ + d1(t>Tb) + τt−1 + ηt (permanent component with possible structural break at Tb)
  - ct = ρ c t−1 + et (temporary/cyclical component)
- Estimation: detect structural break Tb using Perron (2005) procedure; translate model to state-space and estimate via Kalman filter.
- Rationale: HP filter can mis-measure cycles in presence of stochastic trends and structural breaks; Beveridge-Nelson style decomposition with structural break detection yields materially different trend/cycle splits (illustrated for Malaysia, Chile, Nigeria, Burundi).

#### Assessing the cyclicality of fiscal policy
- Estimation equation (real fiscal expenditure in log, gt):
  - gt = β0 + β1 gt−1 + β2 Δτt + β3 Δct + εt
  - Interpretation: β3 negative → counter-cyclical response to temporary shocks; β2 positive → pro-cyclical response to permanent shocks (indicative of fiscal sustainability constraints).

#### Main empirical results (panel regressions)
- Mean Group (MG) regressions (Table 1):
  - Samples: (1) 1980-2009, (2) 1985-2009, (3) 1990-2009.
  - Coefficients (selected, standard errors in parentheses):
    - gt−1: 0.863*** (0.0239); 0.806*** (0.0307); 0.719*** (0.0542)
    - Δτt: 3.608** (1.579); 6.637*** (2.183); 6.671** (3.057)
    - Δct: -0.969* (0.511); -1.824*** (0.638); -2.629*** (1.007)
    - Constant: 0.826*** (0.166); 1.043*** (0.204); 1.260*** (0.277)
  - Observations: 1,330; 1,092; 838
  - Number of countries: 64; 61; 60
  - Significance legend: *** p<0.01, ** p<0.05, * p<0.1
  - Interpretation: β2>0 and significant; β3<0 and significant across samples → positive response to permanent shocks, negative to temporary shocks.
- Common Correlated Effects Mean Group (CCEMG/CCEG) robustness (Table 2):
  - 1985-2009 CCEMG and MG variants confirm signs and significance of β2 and β3; inclusion of time trend produces similar results; coefficients remain of comparable magnitude.
  - Example CCEMG coefficients (1985-2009):
    - gt−1: 0.817*** (0.0370)
    - Δτt: 5.937*** (1.790)
    - Δct: -0.815* (0.447)
- Regressions including debt (Table 3):
  - Debt-to-GDP ratio included (debtt−1); results remain:
    - Δτt > 0 and significant; Δct < 0 and significant in MG and CCEG specifications with and without time trend.
    - Coefficient on debt not consistently significant (examples: -0.000106 (0.000495); 0.000908 (0.000720); -0.000114 (0.000851)).
  - Observations and number of countries vary across specifications (e.g., Observations 1,130; 1,097; 1,097; 1,091; Number of countries 53; 52; 52; 51).
  - Interpretation: debt-to-GDP as a simple proxy for sustainability may be inadequate once permanent vs temporary decomposition is accounted for.
- Cross-country dispersion and income per capita:
  - No clear pattern linking β2 or β3 to log GDP per capita in 1995; poorer countries are not systematically more counter-cyclical.

### Model

#### Standard RBC setup
- Agents: households, firms, government.
- Household problem and first-order conditions summarized with parameters and variables exactly as in the source:
  - Utility discount factor β ∈ (0,1)
  - Inverse intertemporal elasticity σ > 0
  - Weight on leisure η > 0
  - Inverse Frisch elasticity μ > 0
  - Budget constraint, capital accumulation Kt+1 = (1−δ)Kt + It
  - Euler conditions: λt = C−σt; λt = η L μ t (1−τt) wt; λt = β Et λt+1 [(1−τt+1) rk t+1 + 1−δ]; λt = β Et λt+1 rt
- Firms: Cobb-Douglas production Yt = At Kt α Lt 1−α with 0 < α < 1.
- Technology decomposition: At = Att (A p t)1−α with:
  - log(A p t) = log(A p t−1) + ε p t (permanent)
  - log(At t) = (1−ρA) log(A) + ρA log(Att−1) + ε t t (temporary)
  - Shocks ε p t and ε t t iid normal with standard deviations σp and σt.

#### Government fiscal reaction rule (spending rule)
- Spending growth rule:
  - Δ log(Gt) = ρg Δ log(Gt−1) − (1−ρg)[ρg,y Δ log(Yt) + ρg,b log(ζb t / ̄ζb)] + ε g t
- Debt-sustainability term:
  - ζb t = Et ∑_{i=0}^∞ ωt+i b t+i where b t+i ≡ Bt+i / Yt+i
- Two alternative weighting (discount) schemes for future debt:
  - ω1,t = βb t (declining function; βb is government subjective discount factor over future debt ratios)
  - ω2,t = t κ1−1 exp(−t / κ2) / (κ κ1 2 Γ(κ1)) (asymmetric bell-shaped)
- Government budget constraint: Bt κt Rt = Bt−1 − (τt Yt − Gt)
- Tax feedback rule:
  - log(τt / ̄τ) = γτ,g log(gt / ̄g) + γτ,b log(bt / ̄b)
- Interpretation: ρg,y positive reflects pro-cyclical government spending; ρg,b captures aversion to future debt deviations (spending reversals).

### Theoretical results

#### Calibration (baseline)
- Period = 1 year.
- Parameters set as:
  - β = 0.95
  - σ = 1
  - Frisch elasticity of labor supply = 0.75
  - μ chosen so steady-state hours = 0.25
  - δ = 0.1
  - α = 0.36
  - Steady-state government spending share = 0.2
  - Steady-state debt share = 0.4
- Tax rates set to steady-state levels to satisfy government budget constraint.

#### Impulse-response functions and mechanisms
- Simulated shock: one percentage point decline in productivity; three persistence scenarios (ρA = 0.25, ρA = 0.75, permanent) and two fiscal rule variants (with and without debt stabilization; with ω1,t or ω2,t).
- Key dynamics without debt-sensitive rule:
  - Transitory shocks produce short-lived output blips; negative technology shocks yield opposite signs for output and government spending.
  - As shock becomes less persistent, conditional correlation between government spending and output approaches -1.
- Key dynamics with debt-sensitive rule:
  - For mild persistence (low ρA) and with debt reaction, output response similar to no-debt case but debt ratio increases sharply on impact. Example: output declines by about 1.4 percent and debt overshoots by 2.5 percent on impact in one scenario.
  - For high persistence (higher ρA), government spending can switch from positive to negative response (i.e., from counter- to pro-cyclical) because persistent output decline raises debt ratio and future debt expectations dominate stabilization motives.
  - Discounting scheme matters: ω1,t (declining weights) produces lower counter-cyclicality than ω2,t (bell-shaped) because near-future debt aversion induces sharper spending reversal when shocks persist.
- Conditional correlations generated by the calibrated model (Table 4):
  - Corr(Δgt, Δyt | ε t t ) and Corr(Δgt, Δyt | εp t ):
    - Without debt: -0.983 (ρA=0.25) ; -0.980 (ρA=0.75) ; -0.979 (permanent)
    - With debt, ω1,t: -0.655 (ρA=0.25) ; -0.157 (ρA=0.75) ; 0.229 (permanent)
    - With debt, ω2,t: -0.837 (ρA=0.25) ; -0.350 (ρA=0.75) ; 0.242 (permanent)
  - Interpretation: model generates counter-cyclicality for temporary shocks when government reacts only to output; including debt expectations reduces counter-cyclicality and can yield pro-cyclicality for persistent/permanent shocks.
- Sensitivity analysis (ρg,y and ρg,b):
  - Conditional correlation Corr(Δgt, Δyt | ε t t ) and Corr(Δgt, Δyt | ε p t ) vary with ρg,y and ρg,b; introducing debt in the rule dampens negative correlation for temporary shocks and can shift sign for persistent/permanent shocks depending on elasticity values.

#### Filters and mismeasurement of cyclicality
- Simulated exercise: apply HP filter (λ = 100) and growth-rate correlations to model-generated series under varying σp/σt and ρg,b.
- Finding: HP-filter and growth-rate unconditional correlations can deviate substantially from the true model-based conditional correlations in regions where trend shocks are more volatile (higher σp/σt). In such cases, filters may produce pro-cyclical co-movement between government spending and output even though the true conditional relation for temporary shocks is counter-cyclical.
- Implication: in countries with more volatile stochastic trends (e.g., many developing countries), HP-filter-based assessments may mistakenly classify fiscal policy as pro-cyclical.

### Conclusion (key takeaways)
- Empirical decomposition distinguishing permanent and temporary components of GDP yields:
  - Government spending reacts positively to permanent shocks (β2 > 0) and negatively to temporary shocks (β3 < 0).
  - Considerable dispersion across countries in magnitudes and signs of responses; no simple relation with income per capita.
- Theory: an RBC model with an endogenous fiscal rule responding both to output gaps and to the expected discounted sum of future debt explains why governments tend to be counter-cyclical to temporary shocks and pro-cyclical to permanent shocks.
- Mechanisms affecting cross-country dispersion:
  - Degree of aversion to future debt levels (ρg,b)
  - Endogenous persistence (habit formation, costly investment adjustment)
  - Tax-rule responsiveness
  - Learning about shock source explains little of the dispersion.
- Policy and measurement implication: accounting for stochastic trends and structural breaks is crucial — standard HP-filter approaches can mismeasure cyclicality, particularly for countries with volatile permanent shocks.

*Source: IMF Working Paper — chapter "16.    Transitory shock: U.S. versus Morocco" (extracted text).*

### REFERENCES

### REFERENCES

### A.1. Learning
- Setup:
  - Agents observe composite productivity process Zt = log(At) − log(At−1) + εzt, where εzt ∼ N(0, σz).
  - Composite signal can be written Zt = (1−α)εpt + log(Att) − log(Att−1) + εzt.
  - Kalman filter / state-space formulation (measurement matrix M and transition matrix N shown in text).
  - Error covariance V solves V = NVN′ − NVM′(MVM′)−1MVN′ + OΩO′, where Ω is covariance of εt.
  - Kalman gain corresponds to [VM′(MVM′)−1].
- Results and implications:
  - For very low signal noise (σz → 0), Zt fully reveals components and model features perfect information.
  - Impulse responses to a −1 percent transitory and permanent shock are plotted for perfect and imperfect information (Figure 10).
  - Introduction of the learning mechanism "adds little to generate change in the degree of cyclicality." For transitory shocks, impulse responses are qualitatively similar under perfect and imperfect information.
  - Following a trend (permanent) shock, learning reduces long-run responses because agents partially interpret the permanent shock as temporary (Figure 11).
  - Government spending reacts similarly under imperfect information and full information owing to smaller expected effect of debt in following periods.
  - Overall: learning can exacerbate counter-cyclicality under imperfect information but the effect is "very limited."
  - Conclusion: learning alone "does not seem to be a sufficient candidate to interpret the heterogeneous counter-cyclicality across countries" because impulse-response functions are stable following a temporary shock and conditional correlation between output and government spending is moderately affected.

### A.2. Government degree of impatience
- Experiment:
  - Vary parameter κ1 (degree of government patience with respect to future debt fluctuations) with values 5 and 20 (less versus more patient government). Discount-function profiles shown in Figure 12.
- Findings:
  - A negative persistent productivity shock leads to overshooting of government spending when κ1 = 20 since the government cares less about medium term levels of debt.
  - Adjustment path of debt is delayed when κ1 is high because debt aversion is relatively low.
  - Countries with high levels of κ1 are expected to witness a high degree of counter-cyclical behavior.
  - In the case of a permanent negative productivity shock, impulse-response functions of output and government spending are "not very sensitive to the parameter κ1" in the short term; higher κ1 generates higher persistence in the model.
- Note:
  - Footnote: "Although we do not explicitly model a risk premium in this model, an impatient government in this context may reflect this assumption."

### A.3. Endogenous persistence (real rigidities)
- Model extensions:
  - Habit formation in consumption: period utility u(t) = (Ct − ψCt−1)1−σ/(1−σ) − ηLt1+μ/(1+μ), where ψ is degree of habit formation.
  - Investment adjustment costs: Kt+1 = (1−δ)Kt + [1 − Γ(It/It−1)]It, with Γ(1) = 0, Γ′(1) = 0, and χ = Γ′′(1) > 0.
- Mechanism:
  - Habit formation and convex investment adjustment costs amplify sluggishness of real-variable responses.
  - Following a transitory productivity shock, households prefer to smooth via investment, but costly investment leads them to consume more leisure in short term, generating reversed hump-shaped output response with mild impact.
  - Real rigidities can raise government concerns about debt sustainability and discourage counter-cyclical spending after a transitory shock.
- Quantitative illustration:
  - Figure 14 shows responses for (a) ψ = 0 and χ = 0 and (b) ψ = 0.8 and χ = 10.
  - With "reasonably high" habit and adjustment-cost parameters, model yields a positive conditional correlation between output and government spending—contrasting with baseline calibration.
  - Parameter choices noted: "The parameters of habit formation and investment adjustment cost are set to 0.8 and 10, respectively; which are slightly in the upper side of the usual calibration or estimation found in the literature."
- Conclusion:
  - Endogenous real rigidities are a plausible source of heterogeneity in observed fiscal cyclicality; extended model with these mechanisms "can explain the heterogeneity in the degree of cyclicality observed in the empirical analysis."

### A.4. Endogenous tax rules
- Setup:
  - Tax rates modeled as policy feedback rules (function of government spending and debt shares of total output) as in Galí, López-Salido, and Vallés (2007).
  - Compare exogenous tax rates (γτ,g = γτ,b = 0) versus endogenous reactive taxes (γτ,g = γτ,b = 0.5) (Figure 15).
- Findings:
  - Negative transitory shock impact on output is exacerbated because tax rates increase in reaction to higher government spending and immediate debt reactions.
  - Government spending becomes more persistent when taxes are debt stabilizing.
  - Conditional correlation between government spending and total output rises as tax rates become reactive to fiscal variables.
  - The same qualitative result holds following a permanent shock.
- Conclusion:
  - A negative comovement of government spending and total output requires a sufficiently high response of taxes to debt; countries exhibiting more counter-cyclicality may have a higher coefficient on debt in the tax rule.

### A.5. Model-based country comparison (United States versus Morocco)
- Estimation approach:
  - Model estimated using Bayesian techniques; Metropolis-Hastings algorithm with Kalman filter to evaluate likelihood.
  - Annual U.S. data used for 1970–2008. Gross government debt-to-GDP ratios taken from Abbas and others (2010) database.
  - Calibration: subjective discount factor = 0.95; δ = 10 percent; μ adjusted so steady-state hours = 0.25; η = 0.75.
  - Priors: autocorrelations beta(mean = 0.5, s.d. = 0.15); shock s.d. inverse gamma(mean = 0.005, s.d. = 4); government spending rule coefficients uniform(mean = 0, s.d. = 1); distortionary tax rule priors mean = 0, s.d. = 0.25; habit formation prior mean = 0.5, s.d. = 0.15; investment adjustment cost prior normal(mean = 10, s.d. = 5).
  - Estimation details: 200,000 draws, first 40,000 discarded; acceptance rate ~25 percent.
- Likelihood / model comparison:
  - Log-likelihood: 208.733 (US benchmark), 205.287 (US exogenous spending), 120.008 (Morocco benchmark), 116.187 (Morocco exogenous spending).
  - Bayes factor: marginal data density of benchmark model is 3.45 and 3.82 higher on a log-scale for the US and Morocco, respectively — posterior odds favor model with endogenous government spending rule.
- Selected posterior estimates (posterior mode and 90 percent intervals from Table 5; units and parameter labels preserved):
  - Real rigidity parameters
    - ψ (habit formation in consumption): posterior mean 0.2656 [0.111, 0.413] (United States, benchmark model).
    - χ (capital adjustment cost): posterior mean 13.870 [5.326, 22.155] (United States, benchmark model).
  - Fiscal policy parameters
    - ρg,y (gov. spending reaction to output): posterior mean 1.251 [0.784, 1.732] (United States, benchmark model); 0.820 [0.081, 1.731] (Morocco, benchmark model).
    - ρg,b (gov. spending reaction to debt): posterior mean 0.636 [0.035, 1.314] (United States, benchmark model); 1.251 [0.206, 2.327] (Morocco, benchmark model).
    - κ1 (debt discount function): posterior mean 14.462 [6.051, 23.844] (United States, benchmark model); 11.435 [5.230, 17.704] (Morocco, benchmark model).
    - γτ,g (elasticity of taxes to gov. spending): posterior mean 0.559 [0.317, 0.797] (United States, benchmark model); 0.220 [-0.090, 0.511] (Morocco, benchmark model).
    - γτ,b (elasticity of taxes to debt ratio): posterior mean 0.245 [0.060, 0.423] (United States, benchmark model); 0.362 [0.167, 0.543] (Morocco, benchmark model).
  - Exogenous processes and shock s.d. (selected)
    - ρAt (AR term in temp. technology): United States 0.299 [0.165, 0.431]; Morocco 0.219 [0.108, 0.334].
    - ρg (AR term in gov. spending): United States 0.395 [0.222, 0.562]; Morocco 0.339 [0.155, 0.514].
    - σp (σ perm. technology shock): United States 0.059 [0.044, 0.073]; Morocco 0.134 [0.103, 0.166].
    - σt (σ temp. technology shock): United States 0.020 [0.015, 0.025]; Morocco 0.042 [0.032, 0.052].
    - σg (σ gov. spending shock): United States 0.030 [0.023, 0.036]; Morocco 0.106 [0.080, 0.129].
    - σs (σ technology signal noise): United States 0.006 [0.001, 0.013]; Morocco 0.005 [0.001, 0.010].
- Comparative interpretation (model-implied):
  - US government estimated features (relative to Morocco):
    - More reactive to output fluctuations: ρg,y ≈ 1.28 (US) versus 0.82 (Morocco).
    - Less reactive to debt fluctuations: ρg,b ≈ 0.64 (US) versus 1.25 (Morocco).
    - More patient / larger κ1: κ1 ≈ 14.46 (US) versus 11.435 (Morocco); "government highly cares about debt changes after more than 14 years" (US) versus "only 11 years for Morocco."
  - Implications:
    - Government spending is more negatively correlated with output in the US than in Morocco.
    - Posterior mode correlation between government spending and output: −0.6 (US) versus −0.4 (Morocco).
    - Morocco’s government is more concerned with bringing debt back to pre-shock steady state, producing less counter-cyclical spending responses.

*Italic: Source PDF filename: _wp1659 - REFERENCES*

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