## wpiea2020246-print-pdf

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---

### Data
- Dataset and sample:
  - Narrative record constructed by Devries, Guajardo, Leigh, and Pescatori (2011) and extended by Alesina, Favero and Giavazzi (2015).
  - Initial sample of countries: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, the Netherlands, Portugal, Spain, Sweden, United Kingdom, United States.
  - Sample period: 1980 to 2014.
  - After exogeneity tests three countries are dropped: Finland, Netherlands and Sweden.
  - Germany is dropped because data are available after 1991 due to unification and this restricts analysis.
  - Final sample includes a total of 74 episodes for taxes and 101 episodes for government spending.
  - Frequency of observations: annual.
  - Primary data source for fiscal and macro variables: OECD.
  - Government debt series reference: general government debt as percent of GDP from the WEO of the IMF.
- Narrative identification and consolidation classification:
  - Fiscal alterations measured as a percentage of GDP.
  - Focus restricted to fiscal changes exogenous to the economic cycle and motivated by willingness to reduce government deficit; countercyclical policies excluded.
  - Classification: tax-based (TB) versus expenditure-based (EB) consolidations; TB (EB) means adjustments mainly based on tax increases (spending cuts).
  - Only unanticipated and anticipated legislative announcements implemented the same year are considered; future announcement paths are not included.
  - Average horizons: TB plans last around 1.5 years, and EB plans 1.8 years.
- Main macroeconomic and policy variables used (baseline specification):
  - real GDP growth
  - change of log real government spending as a fraction of GDP (primary government spending: total government spending net of interest payments on debt)
  - change of log real government revenues (current receipts) as a fraction of GDP
  - average cost of debt
  - inflation
- Data limitations:
  - Legislative announcements are infrequent, producing many years with no announcement (many zeros).
  - Excluding future announcements reduces additional sparsity because most plans have short horizons.

### Non-Linearities (overview)
- Three non-linear channels:
  - State of the economy: recessions versus expansions.
  - Composition of fiscal consolidation: tax-based versus expenditure-based.
  - Government’s budget constraint / initial debt condition: high-debt ratio versus low-debt ratio.
- Modeling approach:
  - Interacted smooth transition vector autoregression (ISTVAR): blends STVAR and IPVAR.
  - ISTVAR conditions endogenously on government debt and examines instrument relevance (tax versus spending) while accounting for state of the economy.
  - Debt dynamics modelled endogenously as evolution of the government debt-to-GDP ratio as a function of interest payments and the primary government deficit.

### State of the economy (regime indicator and calibration)
- Regime indicator: F(z_it) = exp[−γ_i z_it] / (1 + exp[−γ_i z_it]), γ_i >0.
- s_it = (∆y_it−1 + ∆y_it−2) / 2 (two-years moving average of GDP growth).
- Business cycle index: z_it = (s_it − E(s_it)) / σ(s_it).
- γ_i controls smoothness: larger γ_i → immediate switches; smaller γ_i → smoother transition.
- Calibration target: Pr((z_it) > 0.8) = x_i to match OECD recession frequency.
  - Example: For the US Pr((z_it) > 0.8 = 0.2) and set γ_i = 1.5.
- Average duration in recessionary regime: 18% of time.
- Calibration table (country — recession frequency — γ):
  - AUS 14% 1.14
  - AUT 14% 1.53
  - BEL 14% 1.13
  - CAN 17% 1.09
  - DNK 19% 1.72
  - ESP 25% 1.70
  - FRA 14% 1.59
  - GBR 19% 1.43
  - IRL 14% 1.68
  - ITA 22% 2.24
  - JPN 17% 1.65
  - PRT 22% 1.60
  - USA 17% 1.56

### Type of fiscal consolidation
- Classification rule:
  - Adjustment is "tax-based" if sum of unanticipated and anticipated tax changes > sum of unanticipated and anticipated government spending changes.
  - Adjustment is "expenditure-based" if sum of unanticipated and anticipated government spending changes > sum of unanticipated and anticipated tax changes.
- Approach follows Alesina, Favero and Giavazzi (2015) to account for correlation between nature of changes.

### Government debt identity and channels
- Debt identity (Favero and Giavazzi (2012) approach):
  - Debt_it = (1 + i_it) / ((1 + π_it) (1 + ∆y_it)) · Debt_it−1 + (exp(g_it) − exp(τ_it)).
  - i = average cost of government debt; π = inflation rate; g = primary government spending as fraction of GDP; τ = government revenues as fraction of GDP.
- Decomposition:
  - a. Snowball effect: ( (1 + i_it) / ((1 + π_it) (1 + ∆y_it)) ) Debt_it−1
  - b. Primary balance effect: (exp(g_it) − exp(τ_it))
- Illustrative mechanism:
  - Example: Government reduces expenditure by 1% of GDP → negative output effect and decrease in government spending.
    - Snowball channel (a): decrease in output growth → for given past debt and given i → increase in debt-ratio.
    - Primary balance channel (b): expenditure reduction improves primary balance → reduces debt-ratio.
  - Net debt response depends on which channel dominates.
- Observed vs implicit debt:
  - Simulated implicit series track observed Debt-to-GDP reasonably well; differences arise from seigniorage omission, stock-flow adjustments, measurement error, or approximation from log transformations.

### Key empirical findings and conclusions
- Evidence of state-dependency in fiscal multipliers.
- Debt level functions as a channel explaining heterogeneity in responses to fiscal consolidations.
- When debt is high:
  - Expenditure-based consolidations are more effective in stabilizing the debt-to-GDP ratio.
  - Tax-based consolidations appear self-defeating: they deliver on average higher debt ratios because negative effect on GDP growth is larger.
  - Expenditure cuts stabilize debt independently of state of the cycle.
- Heterogeneity in prior literature (e.g., TB more contractionary than EB) may be driven by periods of low debt; including debt clarifies channels.
- Omitting debt can bias evaluation of output effects of fiscal policy; including debt allows assessment of fiscal sustainability success depending on initial debt, instrument, and state of the economy.

### Model, identification, and empirical strategy
- Identification:
  - Narrative identification of unanticipated and anticipated fiscal actions using Budget Reports, Budget Speeches, Central Bank Reports, Convergence and Stability Programs, IMF Reports, OECD economic surveys.
- ISTVAR builds on:
  - STVAR: regime-switching model with logistic transition (state of cycle).
  - IPVAR: interaction with debt via interaction term.
- Generalized impulse response functions (GIRF) used to derive dynamic responses; fiscal multipliers computed as ratio of integral of output response to integral of policy adjustment.
- Estimation approaches referenced: seemingly unrelated regressions or maximum likelihood.

### ISTVAR specification
- Y_it = (1 − F(z_it)) × [A^E Y_it−1 + Θ^E Debt_it−1 + B^1E e^EB_it + B^2E e^TB_it]
         + F(z_it) × [A^R Y_it−1 + Θ^R Debt_it−1 + B^1R e^EB_it + B^2R e^TB_it]
         + λ_i + χ_t + u_it
- B_j^S = B_S^0 + B_S^1 · Debt_it−1, for S = E, R and j = 1,2.
- Debt_it follows identity in section 3.2.3; F(z_it) as defined with γ_i > 0.
- Variable vector: Y = [∆y ∆τ ∆g i π], where ∆y = GDP growth, ∆τ = change of government revenues (as fraction of GDP), ∆g = change of government spending (as fraction of GDP), i = average cost of government debt, π = inflation rate.
- Fixed effects: λ_i = country fixed effects; χ_t = time fixed effects; u_it ~ N(0, Σ_u).
- Policy variable Debt treated as identity (no error term).
- Narrative shocks:
  - e^EB_it and e^TB_it are narratively identified shocks (unanticipated and anticipated shocks implemented same year).
  - e^EB_it = e^IMF_it · EB_it; e^TB_it = e^IMF_it · TB_it.
  - e^IMF_it = total adjustment; EB_it and TB_it are dummies.
- Linear model is special case for γ = 0.
- Reported initial values for scenarios: Recessionary versus expansionary regime approx. 0.8 versus 0.2. High versus low debt ratio: 0.3 and 0.9.

### Generalized Impulse Response Functions (GIRF) and inference
- GIRF_∆y(h, Ω_t−1, shock_t) = E(∆y_t+h | Ω_t−1, shock_t = 1) − E(∆y_t+h | Ω_t−1, shock_t = 0)
- Simulation steps:
  - Step 1: Simulate forward with structural shock of interest = 1 and others = 0.
  - Step 2: Simulate forward with all shocks = 0.
  - Step 3: Impulse response = difference between Step 1 and Step 2.
  - Step 4: Correlated bootstrap for confidence intervals.
- Bootstrap and confidence intervals:
  - Confidence intervals reported: 16-84%.
  - Bootstrap re-samples residuals of estimated non-linear VAR allowing residual correlation across countries; re-estimate model and derive GIRFs.
  - Number of bootstrap iterations: 1000.
- Scenarios evaluated (2^3 combinations), examples include:
  - TB shock in recession (F(z) = 0.8) when debt is high (0.9);
  - EB shock in recession (F(z) = 0.8) when debt is high (0.9);
  - TB shock in expansion (F(z) = 0.2) when debt is high (0.9);
  - EB shock in expansion (F(z) = 0.2) when debt is high (0.9);
  - TB shock in recession (F(z) = 0.8) when debt is low (0.3); and other combinations completing 2^3 set.

### The Interacted-STVAR: Model and setup
- I-STVAR allows endogenous transition F(z) and endogenous feedback of the debt-ratio.
- Initial illustrative debt states: Debt low = 30%, Debt high = 90%.
- Results represent average country in sample (Japan dropped).
- Convergence targets:
  - Low-debt specification: economy spends on average 20% of time in a recessionary regime.
  - High-debt specification: economy converges to probability of being 50-60% in a recessionary regime.

### Impulse response results — High Debt (Debt = 90%)
- Tax-based consolidation (TB) when debt is high:
  - Tax increases are self-defeating both in recessions and expansions.
  - Tax shock increases public debt, which remains on an upward trajectory.
  - Output growth falls on impact; even if recovery signs after one year, economy remains in recessionary regime.
- Expenditure-based consolidation (EB) when debt is high:
  - Effects depend on business cycle:
    - In recession: effect on output is negative.
    - In expansion: effect on output is not statistically different from zero.
  - Expenditure-based adjustments stabilize debt independently of cycle.
- Qualitative summary: under high debt, TB can raise debt and deepen recessionary dynamics; EB provides stabilizing feedback to debt.

### Impulse response results — Low Debt (Debt = 30%)
- Tax-based consolidation (TB) when debt is low:
  - Effects on output from tax increases are state-dependent.
  - In low-debt regime, TB implemented in recessions lead to increases in debt-to-GDP ratio.
  - TB implemented in boom periods have the most recessionary effect (statistically different from same consolidations in recessions).
- Expenditure-based consolidation (EB) when debt is low:
  - EB stabilize or decrease debt-to-GDP within five horizons, independently of cycle.
  - Effects on output are less harmful than TB.
- Qualitative summary: in low-debt regimes EB effective at stabilizing/reducing debt with milder output costs; TB more state-dependent and can be recessionary when implemented in booms or increase debt when implemented in recessions.

### Output multipliers (I-STVAR estimates)
- High-Debt Regime (Table 2):
  - TB Impact, Expansion: −1.22 (−1.53,−0.93)
  - TB Impact, Recession: −0.65 (−0.89,−0.40)
  - TB 5-year cumulative, Expansion: −1.02 (−1.81,−0.36)
  - TB 5-year cumulative, Recession: −0.68 (−1.16,−0.16)
  - EB Impact, Expansion: −0.046 (−0.23,0.18)
  - EB Impact, Recession: −0.74 (−0.84,−0.61)
  - EB 5-year cumulative, Expansion: −0.042 (−0.40,0.38)
  - EB 5-year cumulative, Recession: −1.12 (−1.48,−0.70)
- Low-Debt Regime (Table 3):
  - TB Impact, Expansion: −1.05 (−1.40,−0.73)
  - TB Impact, Recession: −0.67 (−0.88,−0.51)
  - TB 5-year cumulative, Expansion: −2.42 (−3.27,−1.58)
  - TB 5-year cumulative, Recession: −0.108 (−0.58,0.28)
  - EB Impact, Expansion: −0.21 (−0.56,0.09)
  - EB Impact, Recession: −0.75 (−0.90,−0.60)
  - EB 5-year cumulative, Expansion: −0.69 (−1.41,−0.14)
  - EB 5-year cumulative, Recession: −0.97 (−1.35,−0.63)
- Note: shocks are not pure spending-only or tax-only; constructed multipliers capture direct effects but disentangling indirect effects through revenues or spending is not straightforward.

### Mechanisms and interpretation
- Two factors explaining increases in public debt after TB, especially in recessions:
  1. Negative effect on output growth.
  2. Contemporaneous increase of government spending that offsets higher revenues’ positive effect on primary balance.
- Omitted-variable bias example:
  - If true model includes lagged debt with coefficient β < 0, and e_IMF_t = κ·Debt_{t−1} + υ_t with κ > 0, omitting Debt_{t−1} biases estimated effect of e_IMF_t on ∆y_t via (β/κ + γ). Including debt is important to avoid overestimating fiscal effects.

### Policy implications
- Expenditure-based adjustments:
  - Tend to harm the economy less.
  - Are effective at stabilizing and sometimes reducing debt-to-GDP.
  - A cut in expenditure may reduce distortionary need for taxation and imply a smaller negative demand shock on GDP growth.
- Tax-based adjustments:
  - More distortionary and, on average, most recessionary.
  - When debt is high, increasing taxes fails to stabilize debt-ratio and can be self-defeating.
- Overall: the initial condition — level of debt — plays a relevant role in propagation of consolidation instruments; policy design should account for debt level, composition of adjustment (tax vs expenditure), and business cycle phase.
- Suggested further work: evaluate transmission channels including role of monetary policy, particularly when interest rates are close to the zero lower bound.

### Robustness and diagnostics
- Linearity tests:
  - LM-type test (Terasvirta and Yang, 2014) and likelihood ratio tests favor non-linear (STVAR) over linear VAR.
  - Information criteria: AIC and BIC lower for non-linear model (AIC 4.15 vs 4.06; BIC 4.61 vs 4.5).
- Exogeneity of narrative shocks:
  - Granger causality tests across countries/components show in most cases past output does not predict narrative measures; Sweden and the Netherlands dropped because null could not be rejected.
- Additional figures: cumulative GIRFs for linear VAR, STVAR, and fiscal STVAR with low/high debt confirm qualitative patterns for output, F(z), and debt responses.

*Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020246-print-pdf.pdf*

### 3.1    Data  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 

### 3.1    Data

### Dataset and sample
- Narrative record constructed by Devries, Guajardo, Leigh, and Pescatori (2011) and extended by Alesina, Favero and Giavazzi (2015).
- Initial sample of countries: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, the Netherlands, Portugal, Spain, Sweden, United Kingdom, United States.
- Sample period: 1980 to 2014.
- After exogeneity tests three countries are dropped: Finland, Netherlands and Sweden.
- Germany is dropped because data are available after 1991 due to unification and this restricts analysis.
- Final sample includes a total of 74 episodes for taxes and 101 episodes for government spending.
- Frequency of observations: annual.
- Primary data source for fiscal and macro variables: OECD.
- Government debt series reference: general government debt as percent of GDP from the WEO of the IMF.

### Narrative identification and consolidation classification
- Fiscal alterations are measured as a percentage of GDP.
- Focus restricted to identification of fiscal changes that are exogenous to the economic cycle and motivated by willingness to reduce government deficit; countercyclical policies are excluded.
- Classification: tax-based (TB) versus expenditure-based (EB) consolidations; tax-based (expenditure-based) means the total of adjustments is mainly based on tax increases (spending cuts).
- Only unanticipated and anticipated legislative announcements implemented the same year are considered; future announcement paths are not included.
- Average horizons mentioned: TB plans last around 1.5 years, and EB plans 1.8 years.

### Main macroeconomic and policy variables used (baseline specification)
- real GDP growth
- change of log real government spending as a fraction of GDP (primary government spending: total government spending net of interest payments on debt)
- change of log real government revenues (current receipts) as a fraction of GDP
- average cost of debt
- inflation

### Data visualizations referenced
- Figure 1: Country-specific narrative unanticipated (blue) and anticipated (red) fiscal adjustments. Sample coverage (x-axis), Size of the fiscal adjustments (y-axis).
- Figure 2: EB (blue) and TB (grey) episodes and the per capita GDP growth series (black line).
- Figure 3: The distribution of fiscal position during the period 1980-2014. Debt-to-GDP ratio (x-axis), Frequency in % (y-axis).
- Figure 4: The distribution of TB and EB fiscal consolidations during the period 1980-2014. Fiscal consolidations (x-axis), Frequency in % (y-axis).

### Notes on narrative data limitations
- Legislative announcements are infrequent, producing many years with no announcement (many zeros in the data).
- Excluding future announcements reduces additional sparsity because most plans have short horizons (see average horizons above).

---

### 3.2    Non-Linearities (overview introduced in Section 1 and 2)

### Three non-linear channels the paper focuses on
- State of the economy: recessions versus expansions.
- Composition of fiscal consolidation: tax-based versus expenditure-based.
- Government’s budget constraint / initial debt condition: high-debt ratio versus low-debt ratio (novel channel in this literature).

### Modeling approach to capture non-linearities
- Proposes the interacted smooth transition vector autoregression (ISTVAR) model: a blend of the smooth transition VAR (STVAR) used in the state-dependent literature and the interacted panel VAR (IPVAR) that conditions on debt.
- ISTVAR conditions endogenously on countries’ government debt and examines the relevance of the stabilization instrument (tax versus spending) while accounting for the state of the economy.
- Debt dynamics are explicitly modelled endogenously as the evolution of the government debt-to-GDP ratio as a function of interest rate payments on the debt and the primary government deficit (Favero and Giavazzi (2012)).
- General encompassing framework presented: Yt = f1(Yt−1, Pt−1, shockt; Φ1) + u1t and Pt = f2(Yt−1, Pt−1, shockt; Φ2) + u2t, where Yt are macro variables and Pt are policy variables (debt-to-GDP used as main policy variable).

---

### Key empirical findings and conclusions (as reported)

- Evidence of state-dependency in fiscal multipliers.
- The level of debt functions as a channel explaining heterogeneity in responses to fiscal consolidations seen in prior literature.
- When debt is high:
  - Expenditure-based consolidations are more effective in stabilizing the debt-to-GDP ratio.
  - Tax-based consolidations appear to be self-defeating: they deliver on average higher debt ratios because the negative effect on GDP growth is larger from the budget changes.
  - Expenditure cuts are able to stabilize debt independently of the state of the cycle.
- Heterogeneity documented in prior literature (e.g., that tax-based consolidations are more contractionary than spending-based ones) may be driven by periods of low-debt; including debt clarifies these channels.
- Omitting debt can bias evaluation of output effects of fiscal policy; including debt allows assessment of whether fiscal authority succeeds in improving fiscal sustainability depending on initial debt, instrument of stabilization, and state of the economy.

---

### Model, identification, and empirical strategy (summary)

- Identification approach: narrative identification of unanticipated and anticipated fiscal actions (based on historical official documents: Budget Reports, Budget Speeches, Central Banks Reports, Convergence and Stability Programs, IMF Reports, OECD economic surveys).
- ISTVAR builds on:
  - STVAR: regime-switching model with logistic distribution controlling transition between regimes (state of the cycle).
  - IPVAR: interaction with debt via an interaction term.
- Generalized impulse response functions are used to derive dynamic responses; fiscal multipliers computed as ratio of integral of output response to integral of policy adjustment.
- Estimation approaches referenced: seemingly unrelated regressions or maximum likelihood for the general model.

---

*Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020246-print-pdf.pdf*

### 3.2    Non-Linearities

### 3.2    Non-Linearities

### 3.2.1    State of the Economy
- Regime indicator: F(z_it) = exp[−γ_i z_it] / (1 + exp[−γ_i z_it]), γ_i >0.
- s_it is the two-years moving average of GDP growth: s_it = (∆y_it−1 + ∆y_it−2) / 2.
- Business cycle index: z_it = (s_it − E(s_it)) / σ(s_it).
- γ_i controls smoothness of transitions; larger γ_i → immediate switches; smaller γ_i → smoother transition.
- Calibration target: Pr((z_it) > 0.8) = x_i to match OECD recession frequency.
  - Example: For the US Pr((z_it) > 0.8 = 0.2) and set γ_i = 1.5.
- OECD recession dates: quarterly, not seasonally adjusted, dummy (1: recession, 0: expansion); converted to yearly recession series to match yearly narrative shocks.
- Average duration that an average economy spends in a recessionary regime is 18% of its time.
- Calibration note: years with all quarters recessionary considered extreme recessions; years with half quarters recessionary sometimes randomly classified to account for weak recessions.

- Figure referenced: comparison of constructed transition series F(z) to OECD recession dates for 13 economies (shaded grey = extreme and weak recessions; black line = F(z)).

- Table 1: Calibration of smoothness parameter γ
  - AUS 14% 1.14
  - AUT 14% 1.53
  - BEL 14% 1.13
  - CAN 17% 1.09
  - DNK 19% 1.72
  - ESP 25% 1.70
  - FRA 14% 1.59
  - GBR 19% 1.43
  - IRL 14% 1.68
  - ITA 22% 2.24
  - JPN 17% 1.65
  - PRT 22% 1.60
  - USA 17% 1.56

### 3.2.2    Type of Fiscal Consolidation
- Fiscal adjustment composition classified using narrative record (Devries, Guajardo, Leigh, and Pescatori (2011)):
  - Entire fiscal adjustment comprises tax changes and government spending changes together.
  - An adjustment is "tax-based" if the sum of unanticipated and anticipated tax changes > sum of unanticipated and anticipated government spending changes.
  - An adjustment is "expenditure-based" if the sum of unanticipated and anticipated government spending changes > sum of unanticipated and anticipated tax changes.
- Approach follows Alesina, Favero and Giavazzi (2015) to account for correlation between nature of changes.
- Figure referenced: Debt-to-GDP ratio (Figure from the OECD website) for 1995-2014 showing country comparisons.

### 3.2.3    Government Debt
- Debt identity (Favero and Giavazzi (2012) approach):
  - Debt_it = (1 + i_it) / ((1 + π_it) (1 + ∆y_it)) · Debt_it−1 + (exp(g_it) − exp(τ_it)).
  - i stands for the average cost of government debt; π is the inflation rate; g is primary government spending as fraction of GDP; τ is government revenues as fraction of GDP.
  - exp() used because g and τ are in logarithms.
- Decomposition:
  - a. Snowball effect: ( (1 + i_it) / ((1 + π_it) (1 + ∆y_it)) ) Debt_it−1
  - b. Primary balance effect: (exp(g_it) − exp(τ_it))
- Illustrative mechanism:
  - Example: Government reduces expenditure by 1% of GDP → negative output effect and decrease in government spending.
    - Snowball channel (a): decrease of output growth → for given past debt and given i → increase in debt-ratio.
    - Primary balance channel (b): expenditure reduction improves primary balance → reduces debt-ratio.
  - Net debt response depends on which channel dominates.
- Observed vs. implicit debt:
  - Figure referenced: Observed Debt-to-GDP (data) versus the simulated (implicit) series using equation (3); good tracking but differences due to seigniorage omission, stock-flow adjustments, measurement error, or approximation from log transformations.
  - Country-specific data issues (example: Australia combined data sources may cause mismatch).
  - Stock-flow adjustments common in countries with average budget surpluses and lower debt levels (example: Denmark).

### 4    Model Specification

#### 4.1    An Interacted Smooth Transition VAR
- Model name: Interacted Smooth Transition Vector Autoregression (ISTVAR).
- ISTVAR specification:
  - Y_it = (1 − F(z_it)) × [A^E Y_it−1 + Θ^E Debt_it−1 + B^1E e^EB_it + B^2E e^TB_it]
           + F(z_it) × [A^R Y_it−1 + Θ^R Debt_it−1 + B^1R e^EB_it + B^2R e^TB_it]
           + λ_i + χ_t + u_it
  - B_j^S = B_S^0 + B_S^1 · Debt_it−1, for S = E, R and j = 1,2.
  - Debt_it follows the identity in section 3.2.3.
  - F(z_it) as defined previously with γ_i > 0.
- Variable vector:
  - Y = [∆y ∆τ ∆g i π], where
    - ∆y = GDP growth
    - ∆τ = change of government revenues (as fraction of GDP)
    - ∆g = change of government spending (as fraction of GDP)
    - i = average cost of government debt
    - π = inflation rate
- Fixed effects and errors:
  - λ_i = country fixed effects; χ_t = time fixed effects.
  - u_it ~ N(0, Σ_u).
- Policy variable Debt is treated as an “identity” (no error term included).
- Narrative shocks:
  - e^EB_it and e^TB_it are narratively identified shocks (unanticipated and anticipated shocks implemented same year).
  - Distinction between expenditure-based instrument: e^EB_it = e^IMF_it · EB_it
    and tax-based instrument: e^TB_it = e^IMF_it · TB_it.
  - e^IMF_it = total adjustment; EB_it and TB_it are dummies for expenditure-based or tax-based episodes.
- All macroeconomic variables depend on the cycle through f1, a weighted sum of the logistic function.
- Linear model is a special case of STVAR for γ = 0.
- Rationale for including debt:
  - Debt as “initial condition” affects model dynamics and transmission of consolidations.
  - Short lags of ∆τ, ∆g, i and π alone cannot trace debt evolution accurately; debt exhibits long and non-linear dynamics.
- Reported initial values for scenario comparisons:
  - Recessionary versus expansionary regime approx. 0.8 versus 0.2.
  - High versus low debt ratio: 0.3 and 0.9.
- Hypothesis testing examples:
  - Cycle: B^1E = B^1R ; B^2E = B^2R
  - Composition: B^1E = B^2E ; B^1R = B^2R
  - Initial Condition: B^E_0 = B^E_1 ; B^R_0 = B^R_1

#### 4.2    Generalized Impulse Response Functions
- Impulse responses computed using Generalized Impulse Response Functions (GIRF) (Koop, Pesaran and Potter 1996):
  - GIRF_∆y(h, Ω_t−1, shock_t) = E(∆y_t+h | Ω_t−1, shock_t = 1) − E(∆y_t+h | Ω_t−1, shock_t = 0)
  - Ω_t−1 is the history; h = 0,1,2,...,H; shock_t is either tax-based or expenditure-based narrative shock.
- Computation steps:
  - Step 1: Simulate forward assuming structural shock of interest (EB or TB) = 1 and all other shocks = 0.
  - Step 2: Simulate forward assuming all shocks = 0.
  - Step 3: Impulse response = difference between Step 1 and Step 2 simulated values.
  - Step 4: Run correlated bootstrap for confidence intervals.
- Bootstrap and inference:
  - Confidence intervals reported: 16-84%.
  - Bootstrap method: re-sample residuals of estimated non-linear VAR allowing for correlation between residuals of different countries; re-estimate model and derive GIRFs.
  - Number of bootstrap iterations: 1000.
- Scenarios evaluated (2^3 combinations noted):
  - TB shock in recession (F(z) = 0.8) when debt is high (0.9);
  - EB shock in recession (F(z) = 0.8) when debt is high (0.9);
  - TB shock in expansion (F(z) = 0.2) when debt is high (0.9);
  - EB shock in expansion (F(z) = 0.2) when debt is high (0.9);
  - TB shock in recession (F(z) = 0.8) when debt is low (0.3);
  - [and other combinations implied to complete 2^3 set].
- GIRFs allow endogenous transitions of F(z) and feedback between Debt and the regime, producing impulse responses that account for those dynamics.

*Source: wpiea2020246-print-pdf - 3.2    Non-Linearities*

### 5.1    The Interacted-STVAR

### 5.1    The Interacted-STVAR

### Model and setup
- Econometric specification: an Interacted Smooth-Transition VAR (I-STVAR) that allows endogenous transition F(z) and an endogenous feedback of the debt-ratio.
- Initial debt states used for illustrations: Debt is low at a value of 30%, whereas debt is high at the value of 90%.
- The results represent the behavior of the average country in the sample (Japan dropped from the study).
- The I-STVAR baseline equations include regime-specific lag dynamics, debt lag terms, and identified shocks e_EB and e_TB; the transition function F(z_it) = exp[−γ_i z_it] / (1 + exp[−γ_i z_it]), γ_i > 0.
- Convergence targets of the transition variable:
  - Low-debt specification: the economy spends on average 20% of the time in a recessionary regime.
  - High-debt specification: the economy converges to a more recessionary target with a probability of being 50-60% in a recessionary regime.

### Impulse response results — High Debt (Debt = 90%)
- Tax-based consolidation (TB) when debt is high:
  - Mainly tax increases are self-defeating both in recessions and expansions.
  - The tax shock increases public debt, which remains on an upward trajectory in subsequent horizons.
  - Output growth falls on impact and, even if there is a sign of recovery after one year, the economy remains in a recessionary regime.
- Expenditure-based consolidation (EB) when debt is high:
  - Effects depend on the state of the business cycle:
    - In recession: the effect on output is negative.
    - In expansion: effect on output is not statistically different from zero.
  - Expenditure-based adjustments appear effective in stabilizing debt independently of the state of the cycle.
- Qualitative summary: tax-based consolidations under high debt can raise debt and deepen recessionary dynamics; expenditure-based consolidations provide stabilizing feedback to debt.

### Impulse response results — Low Debt (Debt = 30%)
- Tax-based consolidation (TB) when debt is low:
  - Fiscal effects on output growth generated through increases in taxes are state-dependent.
  - In a low-debt regime, tax-based consolidations implemented in recessions lead to increases in the debt-to-GDP ratio.
  - Adjustments mainly composed through taxes implemented in boom periods have the most recessionary effect (statistically different from the same consolidations in recessions).
- Expenditure-based consolidation (EB) when debt is low:
  - Expenditure-based consolidations stabilize or even decrease the debt-to-GDP ratio within five horizons, independently of the state of the cycle.
  - Effects on output are less harmful than tax-based changes.
- Qualitative summary: in low-debt regimes, expenditure-based consolidations are effective at stabilizing or reducing debt with milder output costs; tax-based consolidations are more state-dependent and can be recessionary when implemented in booms or increase debt when implemented in recessions.

### Output multipliers (I-STVAR estimates)
- Table 2: I-STVAR: Output Multiplier in the High-Debt Regime
  - TB Impact, Expansion: −1.22 (−1.53,−0.93)
  - TB Impact, Recession: −0.65 (−0.89,−0.40)
  - TB 5-year cumulative, Expansion: −1.02 (−1.81,−0.36)
  - TB 5-year cumulative, Recession: −0.68 (−1.16,−0.16)
  - EB Impact, Expansion: −0.046 (−0.23,0.18)
  - EB Impact, Recession: −0.74 (−0.84,−0.61)
  - EB 5-year cumulative, Expansion: −0.042 (−0.40,0.38)
  - EB 5-year cumulative, Recession: −1.12 (−1.48,−0.70)

- Table 3: I-STVAR: Output Multiplier in the Low-Debt Regime
  - TB Impact, Expansion: −1.05 (−1.40,−0.73)
  - TB Impact, Recession: −0.67 (−0.88,−0.51)
  - TB 5-year cumulative, Expansion: −2.42 (−3.27,−1.58)
  - TB 5-year cumulative, Recession: −0.108 (−0.58,0.28)
  - EB Impact, Expansion: −0.21 (−0.56,0.09)
  - EB Impact, Recession: −0.75 (−0.90,−0.60)
  - EB 5-year cumulative, Expansion: −0.69 (−1.41,−0.14)
  - EB 5-year cumulative, Recession: −0.97 (−1.35,−0.63)

- Note on multipliers: shocks are not pure spending-only or tax-only shocks (they are a mix), so constructed multipliers capture direct effects but disentangling indirect effects through revenues or spending is not straightforward.

### Mechanisms and interpretation
- Two factors explaining increases in public debt following tax-based consolidations, especially in recessions:
  1. Negative effect on output growth.
  2. Contemporaneous increase of government spending that offsets higher revenues’ positive effect on the primary balance.
- Debt accumulation and omitted-variable bias:
  - If the true model for output growth includes lagged debt with coefficient β < 0, and the identified fiscal shock e_IMF is related to lagged debt via e_IMF_t = κ·Debt_{t−1} + υ_t with κ > 0, then omitting Debt_{t−1} biases the estimated effect of e_IMF_t on ∆y_t through (β/κ + γ). Thus including debt is important to avoid overestimating fiscal effects.

### Policy implications
- Expenditure-based adjustments:
  - Tend to harm the economy less.
  - Are effective at stabilizing and sometimes reducing the debt-to-GDP ratio.
  - A cut in expenditure may reduce the distortionary need for taxation and imply a smaller negative demand shock on GDP growth.
- Tax-based adjustments:
  - More distortionary and, on average, the most recessionary.
  - When debt is high, increasing taxes fails to stabilize the debt-ratio and can be self-defeating.
- Overall: the “initial condition” — the level of debt — plays a relevant role in the propagation of consolidation instruments; policy design should account for debt level, composition of adjustment (tax vs expenditure), and the phase of the business cycle.
- Suggested further work: evaluate channels of transmission including the role of monetary policy, particularly when interest rates are close to the zero lower bound.

### Robustness and diagnostic checks (selected)
- Linearity tests:
  - LM-type test (Terasvirta and Yang, 2014) and likelihood ratio tests favor the non-linear (STVAR) model over the linear VAR.
  - Information criteria: AIC and BIC are lower for the non-linear model (reported values: AIC 4.15 vs 4.06; BIC 4.61 vs 4.5).
- Exogeneity of narrative shocks:
  - Granger causality tests across countries and components show that in most cases past output does not predict narrative measures; Sweden and the Netherlands were dropped because the null could not be rejected.
- Additional figures: cumulative GIRFs for linear VAR, STVAR, and fiscal STVAR with low/high debt confirm the qualitative patterns reported above for output, F(z), and debt responses.

*Source: 5.1 The Interacted-STVAR — wpiea2020246-print-pdf*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020246-print-pdf.pdf_