## wpiea2021039-print-pdf

## Source details

**Canonical URL:** [wpiea2021039-print-pdf](https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021039-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2021/english/wpiea2021039-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2021/english/wpiea2021039-print-pdf.pdf.json)

---

### 2.1 Empirical Model — specification and estimation
- Model: extension of the factor-augmented interacted panel vector-autoregressive model purified of expectations (FAIPVAR-X) of Amendola et al. (2020), building on the I-VAR framework of Caggiano et al. (2017).
- Panel with fixed effects and heterogeneous slopes estimated using the mean group estimator.
- Single interaction term: G × (r−g), where G is government spending and (r−g) is the interest-growth differential.
- Uniform lag structure: L = 1.
- Reduced-form preserved notation (selected):
  - Y_{i,t} = N∑_{i=1} C_i D_{i,j} + N∑_{i=1} L∑_{k=1} A_{i,k} D_{i,j} Y_{i,t−k} + [N∑_{i=1} L∑_{k=1} A^1_{i,k} D_{i,j} G_{i,t−k} × (r−g)_{i,t−k}] + N∑_{i=1} V_i D_{i,j} f(t|t−1:t−4) + V^1 z_{t−1} + u_{i,t}
  - u_{i,t} ~ normal mean zero, covariance Σ_i.
- Endogenous vector (baseline):
  - Y_{i,t} = [G_{i,t}, GDP_{i,t}, T_{i,t}, SR_t, (r−g)_{i,t}, F_t]′
    - G_{i,t}: real government purchases (government gross fixed capital formation + government consumption).
    - GDP_{i,t}: real gross domestic product.
    - T_{i,t}: real net taxes.
    - SR_t: ECB shadow monetary policy rate (Wu and Xia, 2017), complemented with MRO rate at sample start.
    - (r−g)_{i,t}: ten-year government bond yield minus nominal GDP growth (year-on-year quarter).
    - F_t: 1×5 vector of common factors.
- Exogenous variables:
  - f(t|t−1:t−4): forecast of time-t government spending over the past 12 months (four quarters) from the Economist Intelligence Unit (to mitigate fiscal foresight).
  - z_{t−1}: U.S. variables (U.S. output gap, U.S. inflation, U.S. shadow monetary policy rate). Homogeneous slopes imposed for foreign exogenous variables.
- Data and transformations:
  - Quarterly data 2002Q1 to 2019Q4.
  - Ten EA countries: Austria, Belgium, Finland, France, Germany, Ireland, Italy, Netherlands, Portugal, Spain (Luxembourg excluded).
  - Government spending, GDP and net taxes normalized by real potential GDP (Hamilton (2018) filter; HP filter used in robustness).
  - Five common factors extracted; Bai and Ng (2007) IC_{p2} selects five factors.
- Identification and implementation choices:
  - Mean group estimator for heterogeneous slopes.
  - Single interaction term to avoid instability.
  - Government spending shock for GIRFs identified via Cholesky decomposition; government spending ordered first (Blanchard and Perotti (2002) rationale).
  - Two-step estimation: extract five common factors; then add factors to endogenous vector.
  - Forecast variable f(t|t−1:t−4) addresses anticipated fiscal changes up to one year.

### 3.1 Impulse Responses — dynamics and multipliers
- IRF dynamics to a government spending shock:
  - Government spending remains persistently above baseline and takes about ten quarters to die out.
  - Output responds positively.
  - Net taxes respond positively, though the 90-percent credible set often includes zero.
  - (r−g) responds positively on impact in both regimes; afterwards:
    - If r−g >0 at shock: r−g increases (not statistically significant).
    - If r−g <0 at shock: r−g falls (statistically significant).
  - Differences in IRFs across r−g regimes: the 90-percent credible set of differences for output, net taxes, r−g, and the shadow rate excludes zero up to a horizon of 2.5 years or more; for government spending the 90-percent credible set includes zero in the first year.
- Cumulated government spending multiplier computation:
  - M_H = (∑_{h=0}^H dGDP(h)) / (∑_{h=0}^H dG(h)) using discrete approximations of integrals of median GIRFs over horizon h = 0,...,H.
  - Bayesian inference: empirical distributions of differences between multipliers (r−g <0 minus r−g >0) constructed from 10,000 posterior draws; differences and 90 percent credible sets evaluated.
- Key numerical findings (exact median cumulated multipliers from Table 1):
  - 1 year: r−g >0 = 0.95; r−g <0 = 1.14
  - 2 years: r−g >0 = 0.69; r−g <0 = 1.13
  - 3 years: r−g >0 = 0.61; r−g <0 = 1.18
  - 4 years: r−g >0 = 0.60; r−g <0 = 1.21
  - 5 years: r−g >0 = 0.61; r−g <0 = 1.23
- Statistical assessment:
  - At all horizons, at least 90 percent of the distribution of the difference between multipliers lies above zero (very high probability that multiplier is larger when r−g <0).
  - Time-series of cumulated multipliers (over estimation sample) show an inverse correlation between r−g and multiplier size with correlation coefficient 0.92 on average across horizons.

### 3.3 Impact of business cycle, monetary stance, and public debt — placebo tests and robustness
- Purpose: test whether r−g’s role is confounded by three alternative state-dependent channels by replacing r−g in the interaction term (one at a time):
  1. State of the business cycle (output gap).
  2. Monetary policy stance (shadow rate).
  3. Level of public debt (debt/GDP ratio).
- Empirical strategy: same FAIPVAR-X framework, GIRFs and cumulated multipliers computed conditional on positive and negative r−g regimes using same quarters; differences from 10,000 posterior draws.
- Main result: placebo interaction models produce multipliers "rather different" from baseline; for all three alternatives the 90-percent credible sets of the difference between regimes systematically include zero and widen at longer horizons → r−g has an autonomous role.
- Selected numerical entries reported (Table 2 style; entries presented verbatim as in source):
  - State of the business cycle:
    - 1 year: 40.250.43-0.200.47
    - 2 years: 80.250.42-0.280.61
    - 3 years: 120.340.50-0.710.69
    - 4 years: 160.400.56-1.401.06
    - 5 years: 200.450.60-1.971.86
  - Monetary policy stance:
    - 1 year: 41.541.84-0.100.71
    - 2 years: 81.692.09-0.160.99
    - 3 years: 121.862.31-0.211.20
    - 4 years: 161.982.48-0.321.47
    - 5 years: 202.062.59-0.451.86
  - Level of public debt:
    - 1 year: 41.951.90-0.140
    - 2 years: 81.391.33-0.330.06
    - 3 years: 121.171.14-0.550.17
    - 4 years: 161.231.23-0.690.31
    - 5 years: 201.431.47-0.670.53
- Aggregate median cumulated multipliers across specifications (conclusions and robustness):
  - Medium run (5 years) medians:
    - r−g > 0: range between 0.51 and 1.26.
    - r−g < 0: range between 1.22 and 1.77.
  - Difference between negative and positive r−g regimes is systematically different from zero with high probability and tends to increase beyond the first year.
- Robustness exercises (Table 3 representative entries; medians and differences from 10,000 posterior draws):
  - Including public debt:
    - 1 year: 41.241.410.130.24
    - 5 years: 201.111.620.251.49
  - Alternative potential GDP (HP filter):
    - 1 year: 40.620.750.060.19
    - 5 years: 201.261.770.172.19
  - Alternative measure of r (average cost of financing):
    - 1 year: 40.971.100.110.18
    - 5 years: 200.911.330.201.13
  - Alternative critical threshold for r−g regimes (threshold = −0.11):
    - 1 year: 40.951.150.150.26
    - 5 years: 200.641.250.321.59
  - Private investment included:
    - 1 year: 41.261.470.140.28
    - 5 years: 200.951.710.286.82
  - Private consumption included:
    - 1 year: 40.961.150.140.25
    - 5 years: 200.511.220.371.80
  - Pre-crisis (2002Q1–2008Q2) subsample:
    - 1 year: 41.041.100.050.08
    - 5 years: 200.891.110.110.70
  - Post-crisis (2008Q3–2019Q4) subsample:
    - 1 year: 40.881.160.210.36
    - 5 years: 200.411.280.452.26
  - High-debt subsample (Belgium, Ireland, Italy, Portugal, Spain):
    - 1 year: 40.861.130.200.35
    - 5 years: 200.351.190.442.15
  - Low-debt subsample (Austria, Finland, France, Germany, the Netherlands):
    - 1 year: 40.991.130.110.19
    - 5 years: 200.761.210.241.23
- Transmission evidence:
  - Differences in private consumption responses across r−g regimes exclude zero in several quarters; private investment differences often include zero → consumption drives multiplier differences.
- Subsample patterns:
  - Difference between multipliers larger in post-crisis than pre-crisis subsample.
  - Difference larger in high-debt than low-debt subsample.
  - Negative r−g regime: low-debt and high-debt countries have similar multipliers.
  - Positive r−g regime: low-debt countries exhibit higher multipliers than high-debt countries.

### Appendix B — inference and GIRF implementation details
- Estimation: Bayesian strategy with uninformative independent Normal–Wishart prior; Monte Carlo to recover posterior.
- Three-step estimation procedure:
  - Step 1 (reduced-form): draw Σ(s) from IW(ˆS, T×N−f); draw Φ(s) from MN(ˆΦ, Σ(s) ⊗ (X′X)−1); discard explosive draws.
  - Step 2 (GIRFs via posterior draws):
    - GIRF_{i,y}(h, δ, ω_{i,t−1}) = E[y_{i,t+h} | δ, ω_{i,t−1}] − E[y_{i,t+h} | ω_{i,t−1}].
    - Simulation procedure per initial condition ω_{t−1,j}:
      (a) pick ω_{t−1,j};
      (b) simulate residuals ̃u_{r,t+h} ∼ d(0, ̂Σ);
      (c) recover E[y_{t+h,j} | ω_{t−1,j}]_r;
      (d) recover E[y_{t+h,j} | δ, ω_{t−1,j}]_r with residuals perturbed by government spending shock identified by Cholesky (government spending ordered first);
      (e) compute GIRF as in B.1;
      (f) average GIRF across R = 500 simulations for consistent estimates;
      (g) average across countries for each regime initial condition.
    - Explosive simulated initial conditions discarded.
  - Step 3 (parameter uncertainty and final IRFs): repeat step 2 for 10,000 posterior draws; report median IRFs and 5th/95th percentiles as error bands.

### Policy implications — magnitudes and interpretation
- Interpretation: results consistent with forward-looking agents anticipating fiscal tightening when r−g is high, leading to higher saving out of income increases and smaller multipliers.
- Policy-relevant magnitudes:
  - Over a 5-year horizon, median cumulated multipliers typically between 0.51 and 1.26 when r−g > 0, and between 1.22 and 1.77 when r−g < 0.
- Relevance for large fiscal programs (e.g., EU Recovery Fund):
  - With GDP growth resuming and interest rates remaining low, r−g outcomes should be conducive to relatively high government spending multipliers.
  - If adverse shocks keep r−g significantly above zero in large member countries, multipliers may be significantly more modest.

*Source: wpiea2021039-print-pdf — IMF working paper sections 2.1, 3.1, 3.3, Appendix B*

### 2.1  Empirical Model  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    8

### 2.1  Empirical Model

### Overview of the empirical approach
- The empirical model is an extension of the factor-augmented interacted panel vector-autoregressive model purified of expectations (FAIPVAR-X) of Amendola et al. (2020), which builds on the Interacted Vector Auto-Regressive (I-VAR) framework of Caggiano et al. (2017).
- Panel model with fixed effects and heterogeneous slopes estimated using the mean group estimator.
- Interaction term involving two endogenous variables: a proxy for government spending, G, and the interest-growth differential, (r−g).
- Uniform lag structure of one quarter (L = 1) is used for parsimony.

### Reduced-form specification (preserved notation)
Y_{i,t} = N∑_{i=1} C_i D_{i,j} + N∑_{i=1} L∑_{k=1} A_{i,k} D_{i,j} Y_{i,t−k} + [N∑_{i=1} L∑_{k=1} A^1_{i,k} D_{i,j} G_{i,t−k} × (r−g)_{i,t−k}] + N∑_{i=1} V_i D_{i,j} f(t|t−1:t−4) + V^1 z_{t−1} + u_{i,t}

- t = 1,...,T denotes time; i = 1,...,N denotes country; k = 1,...,L denotes lags.
- Y_{i,t} is the vector of endogenous variables.
- Interaction term: G_{i,t−k} × (r−g)_{i,t−k}.
- f(t|t−1:t−4) and z_{t−1} are two sets of exogenous variables.
- C_i: country-specific intercept; A_{i,k}: autoregressive coefficients; A^1_{i,k}: coefficients on the interaction term; V_i: coefficients on first exogenous set; V^1: pooled coefficients on foreign exogenous variables.
- D_{i,j} is an indicator equal to 1 if i = j, 0 otherwise.
- u_{i,t} is normally distributed residual vector with mean zero and covariance matrix Σ_i.

### Advantages of the FAIPVAR-X setup
- Panel dimension exploits quarterly data across ten EA countries.
- Interaction term captures nonlinearities and estimates responses conditional on endogenized (r−g).
- Augmentation with factors extracted from many macroeconomic variables mitigates limited-information concerns.
- Inclusion of forecasts of government spending formed over the past year as an exogenous variable purges anticipated components of government spending shocks (mitigates fiscal foresight).

### Estimation choices and identification notes
- Mean group estimator chosen for heterogeneous slopes.
- Single interaction term is used to avoid pervasive instability associated with multiple interaction terms.
- Identification of government spending shock for GIRFs uses a Cholesky decomposition; government spending ordered first among endogenous variables (Blanchard and Perotti (2002) ordering rationale).

---

### Vector of endogenous variables (baseline)
Y_{i,t} = [G_{i,t}, GDP_{i,t}, T_{i,t}, SR_t, (r−g)_{i,t}, F_t]′

- G_{i,t}: real government purchases (government gross fixed capital formation + government consumption).
- GDP_{i,t}: real gross domestic product.
- T_{i,t}: real net taxes (government receipts of direct and indirect taxes minus transfers to businesses and individuals).
- SR_t: European Central Bank’s shadow monetary policy rate (Wu and Xia, 2017).
  - Note: SR_t is complemented with the Main Refinancing Operations (MRO) rate for sample start given SR_t availability from 2004Q3; the two are virtually indistinguishable until 2008.
- (r−g)_{i,t}: interest-growth differential (baseline computed as ten-year government bond yield minus nominal GDP growth relative to the quarter of the previous year).
  - Robustness checks replace ten-year yield with average cost of financing the debt (Section 4).
- F_t: 1×5 vector of common factors extracted via principal components from a large set of macroeconomic time series.

### Exogenous variables
- f(t|t−1:t−4): forecast of time-t government spending over the past 12 months (four quarters) from the Economist Intelligence Unit; included to mitigate fiscal foresight.
- z_{t−1}: set of U.S. variables (U.S. output gap, U.S. inflation, U.S. shadow monetary policy rate) to account for international factors.
- Homogeneous slopes estimated for foreign exogenous variables due to data availability constraints.

### Data, sample, and transformations
- Quarterly data covering 2002Q1 to 2019Q4.
- Ten EA countries: Austria, Belgium, Finland, France, Germany, Ireland, Italy, Netherlands, Portugal and Spain.
  - Luxembourg excluded per Auerbach and Gorodnichenko (2013).
- Government spending, GDP and net taxes are divided by real potential GDP of the corresponding country to avoid log transformations and ex-post conversions.
- Real potential GDP computed using Hamilton (2018) filter; robustness checked with HP filter (Section 4).
- Five common factors extracted; Bai and Ng (2007) IC_{p2} information criterion determines five factors.

### Computation of cumulated government spending multipliers
- Multipliers computed as ratio of discrete approximations of the integral of median generalized impulse response functions (GIRFs) of real output and government purchases over horizon h = 0,1,...,H:
  - M_H = (∑_{h=0}^H dGDP(h)) / (∑_{h=0}^H dG(h))
- Technical details on inference and identification in Appendix B.

---

### Key methodological constraints and practical choices
- Uniform lag L = 1 to permit well-behaved generalized impulse response functions.
- One interaction term only.
- Two-step estimation procedure: first extract five common factors; second, add factors to endogenous vector.
- Forecast variable f(t|t−1:t−4) addresses anticipated fiscal changes over the past year but not beyond one year.

*Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021039-print-pdf.pdf*

### 3.1  Impulse Responses

### 3.1  Impulse Responses

### Impulse response function (IRF) patterns to a government spending shock
- Model: FAIPVAR-X allows conditioning IRFs on initial condition represented by r−g being negative or positive.
- General dynamics:
  - A shock to government spending keeps spending itself persistently above baseline and it takes about ten quarters to die out.
  - Output responds positively to the shock.
  - Net taxes respond positively to the shock, although the 90-percent credible set of the responses of net taxes often includes zero.
  - r−g responds positively on impact in both regimes.
    - Afterwards, r−g increases (albeit not in a statistically significant manner) if the fiscal shock occurs when r−g >0.
    - r−g falls (to a statistically significant extent) when r−g <0.
- Statistical assessment of differences across r−g regimes:
  - The left and middle columns of Figure 2 report IRFs for positive and negative r−g regimes; the right column reports the difference in IRFs across the two regimes.
  - The 90-percent credible set of the difference in the median response of output, net taxes, r−g, and the shadow rate, between the negative and positive r−g regimes, excludes zero up to a horizon of 2.5 years or more.
  - The 90-percent credible set includes zero in the case of government spending in the first year after the occurrence of the shock.

### Cumulated Government Spending Multipliers (summary of Section 3.2)
- Method:
  - Cumulated multipliers computed from IRFs as explained in Subsection 2.2 and Equation (3).
  - For Bayesian inference, empirical distributions of differences between multipliers (negative r−g regime minus positive r−g regime) are constructed using 10,000 parameter draws from the posterior distribution; differences and 90 percent credible sets are evaluated.
- Key comparative findings:
  - Both in the short and the medium term the multiplier is systematically higher if the fiscal shock occurs when r−g <0, relative to when r−g >0.
  - At a one-year horizon:
    - Point estimate in the positive r−g regime: 0.95
    - Point estimate in the negative r−g regime: 1.14
  - Beyond one year the difference grows:
    - In the medium-term (5 years) the multiplier is about 0.6 when r−g >0 and about 1.2 when r−g <0.
- Exact median cumulated multipliers (Table 1)
  - Horizon H = 1 year: r−g >0 = 0.95; r−g <0 = 1.14
  - Horizon H = 2 years: r−g >0 = 0.69; r−g <0 = 1.13
  - Horizon H = 3 years: r−g >0 = 0.61; r−g <0 = 1.18
  - Horizon H = 4 years: r−g >0 = 0.60; r−g <0 = 1.21
  - Horizon H = 5 years: r−g >0 = 0.61; r−g <0 = 1.23
- Statistical significance of differences:
  - Figure 3 plots distributions of the difference between the two multipliers cumulated at various horizons with 90 percent credible sets.
  - At all horizons, at least 90 percent of each distribution is located above zero, indicating the difference between the two multipliers is positive with very high probability.
- Time-series analysis across EMU history:
  - FAIPVAR-X IRFs conditional on initial conditions corresponding to specific quarters allow computation of time series of cumulated multipliers over the entire estimation sample.
  - Figure 4 reports time series for five time horizons.
  - These charts show an inverse correlation between r−g and the size of the government spending multiplier.
    - The correlation coefficient is 0.92, on average, across the horizons of cumulated multipliers.
- Economic interpretation:
  - Findings are compatible with theories of forward-looking private agents who expect governments to run larger primary budget balances in the future if r−g is high at the time of a government spending expansion.
  - Anticipating a fiscal tightening (e.g. higher taxes) agents will save a larger fraction of the increase in income generated by the increase in government expenditures, leading to a smaller fiscal multiplier.

*Source: wpiea2021039-print-pdf - 3.1  Impulse Responses*

### 3.3  Impact of the State of the Business Cycle, the Monetary Pol-

### 3.3  Impact of the State of the Business Cycle, the Monetary Policy Stance, and the Level of Public Debt

### Context and purpose
- The subsection evaluates whether the documented role of r−g in shaping government spending multipliers is exaggerated by other state-dependent channels emphasized in the literature.
- Three alternative channels are tested: (1) state of the business cycle, (2) monetary policy stance, and (3) level of public debt.
- The empirical aim is a placebo-style test: replace r−g in the interaction term with indicators for each alternative channel (one at a time) and compare resulting multipliers to the baseline r−g-interacted model.

### Key alternative channels tested
- State of the business cycle: proxied by the output gap measure already present among the endogenous variables.
- Monetary policy stance: proxied by the shadow monetary policy rate (endogenous).
- Level of public debt: proxied by the ratio of public debt to GDP (added to endogenous variables and excluded from factor extraction).

### Empirical strategy
- Use the same sample and model framework as baseline (factor-augmented interacted panel VAR).
- Compute GIRFs and cumulated fiscal multipliers conditional on positive and negative r−g regimes, using the same quarters that define these regimes elsewhere in the paper.
- Differences between multipliers computed as multipliers conditional on r−g < 0 minus multipliers conditional on r−g > 0. Distributions obtained from 10,000 posterior parameter draws.

### Main findings
- Placebo tests using the three alternative channels produce multipliers that are "rather different" from the baseline r−g-interacted multipliers.
- For all three alternative-interaction models, the 90-percent credible sets of the difference between multipliers in the two regimes systematically include zero; credible sets widen at longer horizons.
- Conclusion: the interest-growth differential (r−g) has an autonomous role in affecting the size of the EA government spending multiplier and is not observationally equivalent to the tested alternative channels.

- Selected numerical entries reported (as in Table 2; multipliers computed as in Equation (3); H identifies the number of quarters after the shock):
  - State of the business cycle (Table 2 block as reported):
    - 1 year: 40.250.43-0.200.47
    - 2 years: 80.250.42-0.280.61
    - 3 years: 120.340.50-0.710.69
    - 4 years: 160.400.56-1.401.06
    - 5 years: 200.450.60-1.971.86
  - Monetary policy stance (Table 2 block as reported):
    - 1 year: 41.541.84-0.100.71
    - 2 years: 81.692.09-0.160.99
    - 3 years: 121.862.31-0.211.20
    - 4 years: 161.982.48-0.321.47
    - 5 years: 202.062.59-0.451.86
  - Level of public debt (Table 2 block as reported):
    - 1 year: 41.951.90-0.140
    - 2 years: 81.391.33-0.330.06
    - 3 years: 121.171.14-0.550.17
    - 4 years: 161.231.23-0.690.31
    - 5 years: 201.431.47-0.670.53

- Aggregate summary statistics (from conclusions and robustness exercises):
  - Median cumulated multipliers over the medium run (5 years) across specifications:
    - In the positive r−g regime: range between 0.51 and 1.26.
    - In the negative r−g regime: range between 1.22 and 1.77.
  - The difference between multipliers in negative and positive r−g regimes is systematically different from zero with high probability and tends to increase at horizons beyond the first year.

### Robustness checks and additional empirical evidence
- Robustness experiments conducted (Table 3) include:
  - Including public debt in the specification while keeping r−g in the interaction term.
  - Alternative measure of potential GDP (HP filter vs. Hamilton filter).
  - Alternative measure of r (average cost of financing the public debt: net interest payments over previous-quarter debt stock).
  - Alternative critical threshold for r−g regimes set to the sample median, −0.11 (the mean of 0.19 produces very similar results).
  - Including private investment or private consumption explicitly among endogenous variables (one at a time).
  - Subsample analyses: pre-crisis (2002Q1–2008Q2) and post-crisis (2008Q3–2019Q4).
  - Country subsamples by debt level: high-debt (Belgium, Ireland, Italy, Portugal, Spain) versus low-debt (Austria, Finland, France, Germany, the Netherlands).

- Representative numerical entries from Table 3 (multipliers are median values; differences computed as r−g < 0 minus r−g > 0; distributions from 10,000 posterior draws):
  - Public debt in the specification:
    - 1 year: 41.241.410.130.24
    - 5 years: 201.111.620.251.49
  - Alternative measure of potential GDP:
    - 1 year: 40.620.750.060.19
    - 5 years: 201.261.770.172.19
  - Alternative measure of r:
    - 1 year: 40.971.100.110.18
    - 5 years: 200.911.330.201.13
  - Alternative critical threshold for r−g regimes (threshold = −0.11):
    - 1 year: 40.951.150.150.26
    - 5 years: 200.641.250.321.59
  - Private investment in the specification:
    - 1 year: 41.261.470.140.28
    - 5 years: 200.951.710.286.82
  - Private consumption in the specification:
    - 1 year: 40.961.150.140.25
    - 5 years: 200.511.220.371.80
  - Pre-crisis subsample:
    - 1 year: 41.041.100.050.08
    - 5 years: 200.891.110.110.70
  - Post-crisis subsample:
    - 1 year: 40.881.160.210.36
    - 5 years: 200.411.280.452.26
  - High-debt subsample:
    - 1 year: 40.861.130.200.35
    - 5 years: 200.351.190.442.15
  - Low-debt subsample:
    - 1 year: 40.991.130.110.19
    - 5 years: 200.761.210.241.23

- Transmission channel evidence:
  - Impulse responses show private consumption differences across negative versus positive r−g regimes exclude zero in several quarters, while differences in private investment responses include zero. This suggests consumption responses drive the multiplier differences across regimes.
- Subsample patterns:
  - The difference between multipliers in negative and positive r−g regimes is larger in the post-crisis subsample than in the pre-crisis subsample.
  - Difference is larger in the high-debt subsample than in the low-debt subsample.
  - Low-debt and high-debt countries have similar multipliers in the negative r−g regime; in the positive r−g regime, low-debt countries exhibit higher multipliers than high-debt countries.

### Implications for policy
- The autonomous role of r−g implies that expected future fiscal adjustments associated with higher r−g can dampen government spending multipliers.
- Policy-relevant magnitudes:
  - Over a 5-year horizon, median cumulated multipliers typically lie between 0.51 and 1.26 when r−g > 0, and between 1.22 and 1.77 when r−g < 0.
- Relevance for EU Recovery Fund and large-scale fiscal programs:
  - With GDP growth expected to resume and interest rates remaining low, resulting r−g outcomes should be conducive to relatively high government spending multipliers.
  - If adverse shocks keep r−g significantly above zero in large member countries, multipliers may become significantly more modest.

*Source: IMF working paper chapter 3.3 (wpiea2021039-print-pdf).*

### References

### References

### Bibliographic entries (selected, preserved verbatim)
- Almunia, M., A. Benetrix, B. Eichengreen, K. H. O’Rourke, and G. Rua (2010, April). From Great Depression to Great Credit Crisis: Similarities, Differences and Lessons. Economic Policy 25, 219–265.
- Amendola, A., M. Di Serio, M. Fragetta, and G. Melina (2020). The Euro-Area Government Spending Multiplier at the Effective Lower Bound. European Economic Review 127, 103480.
- Auerbach, A. J. and Y. Gorodnichenko (2012). Measuring the Output Responses to Fiscal Policy. American Economic Journal: Economic Policy 4(2), 1–27.
- Auerbach, A. J. and Y. Gorodnichenko (2013). Fiscal Multipliers in Recession and Expansion. In A. Alesina and F. Giavazzi (Eds.), Fiscal Policy after the Financial Crisis, pp. 63–98. University of Chicago Press.
- Bai, J. and S. Ng (2007). Determining the Number of Primitive Shocks in Factor Models. Journal of Business and Economic Statistics 25(1), 52–60.
- Barrett, P. (2018). Interest-Growth Differentials and Debt Limits in Advanced Economies. IMF Working Paper 18/82.
- Batini, N., G. Callegari, and G. Melina (2012). Successful Austerity in the United States, Europe and Japan. IMF Working Paper 190.
- Bernanke, B. S., J. Boivin, and P. S. Eliasz (2005). Measuring the Effects of Monetary Policy: a Factor-augmented Vector Autoregressive (FAVAR) approach. The Quarterly Journal of Economics 120, 387–422.
- Blanchard, O. (2019, April). Public Debt and Low Interest Rates. American Economic Review 109(4), 1197–1229.
- Blanchard, O. J. and R. Perotti (2002). An Empirical Characterization of the Dynamic Effects of Changes in Government Spending and Taxes on Output. Quarterly Journal of Economics 117(4), 1329–1368.
- Bonam, D., J. De Haan, and B. Soederhuizen (2020). The Effects of Fiscal Policy at the Effective Lower Bound. Macroeconomic Dynamics, 1–37.
- Caggiano, G., E. Castelnuovo, V. Colombo, and G. Nodari (2015). Estimating Fiscal Multipliers: News From A Non-linear World. Economic Journal 125(584), 746–776.
- Caggiano, G., E. Castelnuovo, and G. Pellegrino (2017). Estimating the Real Effects of Uncertainty Shocks at the Zero Lower Bound. European Economic Review 100, 257–272.
- Canova, F. and M. Ciccarelli (2013). Panel Vector Autoregressive Models: A Survey. In VAR Models in Macroeconomics–New Developments and Applications: Essays in Honor of Christopher A. Sims, pp. 205–246. Emerald Group Publishing Limited.
- Cogley, T. and T. J. Sargent (2005). Drift and Volatilities: Monetary Policies and Outcomes in the Post WWII U.S. Review of Economic Dynamics 8, 262–302.
- Crafts, N. and T. C. Mills (2013, December). Rearmament to the Rescue? New Estimates of the Impact of “Keynesian” Policies in 1930s’ Britain. The Journal of Economic History 73(4), 1077–1104.
- Del Negro, M. and F. Schorfheide (2011). Bayesian Macroeconometrics. In J. Geweke, G. Koop, and H. V. Dijk (Eds.), The Oxford Handbook of Bayesian Econometrics. Oxford University Press.
- Di Serio, M., M. Fragetta, and E. Gasteiger (2020, December). The Government Spending Multiplier at the Zero Lower Bound: Evidence from the United States. Oxford Bulletin of Economics and Statistics 82(6), 1262–1294.
- Dickey, D. A. and W. A. Fuller (1979). Distribution of the Estimators for Autoregressive Time Series With a Unit Root. Journal of the American Statistical Association 74(366), 427–431.
- Favero, C. and F. Giavazzi (2007, January). Debt and the Effects of Fiscal Policy. Working Paper 12822, National Bureau of Economic Research.
- Forni, M. and L. Gambetti (2010, May). Fiscal Foresight and the Effects of Goverment Spending. CEPR Discussion Papers 7840, C.E.P.R. Discussion Papers.
- Forni, M., D. Giannone, M. Lippi, and L. Reichlin (2009). Opening the Black Box: Structural Factor Models with Large Cross Sections. Econometric Theory 25, 1319–1347.
- Fotiou, A. (2020, November). Non-Linearities in Fiscal Policy: The Role of Debt. IMF Working Papers 2020/246, International Monetary Fund.
- Fotiou, A., W. Shen, and S.-C. S. Yang (2020). The Fiscal State-Dependent Effects of Capital Income Tax Cuts. Journal of Economic Dynamics and Control 117, 103860.
- Fragetta, M. and E. Gasteiger (2014). Fiscal Foresight, Limited Information and the Effects of Government Spending Shocks. Oxford Bulletin of Economics and Statistics 76(5), 667–692.
- Gordon, R. J. and R. Krenn (2010). The End of the Great Depression 1939-41: Policy Contributions and Fiscal Multipliers. NBER Working Paper 16380, National Bureau of Economic Research.
- Hamilton, J. D. (2018). Why You Should Never Use the Hodrick-Prescott Filter. Review of Economics and Statistics 100(5), 831–843.
- Hodrick, R. J. (2020, February). An Exploration of Trend-Cycle Decomposition Methodologies in Simulated Data. Working Paper 26750, National Bureau of Economic Research.
- Hodrick, R. J. and E. C. Prescott (1997). Postwar U.S. Business Cycles: An Empirical Investigation. Journal of Money, Credit and Banking 29(1), 1–16.
- Huidrom, R., M. A. Kose, J. J. Lim, and F. L. Ohnsorge (2020). Why Do Fiscal Multipliers Depend on Fiscal Positions? Journal of Monetary Economics 114, 109–125.
- Kilian, L. and R. J. Vigfusson (2011). Are the Responses of the US Economy Asymmetric in Energy Price Increases and Decreases? Quantitative Economics 2(3), 419–453.
- Kirchner, M., J. Cimadomo, and S. Hauptmeier (2010). Transmission of Government Spending Shocks in the Euro Area: Time Variation and Driving Forces. Working Paper Series 1219, European Central Bank.
- Klein, M. and R. Winkler (2018, January). The Government Spending Multiplier at the Zero Lower Bound: International Evidence from Historical Data. Working Papers 2018-001, University of Antwerp, Faculty of Business and Economics.
- Koop, G., M. H. Pesaran, and S. M. Potter (1996). Impulse Response Analysis in Nonlinear Multivariate Models. Journal of econometrics 74(1), 119–147.
- Kwiatkowski, D., P. C. B. Phillips, P. Schmidt, and Y. Shin (1992). Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root: How Sure Are We That Economic Time Series Have a Unit Root? Journal of Econometrics 54(1-3), 159–178.
- Leeper, E. M., T. B. Walker, and S. C. S. Yang (2013). Fiscal Foresight and Information Flows. Econometrica 81(3), 1115–1145.
- Lian, W., A. F. Presbitero, and U. Wiriadinata (2020, July). Public Debt and r - g at Risk. IMF Working Papers 2020/137, International Monetary Fund.
- Mauro, P. and J. Zhou (2020, March). r minus g negative: Can We Sleep More Soundly? IMF Working Papers 2020/052, International Monetary Fund.
- Miyamoto, W., T. L. Nguyen, and D. Sergeyev (2018, July). Government Spending Multipliers under the Zero Lower Bound: Evidence from Japan. American Economic Journal: Macroeconomics 10(3), 247–77.
- Nickel, C. and A. Tudyka (2014). Fiscal Stimulus in Times of High Debt: Reconsidering Multipliers and Twin Deficits. Journal of Money, Credit and Banking 46(7), 1313–1344.
- Pesaran, M. H. and R. Smith (1995). Estimating Long-Run Relationships From Dynamic Heterogeneous Panels. Journal of econometrics 68(1), 79–113.
- Ramey, V. A. and S. Zubairy (2018). Government Spending Multipliers in Good Times and in Bad: Evidence from US Historical Data. Journal of Political Economy 126(2), 850–901.
- Rogoff, K. (2020). Falling Real Interest Rates, Rising Debt: A free lunch? Journal of Policy Modeling 42(4), 778 – 790. SI: Growth and Trade in the United States and Europe.
- Sá, F., P. Towbin, and T. Wieladek (2014, apr). Capital Inflows, Financial Structure and Housing Booms. Journal of the European Economic Association 12(2), 522–546.
- Schmitt-Grohé, S. and M. Uribe (2012, November). What’s News in Business Cycles. Econometrica 80(6), 2733–2764.
- Stock, J. H. and M. W. Watson (2005). Implications of Dynamic Factor Models for VAR Analysis. NBER Working Papers 11467.
- Wu, J. C. and F. D. Xia (2016). Measuring the Macroeconomic Impact of Monetary Policy at the Zero Lower Bound. Journal of Money, Credit and Banking 48(2-3), 253–291.
- Wu, J. C. and F. D. Xia (2017). Time-Varying Lower Bound of Interest Rates in Europe. Chicago Booth Research Paper No. 17-06.

### Appendix

### A Data
- A.1 Endogenous Variables
  - Variables of interest: gross domestic product, net taxes, government spending, the European Central Bank’s shadow rate and the interest-growth differential.
  - Net taxes: sum of government receipts of direct and indirect taxes minus transfers to businesses and individuals.
  - Government spending: sum of government gross fixed capital formation and government consumption.
  - Interest-growth differential (r−g): difference between the 10-year Government Bond Yield and year-on-year GDP growth.
  - Alternative measure of r (robustness analysis): ratio between net interest payments and the public debt stock outstanding in the previous quarter.
  - Data source: Eurostat database available on the Thomson Reuters Datastream Economics database.
  - Transformations: Gross domestic product, net taxes and government spending transformed in real terms using the implicit GDP price deflator, then normalized by dividing by real potential GDP.
  - ECB shadow rate: that developed by Wu and Xia (2017).

- A.2 Exogenous Variables
  - Exogenous variables: forecast of the annualized growth rate of total government expenditure over GDP produced by the Economist Intelligence Unit.
  - Construction: in each quarter compute the average forecast for the current year over the past 12 months.
  - Other exogenous variables: the U.S. output gap, and the U.S. inflation downloaded from the Federal Reserve Bank of St. Louis database, and the U.S. Shadow Rate developed by Wu and Xia (2016).

- A.3 Informational Dataset
  - Informational dataset for extracting common factors: composed by 250 series downloaded from the Eurostat database available on the Thomson Reuters Datastream Economics database.
  - Variables downloaded for each country considered:
    - National Account: Domestic Demand; Export of Goods and Services; Imports of Goods and Services; Gross Capital Formation; Final Consumption Expenditure of Households.
    - Government Statistics: Government Consolidated Gross Debt: Central Govt.
    - Output and income: Industrial Production Index (Mining and Quarrying; Manufacturing; Electricity, Gas, Steam and Air Conditioning Supply); Nominal Unit Labor Cost based on persons; Production - Total Industry Excl. Construction; Production of Total Construction; Wages and Salaries; Change in Inventories.
    - Employment and hours: Early Estimates of Labor Productivity - Total Economy; Employees Domestic Concept; Unemployment: Total.
    - Stock prices: S&P BMI - Price Index.
    - Exchange rates: REER: 19 trading partners EA; NEER: 37 Trading Partners.
    - Money and credit quantity aggregates: Money Supply: M1 - Contribution to Euro M1; Money Supply: M2 - Contribution to Euro M2; Money Supply: M3 - Contribution to Euro M3; Official Reserve Assets.
    - Interest Rate: Harmonized Government 10-Year Bond Yield.
  - Stationarity: Where appropriate variables are transformed to guarantee stationarity tested by the Dickey and Fuller (1979) and Kwiatkowski et al. (1992) tests.

### B Inference and Identification
- Model estimated: FAIPVAR-X model presented in equation (1).
- Estimation strategy: three steps adopting a Bayesian strategy for inference with an uninformative independent Normal–Wishart prior; Montecarlo simulation to recover the posterior distribution of the parameters.
- Step 1 (reduced-form estimation):
  - Draw the variance-covariance matrix of the residuals after averaging coefficients across countries, Σ(s), from an inverse Wishart distribution IW(ˆS, T×N−f)—where ˆS is the sum of the squared residuals and f is the number of coefficients in a single equation of the VAR—for drawing Φ(s) from the conditional multivariate normal distribution MN(ˆΦ, Σ(s) ⊗ (X′X)−1), where ˆΦ are the estimated reduced-form parameters and X is the matrix of regressors in the panel VAR.
  - Note: As in Cogley and Sargent (2005); Sá et al. (2014), explosive draws from the unrestricted posterior are discarded.
- Step 2 (GIRFs via posterior draws):
  - Make a draw of the posterior distribution and derive Generalized Impulse Response Functions (GIRFs) following Koop et al. (1996) and Caggiano et al. (2017).
  - GIRF definition (as in the source):
    - GIRF_{i,y}(h, δ, ω_{i,t−1}) = E[y_{i,t+h} | δ, ω_{i,t−1}] − E[y_{i,t+h} | ω_{i,t−1}],
    - where, for each country i, E[y_{i,t+h} | δ, ω_{i,t−1}] represents the expected value of the response of the endogenous variables y to a shock of size δ, at horizon h, conditional on an initial history ω_{t−1}; and E[y_{t+h} | ω_{i,t−1}] represents the expected value of the endogenous variable y at horizon h conditional on an initial history ω_{i,t−1}.
  - Methodological notes:
    - The vector of endogenous variables y also includes the variables in the interaction term.
    - Choice of initial condition ω_{i,t−1} = {y_{t−1}, ..., y_{t−k}} allows discerning among different monetary policy regimes.
  - For each country and each regime, the simulation procedure:
    (a) pick an initial condition ω_{t−1,j};
    (b) simulate the residuals series starting from its empirical distribution ̃u_{r,t+h} ∼ d(0, ̂Σ);
    (c) for each simulation of residuals r, recover the path E[y_{t+h,j} | ω_{t−1,j}]_r;
    (d) simulate the path E[y_{t+h,j} | δ, ω_{t−1,j}]_r starting from the residuals obtained in sub-step (b) perturbated by a government spending shock identified using a Cholesky decomposition (as in Kilian and Vigfusson, 2011 and Caggiano et al., 2017);
      - Following Blanchard and Perotti (2002), government spending is ordered as the first endogenous variable.
    (e) compute the GIRF as in equation B.1;
    (f) compute the average GIRF across R = 500 simulations in order to obtain consistent estimates;
    (g) average across countries once the average GIRF for each initial condition belonging to different regimes has been collected.
  - Notes: If a given initial condition produces an explosive response, it is discarded.
- Step 3 (parameter uncertainty and final IRFs):
  - Repeat step 2 for 10,000 draws from the posterior distribution.
  - Consider the median IRFs across the 10,000 parameter draws.
  - Parameter uncertainty: save the 5th and 95th percentile of the distribution as error bands.

*Source: wpiea2021039-print-pdf - References*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021039-print-pdf.pdf_