## wpiea2019133 — Section 2.1, Appendix A.3, 3.4 (Empirical Model; Multiplier Inference; Correlations) ## Source details **Canonical URL:** [wpiea2019133 — Section 2.1, Appendix A.3, 3.4 (Empirical Model; Multiplier Inference; Correlations)](https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019133.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2019/wpiea2019133.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2019/wpiea2019133.pdf.json) --- ### Empirical model: FAIPVAR-X — purpose and advantages - Framework: factor-augmented interacted panel vector-autoregressive model purified of expectations (FAIPVAR-X), an extension of the IPVAR model by Towbin and Weber (2013) and Sá et al. (2014). - Four stated advantages: - Panel dimension exploits quarterly data of ten euro area (EA) countries (Austria, Belgium, Finland, France, Germany, Ireland, Italy, Netherlands, Portugal and Spain). - Interaction term captures nonlinearities and allows estimating responses to a government spending shock at each percentile of the shadow rate distribution. - Augmenting with factors extracted from a large number of macroeconomic variables addresses limited information concerns (proxies unobserved factors used by agents). - Including forecasts of government spending as an exogenous variable purges anticipated components and addresses fiscal foresight, reducing non-fundamentalness. ### Model structure and identification features - Structural form components (as given in source): - Endogenous vector y_{i,t}. - Interaction term x_t affecting both levels and dynamics through country-specific coefficients κ^1_j and dynamic terms Γ^1_k. - Two sets of exogenous variables: f_{(t−1:t−4)} and z_{t−1}. - Country fixed effects D_{j,i}, country-specific intercepts κ_j, autoregressive coefficient matrices Γ_{j,k}, country-specific coefficients ν_j on first set of exogenous variables, pooled coefficients v_1 on other exogenous variables, and i.i.d. residuals ε_{i,t} assumed uncorrelated across countries. - Matrix B_{i,t} is a (q×q) lower triangular matrix with ones on the main diagonal, imposing a recursive structure that implies a diagonal covariance matrix Σ_ε of the residuals. - Interaction terms and lagged interacted terms x_t y_{i,t−k} present; due to data availability, slopes on lagged interacted terms and on foreign exogenous variables z_{t−1} are estimated as homogeneous across countries. ### Estimation choices - Estimator: mean group estimator applied to a panel model with fixed effects and heterogeneous slopes. - Lag structure: baseline L = 1 (one quarter); robustness checks include L = 2. - Residual assumptions: ε_{i,t} are i.i.d. and uncorrelated across countries by assumption. ### Data coverage and sample - Frequency and sample period: quarterly data covering 2002q2 to 2017q4. - Countries: Austria, Belgium, Finland, France, Germany, Ireland, Italy, Netherlands, Portugal and Spain (Luxembourg excluded per Auerbach and Gorodnichenko (2013)). - Shadow rate availability: Wu and Xia shadow monetary policy rate available from 2004Q3 onward; complemented by the Main Refinancing Operations (MRO) rate for the earliest part where they are virtually indistinguishable. - Interaction term used: lagged shadow rate x_t = sr_{t−1} to avoid reversed causality (predetermined relative to endogenous variables). ### Endogenous variables, transformations, and factors - VAR vector: y_{i,t} = [G_{i,t}, GDP_{i,t}, T_{i,t}]′ with: - 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). - Modifications: - All endogenous variables divided by real potential GDP of the corresponding country to avoid logarithms and ex-post conversions of elasticities to dollar equivalents. - Real potential GDP computed using Hamilton (2018) filter. - Factor augmentation: - Five common factors extracted by principal components as determined by Bai and Ng (2007) IC_{p2}. - Extended endogenous vector: y_{i,t} = [G_{i,t}, GDP_{i,t}, T_{i,t}, F_t]′ where F_t is a 1×5 vector common to all countries. ### Exogenous variables to address fiscal foresight and information - Forecast variable: f_{(t|t−1:t−4)} = Economist Intelligence Unit forecast of time-t government spending over the past 12 months (four quarters). Included as an exogenous regressor to purge anticipated government spending changes. - Foreign exogenous variables: z_{t−1} includes U.S. variables and is included as pooled exogenous regressors with homogeneous slopes. ### Appendix A.3 — informational dataset and additional exogenous controls - Appendix A.3 includes both money and credit quantity aggregates, and the harmonized government ten-year bond yield (used to capture expectations on monetary policy and market sentiment toward government debt dynamics). - Exogenous U.S. variables in z_{t−1}: U.S. output gap, U.S. inflation and U.S. shadow monetary policy rate (Wu and Xia (2016)). - Informational dataset: 250 series from Eurostat/Thomson Reuters Datastream Economics database covering national accounts, government statistics, output and income, employment and hours, stock prices, exchange rates, money and credit aggregates, and harmonized government 10-Year Bond Yield. - Frequency conversions: selected monthly and daily series converted to quarterly as listed in Appendix A.3. - Stationarity: variables transformed where appropriate and tested by Dickey and Fuller (1979) and Kwiatkowski et al. (1992). ### Inference, identification, and computation of cumulated government spending multipliers (seven-step procedure) - Estimation and multiplier computation steps (per Sá et al. (2014)): 1. Estimate structural model equation-by-equation using OLS and adopt a Bayesian strategy with an uninformative independent Normal–Wishart prior; use Montecarlo simulation to recover posterior distribution of structural parameters. 2. Make a draw of the posterior distribution and evaluate it at pre-specified values of the interaction term x_t. 3. Derive reduced form by pre-multiplying equation (1) by B^{-1}_{i,t}. 4. Use sign restriction strategy to identify an unexpected government spending shock and compute IRFs. Define V^{d}_{x} as the Cholesky decomposition of reduced form variance-covariance Σ^{d}_{x}; draw orthonormal Q such that Q′ Q = I, B^{d} = V^{d}_{x} Q. Identification restriction: government spending shock should raise G_{it} and GDP_{it} for at least four quarters (Table 1). 5. Use the median target approach (Fry and Pagan (2011)) to compute IRFs: for every 100 draws of Q satisfying sign restrictions, save the Q implying IRFs closest to the median. 6. Make 20,000 draws from the posterior, discard first 10,000 as burn-in. For remaining 10,000 draws follow step 5 and keep the model producing IRFs nearest the median IRFs. 7. Compute cumulated multipliers following Gordon and Krenn (2010) and Ramey and Zubairy (2018). With variables normalized by real potential GDP: - M_H = (∑_{h=0}^H dGDP(h)) / (∑_{h=0}^H dG(h)). - Parameter uncertainty: save 5th and 95th percentiles of the distribution of the median as error bands. - Explosive draws from the unrestricted posterior are discarded (as in Cogley and Sargent (2005); Primiceri (2005); Sá et al. (2014)). ### Regimes, shadow rate percentiles, and impulse responses - Two EA monetary policy regimes analyzed by conditioning IRFs on shadow rate percentiles: - normal times: period 2002q2 to 2008q3; shadow rate almost coincided with official Eonia rate and both were clearly positive. - ELB (effective lower bound): period 2012q4 to 2017q4; shadow rate systematically negative, capturing unconventional measures including the Asset Purchase Program (APP) and forward guidance. - Representative shadow rate percentiles: - 77th percentile (2.75 percent; 2003q2) as representative of normal times (closest to average shadow rate 2.82 percent). - 16th percentile (-2.23 percent; 2015q3) as representative of the ELB (closest to average shadow rate -2.29 percent). - IRF observations: - Government spending shock keeps spending persistently above baseline, taking about ten quarters to die out. - Output and net taxes respond positively; credible set for net taxes often includes zero. - Comparing normal times vs ELB: output responses mostly larger at ELB, with confidence bands not overlapping in several quarters after the shock (quarters 8-14). ### Reported cumulated government spending multipliers conditional on two shadow rate levels (as reported) - Table 2 — Cumulated Government Spending Multipliers Conditional on Two Levels of the Shadow Rate Representative of Normal Times and the ELB (preserved formatting as in source): - M_H | pctl(sr) M_H | pctl(sr) - Normal Times Effective Lower Bound - Horizon H pctl(sr) = 77 pctl(sr) = 16 - 1 year41,902,17 - 2 years81,482,59 - 3 years120,992,82 - 4 years160,742,96 - 5 years200,753,05 - Interpretation from source: - Short- and medium-term multipliers systematically higher at the ELB relative to normal times. - One-year multipliers comparable across regimes; over medium term (3–5 years) multiplier ~1 in normal times and increases up to 3 at the ELB. - Statistical assessment: empirical distributions of differences (M_H | pctl(sr)=16 minus M_H | pctl(sr)=77) across 10,000 posterior draws show that from horizon three to five, 90 percent of distributions do not include zero (difference positive with high probability). ### Average cumulated multipliers across regimes (as reported) - Table 3 — Average Cumulated Government Spending Multipliers in Normal Times and at the ELB (preserved formatting as in source): - mean(M_H | pctl(sr)) mean(M_H | pctl(sr)) - Horizon H Normal Times Effective Lower Bound - 1 year42,132,10 - 2 years81,572,44 - 3 years121,082,58 - 4 years160,792,73 - 5 years200,642,83 - Additional quantitative findings (from source text): - Averaging across percentiles yields consistent conclusions: one-year multipliers very similar; longer horizons diverge with normal-times multiplier falling below 1 and ELB multiplier much larger in medium term. - Reported examples: three-year multiplier ~1 in normal times and 2.6 at the ELB; five-year multiplier 0.6 in normal times and 2.8 at the ELB. - Distributions of differences between average multipliers: difference non-zero with 90 percent probability at horizons 2, 3 and 4 years (marginal significance at year 2); cannot exclude zero at horizons 1 and 5 years. ### Correlations of multipliers with shadow rate and business cycle (Section 3.4) - Main findings: - Negative correlation between cumulated government spending multiplier and the shadow rate that survives after controlling for business cycle state. - One-year horizon: conditional correlation statistically insignificant; horizons beyond one year: conditional correlation strongly negative and statistically significant. - Multiplier also negatively correlated with business cycle after controlling for shadow rate; correlation statistically different from zero at 1 percent level at all horizons except first year. - Short run: multiplier very similar across normal times and ELB. - Medium run (three years): average multiplier ~1 in normal times and between 1.6 and 2.8 at the ELB (depending on specification). - Multiplier inversely correlated with level of shadow monetary policy rate across percentiles. - Shadow rate and business cycle have autonomous correlations with size of fiscal multiplier in eurozone. - Conditional correlations from Table 4 (preserved formatting and values): - Horizon 1 year (H = 4): corr(ˆε_M_H|bc_t, sr_{t−1}) = 0,2044 (p-value 0,1383); corr(ˆε_M_H|sr_t, bc_{t−1}) = -0,1822 (p-value 0,1873). - Horizon 2 years (H = 8): corr = -0,7200 (p-value 0,0000); corr = -0,3508 (p-value 0,0093). - Horizon 3 years (H = 12): corr = -0,8290 (p-value 0,0000); corr = -0,4052 (p-value 0,0024). - Horizon 4 years (H = 16): corr = -0,8389 (p-value 0,0000); corr = -0,3326 (p-value 0,0140). - Horizon 5 years (H = 20): corr = -0,8258 (p-value 0,0000); corr = -0,2814 (p-value 0,0393). - Note on computation: conditional correlations obtained by regressing cumulated multiplier on a constant and the lagged business cycle indicator (or lagged shadow rate) and correlating residuals with the lagged shadow rate (or lagged business cycle indicator). P-values in parentheses. ### Robustness checks and outcomes - Robustness variants examined: 1. Sign restrictions imposed for 2 quarters (instead of 4). 2. Lag structure L = 2 (instead of 1). 3. Including 10-year government bond yields explicitly in the panel VAR. - General outcome: multipliers under robustness variants remain same order of magnitude as baseline; divergence across regimes increases with horizon with ELB showing substantially higher multipliers in medium run. - Representative robustness entries preserved from source (selected examples): - Horizon 1 year (H = 4) — as reported: Normal Times and ELB include 1,71 and 1,70 (sign restrictions for 2 quarters); 2,22 and 2,91 (lag structure 2 quarters); 2,02 and 1,70 (including 10-year gov. bond yields). - Horizon 3 years (H = 12) — reported values across variants include 1,15 and 2,03; 1,20 and 2,83; 1,39 and 1,63. - Distributional robustness (Table 6): at intermediate horizons (3 and/or 4 years) the 90-percent confidence interval for the difference between multipliers in ELB and normal times excludes zero across estimation variants. - Conditional correlations broadly negative and statistically significant across variants (selected entries): - Sign restrictions for 2 quarters: Horizon 2 years (H = 8) corr(ˆε_M_H|bc_t, sr_{t−1}) = -0,5811 (p-value 0,0000); corr(ˆε_M_H|sr_t, bc_{t−1}) = -0,5592 (p-value 0,0000). - Lag structure L = 2: Horizon 3 years (H = 12) corr(ˆε_M_H|bc_t, sr_{t−1}) = -0,7896 (p-value 0,0000); corr(ˆε_M_H|sr_t, bc_{t−1}) = -0,2828 (p-value 0,0383). - Including 10-year gov. bond yields: Horizon 5 years (H = 20) corr(ˆε_M_H|bc_t, sr_{t−1}) = -0,4140 (p-value 0,0019); corr(ˆε_M_H|sr_t, bc_{t−1}) = -0,5329 (p-value 0,0000). ### Conclusions and policy implications (from source) - Empirical evidence for euro area supports theoretical predictions of larger government spending multipliers at the ELB versus normal times, with qualifications: - (i) The difference between the size of the one-year government spending multiplier at the ELB and in normal times is neither economically nor statistically significant. - (ii) This difference increases over longer horizons, and its distribution is largely away from zero; in the medium term (around 3 years) the multiplier in normal times is about 1, whereas at the ELB it exceeds 2.5. - Contextual note: in the data the ELB coincided with unconventional monetary policy; conditioning on shadow monetary policy rate captures both time-varying ELB effects and effects of unconventional monetary policies. - Practical implication: as monetary policy normalizes (phasing out of asset purchases, prospective normalization of policy rates), estimates suggest a medium-term multiplier for the euro area of about 1 to use in upcoming normal times scenarios. - Methodological implication: FAIPVAR-X enables conditioning multipliers on the shadow monetary policy rate and accounts for limited information and fiscal foresight. ### Appendix highlights (A.1–A.4, B) - Appendix A.1 Endogenous variables: GDP, net taxes, government spending; transformations: deflated by implicit GDP price deflator and normalized by real potential GDP. - Appendix A.2 Exogenous variables: EIU forecast of government expenditure growth; ECB shadow rate (Wu and Xia (2017)); U.S. output gap and U.S. inflation; U.S. shadow rate (Wu and Xia (2016)). - Appendix A.3 Informational dataset: 250 series; variables and frequency conversions listed; money and credit aggregates included; Harmonized Government 10-Year Bond Yield included. - Appendix A.4 Business cycle indicator: aggregate computed as ∑_{i=1}^{10} GDP_i / GDP^{trend}_i (large positive = expansion; large negative = recession). - Appendix B: additional robustness checks on conditional correlations (Tables B.1 and B.2) — conditional correlations and p-values preserved in source tables and summarized above. *Source: IMF Working Paper (wpiea2019133) — Section 2.1 "Empirical Model", Appendix A.3, Section 3.4, and appendices as quoted.* ### 2.1 Empirical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 ### 2.1 Empirical Model ### Purpose and advantages of the FAIPVAR-X approach - Framework: factor-augmented interacted panel vector-autoregressive model purified of expectations (FAIPVAR-X), an extension of the IPVAR model by Towbin and Weber (2013) and Sá et al. (2014). - Four stated advantages: - Panel dimension exploits quarterly data of ten euro area (EA) countries (Austria, Belgium, Finland, France, Germany, Ireland, Italy, Netherlands, Portugal and Spain). - Interaction term captures nonlinearities and allows estimating responses to a government spending shock at each percentile of the shadow rate distribution. - Augmenting with factors extracted from a large number of macroeconomic variables addresses limited information concerns (proxies unobserved factors used by agents). - Including forecasts of government spending as an exogenous variable purges anticipated components and addresses fiscal foresight, reducing non-fundamentalness. ### Model structure and identification features - General structural form (as given in the source) includes: - Endogenous vector y_{i,t}. - Interaction term x_t affecting both levels and dynamics through country-specific coefficients κ^1_j and dynamic terms Γ^1_k. - Two sets of exogenous variables: f_{(t−1:t−4)} and z_{t−1}. - Country fixed effects D_{j,i}, country-specific intercepts κ_j, autoregressive coefficient matrices Γ_{j,k}, country-specific coefficients ν_j on first set of exogenous variables, pooled coefficients v_1 on the other exogenous variables, and i.i.d. residuals ε_{i,t} assumed uncorrelated across countries. - Matrix B_{i,t} is a (q×q) lower triangular matrix with ones on the main diagonal, imposing a recursive structure that implies a diagonal covariance matrix Σ_ε of the residuals. - Interaction terms and lagged interacted terms x_t y_{i,t−k} are present; due to data availability, slopes on lagged interacted terms and on foreign exogenous variables z_{t−1} are estimated as homogeneous across countries. ### Estimation choices - Estimator: mean group estimator applied to a panel model with fixed effects and heterogeneous slopes (to allow heterogeneity across countries). - Lag structure: baseline results use a uniform lag structure of one quarter (L= 1) for parsimony; robustness checks include two lags. - Residual assumptions: ε_{i,t} are i.i.d. and uncorrelated across countries by assumption. ### Data coverage and sample - Frequency and sample period: quarterly data covering 2002q2 to 2017q4. - Countries: ten EA countries included (Austria, Belgium, Finland, France, Germany, Ireland, Italy, Netherlands, Portugal and Spain); Luxembourg excluded per Auerbach and Gorodnichenko (2013). - Shadow rate availability: Wu and Xia shadow monetary policy rate available from 2004Q3 onward; complemented by the Main Refinancing Operations (MRO) rate for the earliest part of the sample where they are virtually indistinguishable. - Interaction term used: lagged shadow rate x_t = sr_{t−1} to avoid reversed causality (predetermined relative to endogenous variables). ### Endogenous variables and transformations - Traditional VAR vector: y_{i,t} = [G_{i,t}, GDP_{i,t}, T_{i,t}]′ where G_{i,t} = real government purchases (government gross fixed capital formation + government consumption), GDP_{i,t} = real gross domestic product, and T_{i,t} = real net taxes (government receipts of direct and indirect taxes minus transfers to businesses and individuals). - Modifications for multiplier computation and nonlinearity: - All endogenous variables divided by real potential GDP of the corresponding country to avoid logarithms and ex-post conversions of elasticities to dollar equivalents. - Real potential GDP computed using Hamilton (2018) filter (chosen to avoid spurious persistence from the HP filter). - Factor augmentation: - Five common factors extracted by principal components as determined by Bai and Ng (2007) IC_{p2}. - Extended endogenous vector: y_{i,t} = [G_{i,t}, GDP_{i,t}, T_{i,t}, F_t]′ where F_t is a 1×5 vector common to all countries but with potentially different country impacts (captures spillovers and unobserved common shocks). ### Exogenous variables to address fiscal foresight and information - Forecast variable: f_{(t|t−1:t−4)} = Economist Intelligence Unit forecast of time-t government spending over the past 12 months (four quarters). Included as an exogenous regressor to purge anticipated government spending changes (addresses fiscal foresight and avoids bias from anticipated fiscal changes). - Foreign exogenous variables: z_{t−1} included as pooled exogenous regressors; slopes on these variables are estimated as homogeneous. *Source: IMF Working Paper (section 2.1 "Empirical Model" from wpiea2019133).* ### Appendix A.3) includes both money and credit quantity aggregates, and the harmonized ### Appendix A.3) includes both money and credit quantity aggregates, and the harmonized ### Model specification and exogenous controls - The model is labeled factor-augmented interacted panel vector-autoregressive model purified of expectations (FAIPVAR-X). - Appendix A.3 includes both money and credit quantity aggregates, and the harmonized government ten-year bond yield (used to capture expectations on monetary policy and market sentiment toward government debt dynamics). - As exogenous variables, the model adds a set of U.S. variables, z_{t−1}, including the U.S. output gap, U.S. inflation and the U.S. shadow monetary policy rate developed by Wu and Xia (2016). ### Inference, identification and computation of cumulated government spending multipliers - The FAIPVAR-X model estimation and multiplier computation follow seven steps (in line with Sá et al. (2014)): 1. Estimate the structural model equation by equation using ordinary least squares (OLS) and adopt a Bayesian strategy for inference utilizing an uninformative independent Normal–Wishart prior, which in turn uses a Montecarlo simulation to recover the posterior distribution of the structural parameters. 2. Make a draw of the posterior distribution and evaluate it at pre-specified values of the interaction term x_t. 3. Derive the model’s corresponding reduced form, by pre-multiplying equation (1) by B^{-1}_{i,t}. 4. Use a sign restriction strategy to identify an unexpected government spending shock and compute the resulting impulse response functions (IRFs). After defining V^{d}_{x} as the Cholesky decomposition of the reduced form variance-covariance matrix Σ^{d}_{x}, draw an orthonormal matrix Q such that Q′ Q = I, from which it follows that B^{d} = V^{d}_{x} Q and Σ^{d}_{x} = B^{d′} B^{d} = V^{d′}_{x} Q′ Q V^{d}_{x}, where d indicates a stable draw from the posterior distributions. To achieve identification, the impulse responses implied by B^{d} have to satisfy the following restrictions: a government spending shock should raise G_{it} and GDP_{it} for at least four quarters (Table 1). 5. Following Fry and Pagan (2011), use the median target approach to compute IRFs. For every 100 draws of the Q matrix satisfying the sign restrictions, save that matrix implying the model with the impulse response functions closest to the median IRFs. 6. Make 20,000 draws from the posterior distribution and discard the first 10,000 parameter draws as burn-in draws. For every remaining draw follow step 5. Among the 10,000 Q matrices, consider again only the model producing the IRFs nearest to the median IRFs. 7. Compute cumulated government spending multipliers following the approach proposed by Gordon and Krenn (2010) and Ramey and Zubairy (2018). Having normalized the variables of interest by real potential GDP, cumulated multipliers are computed simply as the ratio of discrete approximations of the integral of the median IRFs of real output and government purchases over a given time horizon h = 0,1,...,H: - M_H = (∑_{h=0}^H dGDP(h)) / (∑_{h=0}^H dG(h)). - Parameter uncertainty is accounted for by saving the 5th and 95th percentile of the distribution of the median as error bands. - Explosive draws from the unrestricted posterior are discarded (as in Cogley and Sargent (2005); Primiceri (2005); Sá et al. (2014)). ### Regimes, shadow rate percentiles, and impulse responses - Two euro-area monetary policy regimes are analyzed by conditioning IRFs on shadow rate percentiles: - normal times: period 2002q2 to 2008q3; shadow rate almost coincided with official Eonia rate and both were clearly positive. - ELB (effective lower bound): period 2012q4 to 2017q4; shadow rate systematically negative, capturing unconventional measures including the Asset Purchase Program (APP) and forward guidance. - Representative shadow rate percentiles chosen: - 77th percentile (2.75 percent; 2003q2) as representative of normal times (closest to the average shadow rate for that period, 2.82 percent). - 16th percentile (-2.23 percent; 2015q3) as representative of the ELB regime (closest to the average shadow rate for that period, -2.29 percent). - Key IRF observations: - A shock to government spending keeps spending persistently above baseline, taking about ten quarters to die out. - Output and net taxes respond positively to the shock, although the credible set of responses of net taxes often includes zero. - Comparing normal times against ELB: when the economy is at the ELB, the responses of output are mostly larger, with confidence bands not overlapping in several quarters after the shock (quarters 8-14). ### Cumulated government spending multipliers conditional on shadow rate levels (reported results) - Table 2 (as reported in source) — Cumulated Government Spending Multipliers Conditional on Two Levels of the Shadow Rate Representative of Normal Times and the ELB: - M_H | pctl(sr) M_H | pctl(sr) - Normal Times Effective Lower Bound - Horizon H pctl(sr) = 77 pctl(sr) = 16 - 1 year41,902,17 - 2 years81,482,59 - 3 years120,992,82 - 4 years160,742,96 - 5 years200,753,05 - Interpretation and statistical assessment: - Both short- and medium-term multipliers are systematically higher at the ELB relative to normal times. - One-year multipliers are of comparable magnitude across regimes, but dynamics diverge over time: in normal times the multiplier decays so that in the medium term (three to five years) the magnitude is around or less than 1; at the ELB the multiplier increases up to 3. - To assess statistical significance of differences, empirical distributions of differences (M_H | pctl(sr)=16 minus M_H | pctl(sr)=77) are computed across 10,000 posterior draws. From horizon three to five, 90 percent of these distributions do not include zero, indicating the difference is positive with high probability. ### Average cumulated multipliers across regimes (reported results) - Table 3 (as reported in source) — Average Cumulated Government Spending Multipliers in Normal Times and at the ELB: - mean(M_H | pctl(sr)) mean(M_H | pctl(sr)) - Horizon H Normal Times Effective Lower Bound - 1 year42,132,10 - 2 years81,572,44 - 3 years121,082,58 - 4 years160,792,73 - 5 years200,642,83 - Additional findings: - Averaging multipliers across percentiles belonging to each regime yields conclusions consistent with the specific-percentile results: one-year multipliers are very similar across regimes; at longer horizons multipliers diverge with the normal-times multiplier falling below 1 and the ELB multiplier much larger in the medium term. - Quantitatively, the three-year multiplier is reported as around 1 in normal times and 2.6 at the ELB; the five-year multiplier is reported as 0.6 in normal times and 2.8 at the ELB. - Distributions of differences between average multipliers indicate the difference is non-zero with 90 percent probability at horizons 2, 3 and 4 years (marginal significance at year 2); cannot exclude zero at horizons 1 and 5 years. ### Overall conclusions and interpretation - Key summarized conclusions: - (i) The difference between the size of the one-year government spending multiplier at the ELB and in normal times is neither economically nor statistically significant. - (ii) This difference increases over longer horizons, and its distribution is largely away from zero; in the medium term (around 3 years) the multiplier in normal times is about 1, whereas at the ELB it exceeds 2.5. - Contextual note from the source: - These empirical results are consistent with DSGE literature claiming higher fiscal multipliers at the ZLB, but the source stresses that in the data the ELB coincided with unconventional monetary policy; conditioning on the shadow monetary policy rate captures both the time-varying ELB effects and the effects of unconventional monetary policies. *Source: wpiea2019133 - Appendix A.3) includes both money and credit quantity aggregates, and the harmonized (PDF).* ### 3.4 Correlations of the Multiplier with the Shadow Rate and the ### 3.4 Correlations of the Multiplier with the Shadow Rate and the Business Cycle ### Main findings - There exists a negative correlation between the cumulated government spending multiplier and the shadow rate that survives after controlling for the state of the business cycle. - At a one-year horizon the conditional correlation is statistically insignificant; at horizons beyond one year the conditional correlation is strongly negative and statistically significant. - The multiplier is also negatively correlated with the business cycle after controlling for the level of the shadow rate; this correlation is statistically different from zero at a 1 percent level at all horizons except the first year. - Short run (one year): multiplier is very similar across normal times and the ELB. - Medium run (three years): average multiplier is about 1 in normal times and between 1.6 and 2.8 at the ELB, depending on specification. - The multiplier is inversely correlated with the level of the shadow monetary policy rate across percentiles of the shadow rate distribution. - Both the shadow rate and the state of the business cycle have autonomous correlations with the size of the fiscal multiplier in the eurozone. ### Conditional correlations (key statistics from Table 4) - Horizon 1 year (H = 4): corr(ˆε_M_H|bc_t, sr_{t−1}) = 0,2044 (p-value 0,1383); corr(ˆε_M_H|sr_t, bc_{t−1}) = -0,1822 (p-value 0,1873). - Horizon 2 years (H = 8): corr = -0,7200 (p-value 0,0000); corr = -0,3508 (p-value 0,0093). - Horizon 3 years (H = 12): corr = -0,8290 (p-value 0,0000); corr = -0,4052 (p-value 0,0024). - Horizon 4 years (H = 16): corr = -0,8389 (p-value 0,0000); corr = -0,3326 (p-value 0,0140). - Horizon 5 years (H = 20): corr = -0,8258 (p-value 0,0000); corr = -0,2814 (p-value 0,0393). - Note: Conditional correlations obtained by regressing cumulated multiplier on a constant and the lagged business cycle indicator (or lagged shadow rate) and correlating residuals with the lagged shadow rate (or lagged business cycle indicator). P-values in parentheses. ### Robustness checks and their outcomes - Three robustness variants examined: 1. Sign restrictions imposed for 2 quarters (instead of 4). 2. Lag structure of 2 quarters (L = 2) instead of 1 quarter. 3. Including 10-year government bond yields explicitly in the panel VAR. - Multipliers under robustness variants remain in the same order of magnitude as baseline; divergence across regimes increases with horizon, with ELB showing substantially higher multipliers in the medium run. - Representative robustness results on average cumulated multipliers (Table 5, selected): - Horizon 1 year (H = 4): Normal Times and ELB reported values include 1,71 and 1,70 (sign restrictions for 2 quarters); 2,22 and 2,91 (lag structure 2 quarters); 2,02 and 1,70 (including 10-year gov. bond yields). - Horizon 3 years (H = 12): Normal Times and ELB reported values include 1,15 and 2,03; 1,20 and 2,83; 1,39 and 1,63 across variants (values preserved as in table). - Distributional robustness (Table 6): At intermediate horizons (3 and/or 4 years) the 90-percent confidence interval for the difference between multipliers in ELB and normal times excludes zero across estimation variants. - Conditional correlations remain broadly negative and statistically significant across robustness variants (Table 7, selected entries): - Under sign restrictions imposed for 2 quarters: Horizon 2 years (H = 8) corr(ˆε_M_H|bc_t, sr_{t−1}) = -0,5811 (p-value 0,0000); corr(ˆε_M_H|sr_t, bc_{t−1}) = -0,5592 (p-value 0,0000). - Under lag structure of 2 quarters: Horizon 3 years (H = 12) corr(ˆε_M_H|bc_t, sr_{t−1}) = -0,7896 (p-value 0,0000); corr(ˆε_M_H|sr_t, bc_{t−1}) = -0,2828 (p-value 0,0383). - Under including 10-year gov. bond yields: Horizon 5 years (H = 20) corr(ˆε_M_H|bc_t, sr_{t−1}) = -0,4140 (p-value 0,0019); corr(ˆε_M_H|sr_t, bc_{t−1}) = -0,5329 (p-value 0,0000). ### Conclusions and policy implications - Empirical evidence for the euro area supports theoretical predictions of larger government spending multipliers at the ELB versus normal times, with qualifications: - No substantial difference at one-year horizon. - Marked divergence in multipliers at medium-run horizons (around three years), with ELB multipliers materially larger. - Practical implication: as monetary policy normalizes (phasing out of asset purchases, prospective normalization of policy rates), estimates suggest a medium-term multiplier for the euro area of about 1 to use in upcoming normal times scenarios. - The FAIPVAR-X approach enables conditioning multipliers on the shadow monetary policy rate and accounts for limited information and fiscal foresight in estimating multipliers. *Source: wpiea2019133 - 3.4 Correlations of the Multiplier with the Shadow Rate and the Business Cycle* ### References ### References ### Key citations - Aruoba, S. B., P. Cuba-Borda, and F. Schorfheide (2017). Macroeconomic Dynamics near the ZLB: A Tale of Two Countries. The Review of Economic Studies 85(1), 87–118. - 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. - 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. - Bi, H., W. Shen, and S.-C. S. Yang (2016). Debt-dependent effects of fiscal expansions. European Economic Review 88, 142–157. SI: The Post-Crisis Slump. - 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 (2017, July). The effects of fiscal policy at the effective lower bound. DNB Working Papers 565, Netherlands Central Bank, Research Department. - Braun, R. A., L. M. Körber, and Y. Waki (2013). Small and Orthodox Fiscal Multipliers at the Zero Lower Bound. FRB Atlanta Working Paper No. 2013-13. - 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. - 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. - Christiano, L., M. Eichenbaum, and S. Rebelo (2011). When Is the Government Spending Multiplier Large? Journal of Political Economy 119(1), 78–121. - Coenen, G., C. J. Erceg, C. Freedman, D. Furceri, M. Kumhof, R. Lalonde, D. Laxton, J. Lindé, A. Mourougane, D. Muir, S. Mursula, C. de Resende, J. Roberts, W. Roeger, S. Snudden, M. Trabandt, and J. in’t Veld (2012). Effects of Fiscal Stimulus in Structural Models. American Economic Journal: Macroeconomics 4(1), 22–68. - 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. - Cwik, T. and V. Wieland (2011). Keynesian Government Spending Multipliers and Spillovers in the Euro Area. Economic Policy 26(67), 493–549. - Davig, T. and E. M. Leeper (2011). Monetary-Fiscal Policy Interactions and Fiscal Stimulus. European Economic Review 55(2), 211–227. - 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), 27 427–431. - Eggertsson, G. B. (2010). What Fiscal Policy is Effective at Zero Interest Rates? In D. Acemoglu and M. Woodford (Eds.), NBER Macroeconomics Annual, Volume 25, pp. 59–112. Cambridge, MA: MIT Press. - Forni, M. and L. Gambetti (2010, May). Fiscal Foresight and the Effects of Government 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. - 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. - Fry, R. and A. Pagan (2011). Sign Restrictions in Structural Vector Autoregressions: A Critical Review. Journal of Economic Literature 49(4), 938–960. - 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. - Ilzetzki, E., E. G. Mendoza, and C. A. Végh (2013). How Big (Small?) Are Fiscal Multipliers? Journal of Monetary Economics 60(2), 239–254. - Kilponen, J., M. Pisani, S. Schmidt, V. Corbo, T. Hlédik, J. Hollmayr, S. Hurtado, P. Júlio, D. Kulikov, M. Lemoine, M. Lozej, H. Lundvall, J. F. Maria, B. Micallef, D. Papageorgiou, J. Rysanek, D. Sideris, C. Thomas, and G. De Walque (2015). Comparing Fiscal Multipliers Across Models and Countries in Europe. Working Paper Series 1760, European Central Bank. - 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. - 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. - Mertens, K. and M. O. Ravn (2014). Fiscal Policy in an Expectations Driven Liquidity Trap. Review of Economic Studies 81(4), 1637–1667. - 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. - Paustian, M. (2007). Assessing Sign Restrictions. The BE Journal of Macroeconomics 7(1). - Perotti, R. (1999). Fiscal Policy in Good Times and Bad. The Quarterly Journal of Economics 114(4), 1399–1436. - Pesaran, M. H. and R. Smith (1995). Estimating Long-Run Relationships From Dynamic Heterogeneous Panels. Journal of econometrics 68(1), 79–113. - Primiceri, G. E. (2005). Time Varying Structural Vector Autoregressions and Monetary Policy. Review of Economic Studies 72(252), 821–852. - 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. - Rubio-Ramírez, J. F., D. F. Waggoner, and T. Zha (2010). Structural Vector Autoregressions: Theory of Identification and Algorithms for Inference. Review of Economic Studies 77(2), 665–696. - 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. - Stock, J. H. and M. W. Watson (2005). Implications of Dynamic Factor Models for VAR Analysis. NBER Working Papers 11467. - Sutherland, A. (1997). Fiscal crises and aggregate demand: can high public debt reverse the effects of fiscal policy? Journal of Public Economics 65(2), 147–162. - Towbin, P. and S. Weber (2013). Limits of Floating Exchange Rates; the Role of Foreign Currency Debt and Import Structure. Journal of Development Economics 101(C), 179–194. - Woodford, M. (2011). Simple Analytics of the Government Expenditure Multiplier. American Economic Review 3(1), 1–35. - 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. ### Thematic focus of cited literature - Empirical and theoretical analyses of fiscal multipliers, including state-dependent and debt-dependent effects. - Methods for identification and estimation: FAVAR, SVAR, FSVAR, sign restrictions, structural factor models, panel VARs, dynamic factor models. - Fiscal-monetary interactions near the zero lower bound and liquidity traps. - Tests for stationarity and unit roots (Dickey and Fuller (1979); Kwiatkowski et al. (1992)). - Country- and model-comparison studies for European fiscal multipliers. ### Appendix A — Data #### A.1 Endogenous Variables - Variables: gross domestic product, net taxes and government spending. - Net taxes defined as the sum of government receipts of direct and indirect taxes minus transfers to businesses and individuals. - Government spending defined as the sum of government gross fixed capital formation and government consumption. - Source: Eurostat database available on the Thomson Reuters Datastream Economics database. - Transformations: variables transformed in real terms using the implicit GDP price deflator; normalized by dividing by real potential GDP. #### A.2 Exogenous Variables - Exogenous variables include: - 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. - Interaction term: European Central Bank’s shadow rate developed by Wu and Xia (2017). - U.S. output gap and U.S. inflation downloaded from the Federal Reserve Bank of St. Louis database. - U.S. Shadow Rate developed by Wu and Xia (2016). #### A.3 Informational Dataset - Informational dataset composed of 250 series downloaded from the Eurostat database available on the Thomson Reuters Datastream Economics database. - For each country the following variables were downloaded: - 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: Manufacturing; Industrial Production Index: Mig-Intermediate Goods; 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: NEER: 28 Trading Partners; 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. - Frequency conversions: - Converted from monthly to quarterly: Harmonized Government 10-Year Bond Yield; Industrial Production Index: Manufacturing; Industrial Production Index: Mid-Intermediate Goods; Money Supply: M1 - Contribution to Euro M1; Money Supply: M2 - Contribution to Euro M2; Money Supply: M3 - Contribution to Euro M3; Official Reserve Assets. - Converted from daily to quarterly: S&P BMI - Price Index. - Stationarity: variables transformed where appropriate and stationarity tested by the Dickey and Fuller (1979) and Kwiatkowski et al. (1992) tests. #### A.4 Business Cycle Indicator - Aggregate indicator computed as ∑_{i=1}^{10} GDP_i / GDP^{trend}_i, where i are the countries in the sample, and GDP_i, GDP^{trend}_i are the real GDP and the real potential GDP of each country. - Interpretation: large and positive indicates most countries in expansion; large and negative indicates most countries in recession. ### Appendix B — Additional Robustness Checks #### Table B.1: Conditional Correlations of Cumulated Multipliers with the Contemporaneous Shadow Rate and the Contemporaneous Business Cycle - Horizon H | Hcorr(ˆε^M_H | bc_t, sr_t) | corr(ˆε^M_H | sr_t, bc_t) - 1 year | 40,2264 | -0,1329 - (0,0998) (0,3381) - 2 years | 8 | -0,6732 | -0,2688 - (0,0000) (0,0494) - 3 years | 12 | -0,7893 | -0,3265 - (0,0000) (0,0160) - 4 years | 16 | -0,8050 | -0,2697 - (0,0000) (0,0486) - 5 years | 20 | -0,7953 | -0,2327 - (0,0000) (0,0904) - Notes: Conditional correlations obtained by running two auxiliary regressions and computing correlation coefficients between residuals and variables of interest. First regresses cumulated multiplier on a constant and contemporaneous business cycle indicator; correlation computed between residuals and contemporaneous shadow rate. Second regresses cumulated multiplier on a constant and contemporaneous shadow rate; correlation computed between residuals and contemporaneous business cycle indicator. Computations replicated at different horizons H for the cumulated multipliers. P-values in parentheses. #### Table B.2: Robustness Checks on Conditional Correlations of Cumulated Multipliers - Columns: Sign restrictions imposed for 2 quarters; Lag structure of 2 quarters; Including 10-year gov. bond yields. - For each specification the table reports ˆε^M_H | bc_t, sr_t and ˆε^M_H | sr_t, bc_t at horizons. - 1 year: - 40,1730 | -0,3690 - (0,2110) (0,0060) - -0,4435 | -0,3391 - (0,0008) (0,0121) - 0,5022 | -0,4617 - (0,0001) (0,0004) - 2 years: - 8 | -0,5519 | -0,5035 - (0,0000) (0,0001) - -0,7689 | -0,1933 - (0,0000) (0,1614) - 0,1709 | -0,5185 - (0,2167) (0,0001) - 3 years: - 12 | -0,7774 | -0,5346 - (0,0000) (0,0000) - -0,7646 | -0,2190 - (0,0000) (0,1116) - -0,1495 | -0,5249 - (0,2807) (0,0000) - 4 years: - 16 | -0,7785 | -0,5049 - (0,0000) (0,0001) - -0,7235 | -0,1320 - (0,0000) (0,3412) - -0,3283 | -0,5077 - (0,0154) (0,0001) - 5 years: - 20 | -0,7675 | -0,4655 - (0,0000) (0,0004) - -0,6706 | -0,1224 - (0,0000) (0,3778) - -0,3970 | -0,4837 - (0,0030) (0,0002) - Notes: Conditional correlations obtained as in Table B.1. Computations replicated at different horizons H for the cumulated multipliers. P-values in parentheses. *Content derived from the "References" and appended "Appendix" sections of the source PDF.* --- _Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019133.pdf_