2.1 Empirical Model

The empirical model builds on the Interacted Panel Vector Auto-Regressive (IPVAR) framework developed by Towbin and Weber (2013) and Sá et al. (2014). This model is well suited to our purposes because the presence of interaction terms allows us to capture nonlinearities in the reaction of variables of interest to government spending shocks conditional on the whole distribution of the nominal shadow interest rate.

The model specification takes the following general structural form:

where t = 1, …,T denotes the time dimension; j = 1,…, N denotes the country dimension; and k = 1,…, L represents the lag structure. The vector of endogenous variables is denoted by y_i,tt; the interaction term is represented by x_t; while the vectors of two sets of exogenous variables are denoted by f_(t-1-:t-A) (discussed in Subsection 2.2) and z_t-1 (foreign exogenous variables, also discussed in Subsection 2.2). Furthermore, coefficient κj is the country-specific intercept of country $$j:{\kappa }_j^1$$ is the country-specific coefficient of the interaction term; Γj,k is the matrix of autoregressive coefficients attached to the endogenous variables; ν_j is the matrix of country-specific coefficients attached to the first set of exogenous variables; v¹ represents the pooled estimated coefficients of the another set of exogenous variables; D_j,t is an indicator variable for each country (equal to 1 if i = j, and 0 otherwise); and, lastly, ε_i,t is a vector of i.i.d. residuals, which are uncorrelated across countries by assumption. It is noteworthy that the interaction term, x_t, affects both the level and the dynamic relationship across endogenous variables through $${\kappa }_j^{1}and{\mathrm{\Gamma }}_k^1$$ (for more technical details on the IPVAR framework see, e.g., Sá et al., 2014).

To allow for as much heterogeneity as possible, we utilize a panel model with fixed effects and heterogeneous slopes, which we estimate using the mean group estimator. This estimator has been shown to perform better than alternative estimators in dynamic panels (see, e.g., Pesaran and Smith, 1995 and Canova and Ciccarelli, 2013, among others). Due to data availability constraints we estimate homogeneous slopes of lagged interacted terms, X_ty_i,t-k, and foreign exogenous variables, z_t-1.

Matrix B_i,t is a (q x q) lower triangular matrix with ones on the main diagonal. The recursive structure imposed on matrix B_i,t implies that the covariance matrix of the residuals, Σ_ε, is diagonal. Given that the FAIPVAR-X model requires the estimation of a large number of parameters, for the sake of parsimony, we produce the baseline results with a uniform lag structure on one quarter (L = 1). We re-run the estimation also with two lags for robustness (Section 4).

2.2 Data and Baseline Specification

Our dataset is composed of quarterly data and covers the period from 2002q2 to 2017q4.⁵ We consider ten of the eleven countries that joined the EA when it came into existence: Austria, Belgium, Finland, France, Germany, Ireland, Italy, Netherlands, Portugal and Spain. In line with Auerbach and Gorodnichenko (2013), we exclude Luxembourg being a small economy, which exhibits large and volatile changes in government spending series. For details on the construction of the dataset, see Appendix A.

To examine the macroeconomic effects of fiscal shocks, the VAR literature has traditionally used variations of the following vector of endogenous variables:

where G_i,t, GDP_i,t and T_i,t represent real government purchases (the sum of government gross fixed capital formation and government consumption), real gross domestic product and real net taxes (the sum of government receipts of direct and indirect taxes minus transfers to businesses and individuals), respectively. We make a number of modifications to this simple specification to overcome a series of issues we discuss below.

First, to simplify the procedure related to the computation of government spending multipliers, we divide all endogenous variables by the real potential GDP of the corresponding country. This way there is no need to take the logarithm of the variables and perform ex-post conversions of the estimated elasticities to dollar equivalents, avoiding potential biases. In fact, ex-post conversion requires the use of constant sample averages of the ratios of fiscal variables to GDP, which may instead vary over time, potentially biasing the size of the multipliers. This problem is even more acute in nonlinear models, such as that adopted in this paper (for more details on this issue see, e.g., Gordon and Krenn, 2010 and Ramey and Zubairy, 2018, among others). We compute real potential GDP using the filter recently proposed by Hamilton (2018), which avoids the spurious persistence in the cyclical component implied by the traditional Hodrick-Prescott (HP) filter.

Second, to the basic set of endogenous variables listed in vector y_i,t, we add common factors, via principal components, extracted from a large number of macroeconomic times series. In fact, VAR models are characterized by a trade-off between parsimony and omission of relevant variables, which can give rise to nonfundamentalness of the identified shocks (see, e.g., Forni et al., 2009). Extracting information from a large set of macroeconomic variables overcomes the limited information problem because the principal components proxy the unobserved factors affecting most macroeconomic variables (see Fragetta and Gasteiger, 2014 for further details). Similar to Bernanke et al. (2005), we implement a two-step estimation procedure. As a first step, we extract five common factors, as established by the Bai and Ng (2007) IC_p2 information criterion. The second step is adding the five factors to our vector of endogenous variables, which reads as follows:

where F_t is a 1 × 5 vector common to all countries, but that may have a different impact in each country, capturing also potential spillovers across countries.

Third, among the exogenous variables, we add the f_(t|t-1:t-4) series. This represents the forecast of time-t government spending over the past 12 months (four quarters), published by the Economist Intelligence Unit. The addition of this variable represents a way to purge our structural government spending shocks from the change in government spending already anticipated by economic agents, and hence to solve the problem known in the literature as fiscal foresight. Fiscal foresight is the phenomenon by which private agents, mainly due to legislative and implementation lags, can anticipate future movements in government spending. Failing to account for them in the identification of what are meant to be unanticipated government spending shocks may give rise to endogeneity and bias the results (see, e.g., Forni and Gambetti, 2010 and Leeper et al., 2013 among others for further details).

Fourth, we use as interaction term, x_t, the European Central Bank’s shadow monetary policy rate developed by Wu and Xia (2017) (discussed in Section 1), which allows us to control for the overall monetary policy stance in the eurozone, and study the effects of a government spending shock at each percentile of the shadow rate distribution. Since this rate is available from 2004Q3 onward, for the very beginning of the sample, we complement it with the Main Refinancing Operations (MRO) rate, given that the two, until 2008, are virtually indistinguishable. To avoid potential reversed causality issues, we use the first lag of the shadow rate (i.e. x_t = sr_t-1), such that it is predetermined relative to the endogenous variables. It must be said, however, that the five factors do contain information on the contemporaneous monetary policy stance although, for our purposes, there is no need to explicitly identify a monetary policy shock. In particular, the informational dataset (see Appendix A.3) includes both money and credit quantity aggregates, and the harmonized government ten-year bond yield. The latter captures both expectations on monetary policy and market sentiment toward government debt dynamics. In a robustness check (Section 4) we also include the harmonized government ten-year bond yield explicitely in the panel VAR specification rather than in the computation of the factors.

Lastly, in order to account for international factors which may influence our variables of interest, we add as exogenous variables also 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).

All the abovementioned modifications made to the traditional fiscal VAR specification are implemented within the IPVAR model. Therefore we label the model used in this paper as a factor-augmented interacted panel vector-autoregressive model purified of expectations (FAIPVAR-X).

2.3 Inference, Identification and Computation of Cumulated Government Spending Multipliers

In line with Sá et al. (2014), we estimate the FAIPVAR-X model presented in equation (1) and compute cumulated government spending multipliers adopting the following seven steps:

• 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 $$B_{i,t}^{-1}$$

• 4. Use a sign restriction strategy to identify an unexpected government spending shock and compute the resulting impulse response functions (IRFs). More specifically, follow the same procedure of Sá et al. (2014), which in turn uses the algorithm developed by Rubio-Ramírez et al. (2010): after defining $$V_x^d$$ as the Cholesky decomposition of the reduced form variance-covariance matrix $${\mathrm{\Sigma }}_x^d$$, draw an orthonormal matrix Q such that Q’Q = I, from which it follows that $$B^d=V_x^{d}Qand{\mathrm{\Sigma }}_x^d=B^{d\prime }{B}^d=V_x^{d\prime }{Q}^{\prime }Q{V}_x^d$$ where d indicates a stable draw from the posterior distributions.⁶ To achieve identification, the impulse responses implied by B^d have to satisfy the following restrictions: a government spending shock should raise GDP_it and G_it for at least four quarters (Table 1).⁷

Table 1:

Sign Restrictions for Identifying the Government Spending Shock.

Table 1:

Sign Restrictions for Identifying the Government Spending Shock.

Variable	Sign	Periods
Git	+	4
GDPit	+	4
Tit	None	None
Ft	None	None

View Table

Table 1:

Sign Restrictions for Identifying the Government Spending Shock.

Variable	Sign	Periods
Git	+	4
GDPit	+	4
Tit	None	None
Ft	None	None

• 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.⁸

• 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). As already discussed, having normalized the variables of interest by real potential GDP, circumvents any concerns related to ex-post conversion. Thus, 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:

  $$\begin{array}{cc}{M}_H=\frac{{\mathrm{\Sigma }}_{h=0}^{H}dGDP\left(h\right)}{{\mathrm{\Sigma }}_{h=0}^{H}dG\left(h\right)}\mathrm{.}& \left(4\right)\end{array}$$

  We account for parameter uncertainty by saving the 5th and 95th percentile of the distribution of the median as error bands.⁹

3.1 Impulse Responses Conditional on the Shadow Rate

In this subsection we report impulse response functions (IRFs) of government spending, output, and net taxes—all in real terms—to an unexpected shock to government spending. One of the advantages of the FAIPVAR-X model is that it allows conditioning the IRFs on a specific percentile of the distribution of the shadow interest rate. In other words, we can interpret the IRFs as the dynamic reaction of macroeconomic variables to a shock to government spending occurring when the shadow rate takes a value corresponding to a given percentile of its own distribution.

For expositional ease we report IRFs conditional on two percentiles that are representative of two euro-area monetary policy regimes.We label the first regime normal times. This regime corresponds to the period between the beginning of our sample (2002q2) and the bankruptcy of Lehman Brothers (2008q3). In this period the shadow rate almost coincided with the official Eonia rate and the two were clearly positive (see Figure 1). We label the second regime ELB. This regime is comprised between 2012q4, the quarter following ECB President Mario Draghi’s famous ‘Whatever it takes’ speech, to the end of the sample (2017q4). During this period, the ECB lowered the monetary policy rate first to the ZLB and then to negative values. In other words, this period is characterized by a binding but time-varying ELB. The systematically negative shadow rate captures a series of unconventional measures including the Asset Purchase Program (APP) and forward guidance.¹⁰

The choice of regimes translates into a number of choices as far as shadow rate percentiles are concerned. We pick the 77th percentile (2.75 percent; 2003q2) as a representative of the normal times regime, being the closest to the average shadow rate for that period (2.82 percent); and the 16th percentile (-2.23 percent; 2015q3) as a representative of the ELB regime, again being the closest to the average shadow rate for that period (-2.29 percent). Figure 2 contrasts the IRFs for the normal times regime with those for the ELB regime.

Impulse Responses to a Government Spending Shock in Normal Times and at the ELB.

Citation: IMF Working Papers 2019, 133; 10.5089/9781498314947.001.A001

Notes: Impulse responses in percent to a shock of size one standard deviation. Bold lines represent median responses. Shadowed areas and dashed lines represent 90 percent confidence bands.

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Impulse Responses to a Government Spending Shock in Normal Times and at the ELB.

Citation: IMF Working Papers 2019, 133; 10.5089/9781498314947.001.A001

Notes: Impulse responses in percent to a shock of size one standard deviation. Bold lines represent median responses. Shadowed areas and dashed lines represent 90 percent confidence bands.

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Impulse Responses to a Government Spending Shock in Normal Times and at the ELB.

Citation: IMF Working Papers 2019, 133; 10.5089/9781498314947.001.A001

Notes: Impulse responses in percent to a shock of size one standard deviation. Bold lines represent median responses. Shadowed areas and dashed lines represent 90 percent confidence bands.

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A few remarks are in order. First, in all cases a shock to government spending keeps spending itself persistently above baseline and it takes about ten quarters to die out. Second, output and net taxes respond positively to the shock, although the credible set of responses of net taxes often includes zero. Third, comparing the results for normal times against those for ELB unveils that, when the economy is at the ELB, the responses of output are for the most part larger, with their confidence bands not overlapping in several quarters after the occurrence of the shock (quarters 8–14).

3.2 Cumulated Government Spending Multipliers Conditional on Shadow Rate Levels

Based on these impulse responses, we can compute the cumulated government spending multipliers at several time horizons explained in Subsection 2.3. Results are reported in Table 2. Both in the short and the medium term the multiplier is systematically higher when the economy is at the ELB, relative to normal times. Importantly though, while the one-year multiplier is of comparable magnitude, there is a stark difference in the dynamics of the multiplier across the two regimes as times goes by. 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 magnitude of the multiplier increases up to 3.

Table 2:

Cumulated Government Spending Multipliers Conditional on Two Levels of the Shadow Rate Representative of Normal Times and the ELB.

Notes: Multipliers computed as in Equation (4). Percentiles refer to chosen percentiles of the shadow rate. The 77th percentile is representative of the normal times regime. The 16th percentile is representative of the ELB regime. H identifies the number of quarters after the shock.

Table 2:

Cumulated Government Spending Multipliers Conditional on Two Levels of the Shadow Rate Representative of Normal Times and the ELB.

Horizon	H	MH|pctl(sr) Normal Times pctl (sr) = 77	MH|pctl(sr) Effective Lower Bound pctl (sr) = 16
1 year	4	1,90	2,17
2 years	8	1,48	2,59
3 years	12	0,99	2,82
4 years	16	0,74	2,96
5 years	20	0,75	3,05

Notes: Multipliers computed as in Equation (4). Percentiles refer to chosen percentiles of the shadow rate. The 77th percentile is representative of the normal times regime. The 16th percentile is representative of the ELB regime. H identifies the number of quarters after the shock.

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Table 2:

Cumulated Government Spending Multipliers Conditional on Two Levels of the Shadow Rate Representative of Normal Times and the ELB.

Horizon	H	MH|pctl(sr) Normal Times pctl (sr) = 77	MH|pctl(sr) Effective Lower Bound pctl (sr) = 16
1 year	4	1,90	2,17
2 years	8	1,48	2,59
3 years	12	0,99	2,82
4 years	16	0,74	2,96
5 years	20	0,75	3,05

Notes: Multipliers computed as in Equation (4). Percentiles refer to chosen percentiles of the shadow rate. The 77th percentile is representative of the normal times regime. The 16th percentile is representative of the ELB regime. H identifies the number of quarters after the shock.

A fair question is whether the difference between the two sets of multiplier is statistically significant. Bayesian inference does not allow us to construct a test as in the frequentist approach. Therefore we follow an approach analogous to Caggiano et al. (2015). We compute empirical distributions of the differences computed as multipliers conditional on the level of the shadow rate representative of the ELB regime (M_{H|pctl(sr)=16}) minus multipliers conditional on the level of the shadow rate representative of the normal times regime M_{H|pctl(sr)=77} and verify whether a very large part of the distributions include zero or not. In particular, for each of the 10,000 parameter draws from the posterior distribution, we compute the multipliers as in Equation (4), evaluate them at the two percentiles of interest, and compute the difference between the two. Figure 3 plots the distributions of the difference between the two multipliers cumulated at various time horizons together with 90 percent confidence bands. It turns out that, from horizon three to five, 90 percent of these distribution do not include zero, indicating that the difference between the two multiplier is positive with high probability.

Distributions of Differences in Cumulated Government Spending Multipliers between Two Levels of the Shadow Rate Representative of Normal Times and the ELB.

Citation: IMF Working Papers 2019, 133; 10.5089/9781498314947.001.A001

Notes: Empirical distributions of the differences computed as multipliers conditional on the level of the shadow rate representative of the ELB regime M_{H|pctl(sr)=16} minus multipliers conditional on the level of the shadow rate representative of the normal times regime M_{H|pctl(sr)=77}. Multipliers are computed as in Equation (4) for each of the 10,000 parameter draws from the posterior distribution. Vertical dotted lines represent the 5th and the 95th percentiles of the distribution of differences. H identifies the number of quarters after the shock.

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Distributions of Differences in Cumulated Government Spending Multipliers between Two Levels of the Shadow Rate Representative of Normal Times and the ELB.

Citation: IMF Working Papers 2019, 133; 10.5089/9781498314947.001.A001

Notes: Empirical distributions of the differences computed as multipliers conditional on the level of the shadow rate representative of the ELB regime M_{H|pctl(sr)=16} minus multipliers conditional on the level of the shadow rate representative of the normal times regime M_{H|pctl(sr)=77}. Multipliers are computed as in Equation (4) for each of the 10,000 parameter draws from the posterior distribution. Vertical dotted lines represent the 5th and the 95th percentiles of the distribution of differences. H identifies the number of quarters after the shock.

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Distributions of Differences in Cumulated Government Spending Multipliers between Two Levels of the Shadow Rate Representative of Normal Times and the ELB.

Citation: IMF Working Papers 2019, 133; 10.5089/9781498314947.001.A001

Notes: Empirical distributions of the differences computed as multipliers conditional on the level of the shadow rate representative of the ELB regime M_{H|pctl(sr)=16} minus multipliers conditional on the level of the shadow rate representative of the normal times regime M_{H|pctl(sr)=77}. Multipliers are computed as in Equation (4) for each of the 10,000 parameter draws from the posterior distribution. Vertical dotted lines represent the 5th and the 95th percentiles of the distribution of differences. H identifies the number of quarters after the shock.

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3.3 Average Cumulated Government Spending Multipliers in Normal Times and the ELB

With the FAIPVAR-X model, IRFs can be computed conditional on all percentiles of the shadow rate distribution, which can be easily reconducted to a specific quarter in the history of the EMU. This allows us to compute time series of the cumulated government spending multipliers. Table 3 reports the average of these time series over the periods we identify as normal times and ELB. This way we can check whether the results based on the two specific percentiles representative of the two periods, hold also on average. It turns out that using average multipliers does not alter our conclusions. One-year multipliers are still very similar across the two regimes while, at longer time horizons, multipliers still diverge across regimes as the horizon increases, with the normal-times multiplier falling below 1 and the ELB multiplier being much larger in the medium term. Also from a quantitative viewpoint estimates are similar, with the three-year multiplier around one in normal times and 2.6 at the ELB, and the five-year multiplier 0.6 in normal times and 2.8 at the ELB.

Table 3:

Average Cumulated Government Spending Multipliers in Normal Times and at the ELB.

Notes: Multipliers are computed as in Equation (4) for each percentile of the shadow rate distribution and averaged across the percentiles belonging to the normal times and the ELB regime. H identifies the number of quarters after the shock.

Table 3:

Average Cumulated Government Spending Multipliers in Normal Times and at the ELB.

Horizon	H	mean MH|pctl(sr) Normal Times	mean MH|pctl(sr) Effective Lower Bound
1 year	4	2,13	2,10
2 years	8	1,57	2,44
3 years	12	1,08	2,58
4 years	16	0,79	2,73
5 years	20	0,64	2,83

Notes: Multipliers are computed as in Equation (4) for each percentile of the shadow rate distribution and averaged across the percentiles belonging to the normal times and the ELB regime. H identifies the number of quarters after the shock.

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Table 3:

Average Cumulated Government Spending Multipliers in Normal Times and at the ELB.

Horizon	H	mean MH|pctl(sr) Normal Times	mean MH|pctl(sr) Effective Lower Bound
1 year	4	2,13	2,10
2 years	8	1,57	2,44
3 years	12	1,08	2,58
4 years	16	0,79	2,73
5 years	20	0,64	2,83

Notes: Multipliers are computed as in Equation (4) for each percentile of the shadow rate distribution and averaged across the percentiles belonging to the normal times and the ELB regime. H identifies the number of quarters after the shock.

Also in the context of average multipliers we construct distributions of the difference between average multipliers analogous to those constructed on the multipliers conditional on specific shadow rate percentiles (reported on Subsection 3.2). In this case, for each of the 10,000 parameter draws from the posterior distribution, we compute the average multiplier for the two regimes and save the difference between the two. Figure 4 plots the distributions of these differences with 90 percent confidence bands, at various time horizons. The interpretation of each subplot is identical to that of Figure 3. It turns out that the difference between average multipliers is non-zero with 90 percent probability at horizons 2, 3 and 4 years (on the margin of significance at year 2). We cannot exclude zero at horizons 1 and 5 years.

Distributions of Differences in Average Cumulated Government Spending Multipliers between Normal Times and the ELB.

Citation: IMF Working Papers 2019, 133; 10.5089/9781498314947.001.A001

Notes: Empirical distributions of the differences computed as average multipliers in the ELB regime minus average multipliers in the normal times regime. Multipliers are computed as in Equation (4) for each of the 10,000 parameter draws from the posterior distribution. Vertical dotted lines represents the 5th and the 95th percentiles of the distribution of differences. H identifies the number of quarters after the shock.

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Distributions of Differences in Average Cumulated Government Spending Multipliers between Normal Times and the ELB.

Citation: IMF Working Papers 2019, 133; 10.5089/9781498314947.001.A001

Notes: Empirical distributions of the differences computed as average multipliers in the ELB regime minus average multipliers in the normal times regime. Multipliers are computed as in Equation (4) for each of the 10,000 parameter draws from the posterior distribution. Vertical dotted lines represents the 5th and the 95th percentiles of the distribution of differences. H identifies the number of quarters after the shock.

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Distributions of Differences in Average Cumulated Government Spending Multipliers between Normal Times and the ELB.

Citation: IMF Working Papers 2019, 133; 10.5089/9781498314947.001.A001

Notes: Empirical distributions of the differences computed as average multipliers in the ELB regime minus average multipliers in the normal times regime. Multipliers are computed as in Equation (4) for each of the 10,000 parameter draws from the posterior distribution. Vertical dotted lines represents the 5th and the 95th percentiles of the distribution of differences. H identifies the number of quarters after the shock.

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Taken together, Subsections 3.2 and 3.3 suggest that in the EA (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, and its distribution is largely away from zero, at longer time horizons; in the medium-term (say 3 years) while the multiplier in normal times is about 1, at the ELB it exceeds 2.5.

These results are in line with a strand of the theoretical (DSGE) literature that claims the fiscal multiplier to be much higher at the ZLB (Christiano et al., 2011; Coenen et al., 2012; Davig and Leeper, 2011; Eggertsson, 2010; Kilponen et al., 2015; Woodford, 2011, among others).¹¹ However, it has to be stressed that while in DSGE models it is possible to quantify the effects of the ZLB in isolation, in the data this task is much more difficult. In fact, when the policy rate reached the ELB, monetary policy did not simply cease to operate; it continued to function in an unconventional manner. By conditioning the computation of the multiplier on the shadow monetary policy rate, we simultaneously capture not only the effects of the time-varying ELB in the EA, but also those of all unconventional monetary policies.

3.4 Correlations of the Multiplier with the Shadow Rate and the Business Cycle

There is a strand of the literature that finds government spending multipliers to be dependent on the business cycle in advanced economies, and in particular to be higher in recessions relative to expansions (see Auerbach and Gorodnichenko, 2012, 2013; Batini et al., 2012, among others).¹² Given that the shadow rate typically becomes smaller when the economy is below trend and viceversa, in this subsection we verify that the correlation between the fiscal multiplier and the shadow rate survives also when we control for the state of the business cycle.

The FAIPVAR-X model, by conditioning on the various percentiles of the shadow rate, allows us to compute time series of multipliers. Thus, we compute the correlation between the cumulated multipliers and the lagged shadow rate, conditional on a lagged business cycle indicator (see Appendix A for details). In practice, we run the following regression:

where M_H,t is the series of multipliers at horizon H, c is a constant, bc_t is the business cycle indicator, b₀ and b₁ are regression coefficients, and $${ϵ}_t^{M_{H|bc}}$$ is the error term. From this regression, we save the residuals, $${\widehat{ϵ}}_t^{M_{H|bc}}$$, and we compute the correlation coecient with the lagged shadow rate (sr_t-1). Results are reported in Table 4. While statistically insignificant at a one-year horizon, the conditional correlation is strongly negative and significant at all other horizons. This finding gives assurance that the association of the shadow rate with the size of the government spending multiplier at horizons greater than one is autonomous from that of the business cycle. Results based on contemporaneous variables are virtually the same and reported in Appendix B (Table B.1).

Table 4:

Conditional Correlations of Cumulated Multipliers with the Lagged Shadow Rate and the Lagged Business Cycle.

Notes: Conditional correlations are obtained running two auxiliary regressions and computing the correlation coefficients between the residuals of these regressions and the variables of interest. The first regresses the cumulated multiplier on a constant and the lagged business cycle indicator. The correlation is computed between the residuals of this regression and the lagged shadow rate. The second regresses the cumulated multiplier on a constant and the lagged shadow rate. The correlation is computed between the residuals of this regression and the lagged business cycle indicator. These computations are replicated at different horizons H for the cumulated multipliers. P-values are in parentheses.

Table 4:

Conditional Correlations of Cumulated Multipliers with the Lagged Shadow Rate and the Lagged Business Cycle.

Horizon	H	$$corr({\widehat{ϵ}}_t^{M_{H|bc}},s{r}_{t-1})$$	$$corr({\widehat{ϵ}}_t^{M_{H|sr}},{bc}_{t-1})$$
1 year	4	0,2044 (0,1383)	-0,1822 (0,1873)
2 years	8	-0,7200 (0,0000)	-0,3508 (0,0093)
3 years	12	-0,8290 (0,0000)	-0,4052 (0,0024)
4 years	16	-0,8389 (0,0000)	-0,3326 (0,0140)
5 years	20	-0,8258 (0,0000)	-0,2814 (0,0393)

Notes: Conditional correlations are obtained running two auxiliary regressions and computing the correlation coefficients between the residuals of these regressions and the variables of interest. The first regresses the cumulated multiplier on a constant and the lagged business cycle indicator. The correlation is computed between the residuals of this regression and the lagged shadow rate. The second regresses the cumulated multiplier on a constant and the lagged shadow rate. The correlation is computed between the residuals of this regression and the lagged business cycle indicator. These computations are replicated at different horizons H for the cumulated multipliers. P-values are in parentheses.

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Table 4:

Conditional Correlations of Cumulated Multipliers with the Lagged Shadow Rate and the Lagged Business Cycle.

Horizon	H	$$corr({\widehat{ϵ}}_t^{M_{H|bc}},s{r}_{t-1})$$	$$corr({\widehat{ϵ}}_t^{M_{H|sr}},{bc}_{t-1})$$
1 year	4	0,2044 (0,1383)	-0,1822 (0,1873)
2 years	8	-0,7200 (0,0000)	-0,3508 (0,0093)
3 years	12	-0,8290 (0,0000)	-0,4052 (0,0024)
4 years	16	-0,8389 (0,0000)	-0,3326 (0,0140)
5 years	20	-0,8258 (0,0000)	-0,2814 (0,0393)

Notes: Conditional correlations are obtained running two auxiliary regressions and computing the correlation coefficients between the residuals of these regressions and the variables of interest. The first regresses the cumulated multiplier on a constant and the lagged business cycle indicator. The correlation is computed between the residuals of this regression and the lagged shadow rate. The second regresses the cumulated multiplier on a constant and the lagged shadow rate. The correlation is computed between the residuals of this regression and the lagged business cycle indicator. These computations are replicated at different horizons H for the cumulated multipliers. P-values are in parentheses.

Nonetheless, EA data still give support to the view that the multiplier is larger in periods of economic slack. We arrive at this conclusion by running a second regression of the multiplier on a constant and the lagged shadow rate:

and by computing the correlation between the residuals of this regression, $${\widehat{ϵ}}_t^{M_{H|bc}}$$, and the lagged business cycle indicator. This correlation coefficient is negative and statistically different from zero at a 1 percent level at all time horizons except for the first year. In other words, after controlling for the level of the shadow rate, the multiplier is still negatively correlated with the business cycle. Also in this case, results based on contemporaneous variables, reported in Appendix B (Table B.1), are virtually the same. In sum, both the shadow rate and the state of the business cycle have an autonomous correlation with the size of the fiscal multiplier in the eurozone.