In Search of the Transmission Mechanism of Fiscal Policy in the Euro Area

This paper applies the DSGE-VAR methodology to assess the size of fiscal multipliers in the data and the relative contributions of two transmission mechanisms of government spending shocks, namely hand-to-mouth consumers and Edgeworth complementarity. Econometric experiments show that a DSGE model with Edgeworth complementarity is a better representation of the transmission mechanism of fiscal policy as it yields dynamic responses close to those obtained with the flexible DSGE-VAR model (i.e. an impact output multiplier larger than one and a crowding-in of private consumption). The estimated share of hand-to-mouth consumers is too small to replicate the positive response of private consumption.

ABSTRACT : This paper applies the DSGE-VAR methodology to assess the size of fiscal multipliers in the data and the relative contributions of two transmission mechanisms of government spending shocks, namely hand-to-mouth consumers and Edgeworth complementarity. Econometric experiments show that a DSGE model with Edgeworth complementarity is a better representation of the transmission mechanism of fiscal policy as it yields dynamic responses close to those obtained with the flexible DSGE-VAR model (i.e. an impact output multiplier larger than one and a crowding-in of private consumption). The estimated share of hand-to-mouth consumers is too small to replicate the positive response of private consumption. JEL CLASS.: C32, E32, E62.

NON TECHNICAL SUMMARY
Due to concerns about high levels of public debt, many European countries have engaged in large consolidation programs in recent years. The issue of the effectiveness of these programs has initiated a vivid debate on the evaluation of government spending multipliers. Two classes of models have been extensively used to assign a quantitative value to this concept: dynamic stochastic general equilibrium (DSGE) models and vector autoregressions (VARs). However, each approach has disadvantages. While DSGE models potentially face a misspecification problem in imposing excessive restrictions on the data, structural VARs might be sensitive to identification strategies (Ramey, 2011a). The implications drawn from one model or another might not reveal the true policy effects. This paper combines DSGE and VAR models to provide new insights into the transmission mechanisms of fiscal shocks in the euro area. This DSGE-VAR approach relaxes the strong cross-equation restrictions created by the DSGE model and thus allows possible misspecification of the structural model to be considered (Del Negro and Schorfheide, 2004, Del Negro, Schorfheide, Smets, and Wouters, 2007, and Del Negro and Schorfheide, 2009).
Armed with this original tool, we evaluate the relative contributions to the size of estimated fiscal multipliers of two transmission mechanisms of government spending shocks advanced previously in the literature. The first relies on the presence of hand-to-mouth (HtM) consumers in the population, i.e., households that do not have access to financial markets and simply consume their disposable income in each period. Galí et al. (2007) and Forni et al. (2009) show that the interaction of such agents with both real and nominal rigidities increases the government spending multiplier. The second transmission mechanism allows government spending to enter -in a non-separable way-the household's utility function (GiU), such that government activity directly affects the marginal utility of consumption. Bouakez and Rebei (2007), Fève et al. (2013) and Coenen et al. (2013) show that when private consumption and public expenditures display a sufficient amount of Edgeworth complementarity, households have incentives to consume and to work more, thereby generating larger fiscal multipliers. Because each competing model nests the Smets-Wouters specification (Baseline), one can vary the magnitude of the parameter summarizing one of the mechanisms to understand how government spending shocks propagate into the model economy.
Our main findings are the following. First, the Bayesian estimation shows that a DSGE model with non-separable government spending in the utility function outperforms a model with hand-to-mouth consumers in terms of fit and yields larger fiscal multipliers. The version with Edgeworth complementarity provides a multiplier of approximately 1.75, whereas the version with hand-to-mouth consumers yields a value lower than one. Second, we use the DSGE-VAR approach to assess the performance of each version. On impact, the multiplier is approximately 1.5, and its estimated value is weakly affected by the model's specification. This result can be interpreted as suggesting that the data want a multiplier larger than one. Specifically, the DSGE-VAR framework is used to investigate whether the fiscal multipliers obtained in a constrained DSGE model are far from those obtained in a DSGE-VAR model with the same structural features. We obtain a sizable increase in the estimated value of the multiplier in the Baseline and HtM versions, while it remains very similar in the GiU specification and in the model including both mechanisms. This supports our claim that Edgeworth complementarity is a better representation of the transmission mechanism of fiscal policy in the euro area.

INTRODUCTION
Due to concerns about high levels of public debt, many European countries have engaged in large consolidation programs in recent years. The issue of the effectiveness of these programs has initiated a vivid debate on the evaluation of government spending multipliers. Two classes of models have been extensively used to assign a quantitative value to this concept: dynamic stochastic general equilibrium (DSGE) models and vector autoregressions (VARs). 1 However, each approach has disadvantages. While DSGE models potentially face a misspecification problem in imposing excessive restrictions on the data, structural VARs might be sensitive to identification strategies (Ramey, 2011a). The implications drawn from one model or another might not reveal the true policy effects. This paper combines DSGE and VAR models to provide new insights into the transmission mechanisms of fiscal shocks in the euro area. This DSGE-VAR approach relaxes the strong cross-equation restrictions created by the DSGE model and thus allows possible misspecification of the structural model to be considered (Del Negro and Schorfheide, 2004, Del Negro, Schorfheide, Smets, and Wouters, 2007, and Del Negro and Schorfheide, 2009).
Armed with this original tool, we evaluate the relative contributions to the size of estimated fiscal multipliers of two transmission mechanisms of government spending shocks advanced previously in the literature. The first relies on the presence of hand-to-mouth (HtM) consumers in the population, i.e., households that do not have access to financial markets and simply consume their disposable income in each period. Galí et al. (2007) and Forni et al. (2009) show that the interaction of such agents with both real and nominal rigidities increases the government spending multiplier. The second transmission mechanism allows government spending to enter -in a non-separable way-the household's utility function (GiU), such that government activity directly affects the marginal utility of consumption. Bouakez and Rebei (2007), Fève et al. (2013) and Coenen et al. (2013) show that when private consumption and public expenditures display a sufficient amount of Edgeworth complementarity, households have incentives to consume and to work more, thereby generating larger fiscal multipliers. 2 Because each competing model nests the Smets-Wouters specification (Baseline), one can vary the magnitude of the parameter summarizing one of the mechanisms to understand how government spending shocks propagate into the model economy.
Our main findings are the following. First, the Bayesian estimation shows that a DSGE model with non-separable government spending in the utility function outperforms a model with hand-to-mouth consumers in terms of fit and yields larger fiscal multipliers. The version with Edgeworth complementarity provides a multiplier of approximately 1.75, whereas the version with hand-to-mouth consumers yields a value lower than one. Second, we use the DSGE-VAR approach to assess the performance of each version. On impact, the multiplier is approximately 1.5, and its estimated value is weakly affected by the model's specification. This result can be interpreted as suggesting that the data want a multiplier larger than one. Specifically, the DSGE-VAR framework is used to investigate whether the fiscal multipliers obtained in a constrained DSGE model are far from those obtained in a DSGE-VAR model with the same structural features. We obtain a sizable increase in the estimated value of the multiplier in the Baseline and HtM versions, while it remains very similar in the GiU specification and in the model including both mechanisms. This supports our claim that Edgeworth complementarity is a better representation of the transmission mechanism of fiscal policy in the euro area. 2 There exist many concrete examples for which private consumption and public expenditures are complements (health care, education, etc.). As discussed in Fiorito and Kollintzas (2004), the complementarity may reveal relative inefficiency in the provision of public goods. Let us consider the case of education. One may observe the coexistence of public schools and private tutors if private agents consider the quality of public teachers to be too low. In addition, the complementarity may occur because public education allows a higher level of income and thus increases the demand for other goods (similar arguments hold for health care). Though difficult to grasp at the aggregate level, this mechanism is a useful shortcut to quantify the influence of public spending on private decisions. This paper is related to recent studies investigating the size of fiscal multipliers, the transmission mechanism of government spending shocks, and fiscal strategies in DSGE models of the euro area (see, e.g., Cwik and Wieland, 2011, Coenen et al., 2012, and Cogan et al., 2013. In particular, Coenen et al. (2013) consider HtM consumers and government spending in the utility to gauge the effects of government activity in the euro area. Our paper extends these works in two directions. First it considers that the DSGE model (even if it includes relevant propagation mechanisms of government spending) can be misspecified and thus uses the DSGE-VAR approach to quantify the size of the fiscal multiplier and the dynamic effects of government spending shocks. Second, based on empirical evidence and the estimation of different versions of the model, it investigates how the mechanisms interact at the estimation stage and highlights the most relevant channel.
The paper is organized as follows. In the next section, we expound the Baseline DSGE model and the two competing propagation mechanisms. We also conduct a prior predictive analysis.
In section 3, we present empirical results from a Bayesian estimation of different model versions. In section 4, we report the estimation of different DSGE-VAR models. The last section concludes.

MEDIUM-SCALE MODELS FOR THE EURO AREA
Our investigation is based on the canonical medium-scale New-Keynesian framework described by Christiano et al. (2005) and Smets and Wouters (2007), which is currently considered sufficiently rich to fit the data well. It features utility-maximizing households, profitmaximizing firms, a fiscal authority financing public spending with lump-sum taxes, and a central bank setting short-term nominal interest rates according to a Taylor-type rule. The model incorporates a number of real and nominal rigidities, including habits in consumption, investment adjustment costs, variable capacity utilization, and monopolistic competition in goods and labor markets and wage and nominal price and wage rigidities with indexation. 3 This setup is extended in two directions: (i) the introduction of households being hand-tomouth consumers and (ii) the introduction of government spending in the household utility function in a non-separable way.
2.1. Alternative specifications. A first specification (labelled 'Baseline') is similar to Justiniano et al. (2010). This setup is then extended to introduce different transmission mechanisms of government spending shocks.
As in Galí et al. (2007), we assume in a second specification (labelled 'HtM') (i) that a fraction ω of households, called hand-to-mouth consumers, do not have access to financial markets and simply consume their disposable income in each period, (ii) employment agencies do not discriminate between household types in their labor demands, such that the number of hours worked N t is the same for all households. It follows that, in symmetric equilibrium, all households have the same wage rate W t . Therefore, the hand-to-mouth consumers set nominal consumption expenditure C r,t equal to their disposable wage income less lump-sum taxes T r,t .
This results in the following period-by-period budget constraint: The consumption of households that have access to financial markets is denoted C 0,t . Accordingly, total private consumption is then defined as A third specification (labelled 'GiU') augments the Baseline model by including government spending in the utility function. As in Bouakez and Rebei (2007), we allow for complementarity/substitutability between private consumption and public expenditures. Formally, the consumption bundle C * t is now defined as: where the parameter α g measures the degree of complementarity/substitutability between private consumption C t and public expenditures G t . The specification adopted here follows Christiano and Eichenbaum (1992), McGrattan (1994), andFinn (1998), among others. 4 If α g > 0, government spending substitutes for private consumption, with perfect substitution if α g = 1, as in Christiano and Eichenbaum (1992). In this case, a permanent increase in government spending has no effect on output or hours but reduces private consumption, through a perfect crowding-out effect. In the special case in which α g = 0, we recover the standard business cycle model, with government spending operating through a negative wealth effect on labor supply (see Aiyagari et al., 1992, Baxter andKing, 1993). When the parameter α g < 0, government spending complements private consumption. Then, it can be the case (depending on the labor supply elasticity) that private consumption will react positively to an unexpected increase in government spending.
A last specification (labelled 'Full') embeds both hand-to-mouth consumers and government spending in the utility function in the Baseline model.
where ζ t ∼ i.i.d.N 0, Σ ζ is the q-dimensional vector of innovations to the structural shocks, and A(θ) and B(θ) are complicated functions of the model's parameters θ. The measurement equation is given by: where x t is an n-dimensional vector of observed variables, D and E are selection matrices, e t is a vector of measurement errors, and C(θ) is a vector that is a function of the structural parameters.
2.3. Prior predictive analysis. Conditional on a given model specification M i , i ∈ {0 (Baseline), 1 (HtM), 2 (GiU), 3 (Full)}, the prior distribution of θ is p (θ|M i ) and the likelihood function associated with the vector of ex ante observablesX T ≡ {x t } T t=1 is L X T |θ, M i . Regardless of how the conditional distribution of observables and the prior distribution of unobservables are formulated, together they provide a distribution of observables with density: known as the prior predictive density. It summarizes the whole range of phenomena consistent with the model M i and is very easy to access by means of simulations. The prior predictive distribution summarizes the substance of the model and emphasizes that the prior distribution and the conditional distribution of observables are inseparable components, a point forcefully argued by Box (1980). As explained by Canova (1995), Lancaster (2004) and Geweke (2005), prior predictive analysis is a powerful tool to shed light on complicated objects that depend on both the joint prior distribution of parameters and the model specification. In our context, this analysis delivers the possible range of the government spending multiplier conditional on a specific model. As our alternative versions differ only by a parameter, prior predictive analysis offers precise statements concerning how a particular mechanism affects the multiplier.
In all model specifications, we calibrate few parameters: The discount factor β is set to 0.99, the inverse of the Frisch labor supply elasticity ν = 2, the capital depreciation rate δ is equal to 0.025, the parameter α in the Cobb-Douglas production function is set to 0.30 to match the average capital share in net (of fixed costs) output (McAdam and Willman, 2013), the steadystate price and wage markups ε p and ε w are set to 1.20 and 1.35, respectively (Everaert and Schule, 2008), and the steady-state share of government spending in output is set to 0.20 (the average value over the sample period).
Our choice of priors is in line with the literature, especially with Smets and Wouters (2007), Sahuc and Smets (2008) and Justiniano et al. (2010). We impose Beta distributions for all of the parameters, the theoretical support of which is the compact [0,1]. We use Gamma distributions for positive parameters. Finally, we use Inverse Gamma distributions for the standard errors of shocks. Importantly, we are agnostic about the share of non-Ricardian households (ω) and the degree of complementarity/substitutability between private consumption and public expenditures (α g ). We assume U niform priors for these two parameters: ω is distributed on [0, 1], and α g is distributed on [−2, 2]. 5 We take 1,000 draws from our prior distributions and calculate the resulting government spending multipliers. Fiscal multipliers are defined as the present value multipliers: where E t denotes the mathematical expectation operator conditional upon information available at t,β ≡ β/γ z is the inverse of the steady-state real interest rate, s is the selected horizon, and Ψ t = Y t (output), C t (private consumption), I t (private investment). At s = 0, the present value multiplier equals the impact multiplier. As in Leeper et al. (2011), Table I  The middle and lower panels report the probabilities that multipliers for consumption and investment, respectively, are positive at various horizons. 5 The interval has been set such that the minimum value of α g does not imply a negative value of the marginal utility of consumption around the deterministic steady state. By anticipating our estimation results, this prior ensures almost certainly that the marginal utility is positive because it depends on calibrated parameters (α, β, δ), with the only exception being the growth rate of TFP, which is estimated. However, this parameter marginally affects the great ratios (see Section B of the online appendix) and is estimated with precision. 6 In the context of the prior predictive analysis, we follow Leeper et al. (2011) in choosing a prior density for ρ g defined as B[0.70,0.20]. First, we observe that all models, even the Baseline specification, can generate impact output multipliers greater than one. This is because greater price stickiness implies that more firms respond to higher government spending by increasing production rather than prices, and hence markups respond more strongly. However, it is impossible for the Baseline model to produce positive consumption multipliers at any horizon. The negative wealth effect is indeed strong because households decrease their consumption and work more. This decline in private demand offsets most of the increased public demand, causing output to increase by less than the increase in government consumption.
Fiscal multipliers increase substantially when introducing hand-to-mouth consumers or Edgeworth complementarity/substitutability. Intuitively, as non-Ricardian households automatically consume their entire income, they ignore the wealth effects of future taxes and therefore increase their consumption when government expenditures rise. The larger the share of these agents, the lower the overall negative wealth effect on consumption. If wages are sticky, such that real wages increase in the very short run, then non-savers' consumption also increases.
With sufficient non-savers in the economy, the increase in their consumption can cause total consumption to increase, leading to larger output multipliers as well. The version including Edgeworth complementarity/substitutability yields multipliers in line with the HtM consumers version, although smaller. 7 This result originates from our choice of priors for ω and α g . Indeed, the prior mean for ω implies a sizable share of hand-to-mouth consumers, thereby allowing for a positive consumption response. Conversely, the prior uniform distribution for α g is centered on zero (i.e., the value from the Baseline model version), meaning that our prior does not favor this version. Edgeworth complementarity/substitutability allows us to cover a large range of situations for which consumption reacts positively and output multipliers are above one. These two transmission mechanisms are by themselves sufficient to generate high multipliers. Indeed, when we set all other parameters to their respective prior means and let ω and α g be drawn from their respective prior distributions, we obtain similar probabilities as those displayed in Table 1. 8

DSGE MODELS COMPARISON
In this section, we discuss the estimation results of the different specifications of the structural model and present the government spending multipliers inherited from each set of estimates. 9 3.1. Data description. The quarterly euro area data run from 1980Q1 to 2007Q4 and are extracted from the AWM database compiled by Fagan et al. (2005), except hours worked and the working age population. The reason for ending in 2007 is not to blur the results with the zero 7 The reason is that the prior distribution of α g is symmetric at zero. 8 See Table OA2 of the online appendix. 9 A robustness analysis is provided in Section I of the online appendix. lower bound episode in the aftermath of the financial crisis. Inflation π t is measured by the first difference of the logarithm of the GDP deflator (YED), the short-term nominal interest rate R t is a three-month rate (STN), and real wage growth ∆ log (W t /P t ) is the first difference of the logarithm of the nominal wage (WRN) divided by the GDP deflator. Output growth ∆ log Y t is obtained as the first difference of the logarithm of real GDP (YER), private consumption growth ∆ log C t is constructed by multiplying real private consumption (PCR) by the private consumption deflator (PCD), divided by the GDP deflator and transformed into the first difference of the logarithm; private investment growth ∆ log I t is defined as the aggregate euro area total economy gross investment minus general government investment, scaled by the GDP deflator and transformed into the first difference of the logarithm; and government spending growth ∆ log G t is defined as the nominal general government final consumption expenditure (GCN), scaled by the GDP deflator and transformed into the first difference of the logarithm.
Real variables are divided by the working age population, extracted from the OECD Economic Outlook. Ohanian and Raffo (2012) constructed a new dataset of quarterly hours worked for 14 OECD countries. We then derived a weighted (by country size) average of their series of hours worked for France, Germany and Italy to obtain a series of total hours for the euro area.
Interestingly, the series thus obtained is very close to that provided by the ECB on the common sample, i.e. 1995-2007. Total hours worked log N t are taken in logarithms. We use growth rates for the non-stationary variables in our data set (GDP, private consumption, private investment, government spending and the real wage) and express gross inflation, gross interest rates and the first difference of the logarithm of hours worked in percentage deviations from their sample means. The vector of eight observable variables is then given by: Our model abstracts from net exports, public investment and changes in inventories in GDP.
We then introduce a measurement error i.i.d.N 0, σ 2 x e that is directly associated with output growth.
3.2. Estimation results. We follow the Bayesian approach to estimate the models (see An and , for an overview). The posterior distribution associated with the vector of observables X T ≡ {x t } T t=1 cannot be recovered analytically but may be computed numerically using a Monte Carlo Markov chain (MCMC) sampling approach. Specifically, we rely on the Metropolis-Hastings algorithm to obtain a random draw of size 1,000,000 from the posterior distribution of the parameters.
For the sake of comparing different model versions, we resort to the following two standard criteria. First, from p (θ|X T , M i ), one can compute the marginal likelihood of specification M i , which is defined as: Second, given a prior probability p i on a given model specification M i , the posterior odds ratio is defined as: where M is the number of competing models. Table II reports information on the posterior distribution of the share of hand-to-mouth consumers ω and the degree of complementarity/substitutability between private consumption and government expenditures α g for each model version: the mean and the 90 percent confidence interval for each model version. 10 Several results are worth commenting on. The first notable result is that the two propagation mechanisms considered here are essential because they substantially improve the fit of the model ( The estimated value for α g is negative, suggesting a strong complementarity between private 11 See Table OA5 of the online appendix for a sensitivity analysis to the set of observables. consumption and public expenditures. This result is in line with that obtained in Coenen et al. (2013) for the euro area. 12 Using again an uninformative prior with zero mean, we obtain the confidence interval [−1.86; −1.42] for α g .
The largest marginal likelihood is obtained when the two mechanisms are combined (see also the high posterior odds ratio in Table 2). In this Full model version, we obtain a lower share of hand-to-mouth consumers (ω = 0.14) and a slightly lesser complementarity between private consumption and public expenditures (α g = −1.51). Thus, the estimation of the Full model specification on actual data highlights a substitution between these two mechanisms.
It is worth noting that the mean value of ω in the HtM specification is outside the 90 percent confidence interval of the Full model version. This is not the case when we consider the GiU specification. Therefore, we can infer that a model version with Edgeworth complementarity suffers less than a specification with hand-to-mouth consumers from the presence of a competing propagation mechanism. 13 To better illustrate the trade-off between the two transmission mechanisms of fiscal shocks, we plot draws from the posterior distributions of ω and α g in the Full model version. Figure   1 reports the outcome of this exercise. The thick plain line is the nonparametric regression, and the thick dashed lines delineate the 90 percent confidence interval obtained by standard bootstrap techniques. The scatter diagram corresponds to the estimation of the Full model.
Crosses indicates the average parameter values for ω and α g . This figure clearly reveals, in the neighborhood of the posterior means (ω = 0.14 and α g = −1.51), that the two mechanisms substitute. Importantly, a small variation in α g has strong implications for the estimated share of hand-to-mouth consumers. For example, moving α g from -1.51 to -1.60 implies a change in 12 Notice that our estimated degree of Edgeworth complementarity is equivalent to an elasticity of substitution between private and public expenditures in a CES aggregate of µ ces = 0.172, when assuming a private consumption share of κ = 0.75. Indeed, µ ces ≡ (1 − κ) s c /(s c + α g s g ) − κ , where s c and s g denote the consumption to output ratio and government expenditures to output ratio, respectively. 13 Comparing a set of moments from actual data to those generated by alternative specifications yields additional evidence. We observe that the Full model yields a better fit than the Smets-Wouters type model, especially with respect to the volatility and persistence of aggregate variables (output, consumption, government expenditures, inflation, etc.). ω from 0.14 to 0.07. In other words, the GiU specification appears more robust to a model's perturbation, i.e., the introduction of a competing transmission mechanism, than does the HtM specification.
Moreover, the estimated share of hand-to-mouth consumers is too low to generate a positive private consumption multiplier for government consumption shocks in a standard New-Keynesian DSGE model (see, e.g., Coenen and Straub, 2005;Galí et al., 2007). This is confirmed by Panel (a) Figure 3 reports the empirical distribution of the impact output multiplier for the Full specification, and the average value of this multiplier for the Baseline, HtM and GiU models. The figure makes clear that the estimated multiplier differs considerably between the two model versions. In the presence of hand-tomouth consumers, the average output multiplier is approximately 0.80, while it is twice larger (approximately 1.75) when government expenditures enter the household utility function. The estimated multiplier in the HtM case slightly exceeds that obtained in the Baseline model (approximately 0.60). Moreover, the GiU and Full models yield similar multiplier values. 14 Furthermore, the four model versions display nearly identical estimated values for the common structural parameters. Most of the parameter estimates are in line with previous results (Smets and Wouters, 2003, Sahuc and Smets, 2008, Coenen et al., 2013. Neither the parameters related to real rigidities nor those related to nominal rigidities are affected by the presence of ω and α g . 15 In addition, the parameters that govern the driving force and those describing the monetary policy are left unaffected. This means that our additional features improve the fit of a standard DSGE model without altering its propagation mechanisms. 16

THE SIZE OF THE MULTIPLIER: A DSGE-VAR APPROACH
The DSGE-VAR approach has been suggested as a tool for studying the misspecification of a DSGE model and allowing the cross-equation restrictions of the DSGE model to be relaxed in a flexible manner (Del Schorfheide, 2004, Del Negro et al., 2007). 17 The basic idea is to (i) use a VAR model as an approximating model for the DSGE model and (ii) construct a mapping from the DSGE model to the VAR parameters, leading to a set of cross-restrictions for the VAR model. Deviations from these restrictions may be interpreted as evidence for DSGE model misspecification. In a Bayesian framework, one can specify a prior distribution for deviations from the DSGE model restrictions, the tightness of which is scaled by a single hyperparameter λ. By varying this parameter from infinity to zero, we create a continuum of models with the VAR approximation of the DSGE model at one end and an unrestricted VAR at the other end.
The marginal likelihood function of this parameter then provides an overall assessment of the DSGE model restrictions that is more robust and informative than a comparison of the two polar cases (unconstrained VAR model vs. DSGE model). 18 4.1. Assessing the fiscal multiplier from DSGE-VARs. The first step consists in selecting the best DSGE-VAR(λ, p) model, where p denotes the number of lags. An approach consists in choosing the model with the largest marginal likelihood over all pairs (λ, p) and the specification of the DSGE model. We consider lag orders ranging from one to four. Several features emerge. First, the data favor the Full specification. 19 Second, for any value of λ, the logmarginal likelihood with one lag is always greater than that with two, three or four lags, indicating that reducing the number of lags, and hence the number of free parameters, increases the fit of the empirical model. 20 Third, the estimates of λ are positively related to the selected lag order. The DSGE model restrictions help in part because they reduce the number of free parameters, and this reduction becomes more valuable the larger the lag length. In the following, we therefore decide to examine the usefulness of DSGE-VAR models with two lags. 21 As before, we take draws from the posterior distributions of the DSGE-VAR model with two lags and calculate the resulting government spending multipliers. Following Del Negro and Schorfheide (2004), it is natural to use the theoretical DSGE model to provide the prior information that enables the identification of shocks. Indeed, the contemporaneous relationships between the DSGE model variables allow us to orthogonalize the shocks that affect the model dynamics. The mapping between the canonical residuals and the structural shocks is obtained by multiplying the Choleski decomposition of the covariance matrix of the canonical residuals 18 Section J of the online appendix offers a presentation of the DSGE-VAR methodology. 19 See Figure OA4 of the online appendix. 20 The estimated log-marginal likelihood values for a range of DSGE-VAR models are displayed in Figure OA5 of the online appendix. 21 In Figure OA6 of the online appendix, we report the empirical distribution of the DSGE-VAR with two and four lags and show that the two posterior means are very close. To assess the invertibility issue, we plot the response functions to a government spending shock associated with the DSGE-VAR(∞,2) and with DSGE models and show that they are nearly identical ( Figure OA8 of the online appendix). by the factorization of a standardized version of the matrix B(θ) in the DSGE state-space representation. 22 With such a mapping and the moving-average representation of the VAR model, dynamic responses of endogenous variables to structural shocks (especially the government spending shock) can be computed.  (2009) for a refinement of the identification procedure. 23 Notice that the unit-root technology shock in the theoretical DSGE model induces a common stochastic trend in the levels of all real variables. We also estimated a vector error correction model (VECM) with a DSGE-based prior by simply adding the cointegrating relations of the DSGE model to the VAR model. Although the VECM helps to improve the approximation of the DSGE model (via a slightly higher marginal likelihood), we found that the DSGE model maps quite well into the VAR model. Figure OA7 of the online appendix displays the empirical distributions of the output multiplier associated with DSGE-VECMs with 2 and 4 lags. The estimated impact multipliers are very close to those obtained from the DSGE-VARs. sufficiently to a government shock. Notice again that the GiU and Full versions must be preferred, as the response of investment is more pronounced. Finally, the responses of the real interest rate are fairly well reproduced by the GiU and Full versions, whereas this is not the case for models that do not include this feature.
Why do the GiU and/or Full versions match these responses well? Part of the answer lies in the dynamics of the marginal utility of consumption. 24 In the presence of government spending in the utility function, Edgeworth complementarity between private and public consumption plays the same role as a positive preference shock that increases private consumption after a positive government spending shock. While a government spending shock makes households persistently poorer (the estimated persistence of the shock is large), they seek to consume more.
The only choice they have is then to offer more labor to sustain their consumption plan. This is why output can increase substantially in the short run. For their part, firms use more labor input such that the marginal productivity of capital increases, creating incentives to invest more. 25 The large increase in the real interest rate simply reflects the persistent rise in capital productivity. The Baseline and HtM versions cannot generate these patterns because private consumption falls and the increase in labor supply is not sufficient to yield a large positive response of the marginal productivity of capital.

DSGE misspecification and fiscal multipliers.
The DSGE-VAR approach allows us to determine whether the fiscal multipliers obtained in a constrained DSGE model are far from those obtained in a DSGE-VAR model. If they are close, this means that the features included in the DSGE model are consistent with empirical evidence. We complete the previous exercise by comparing each DSGE specification with the associated DSGE-VAR model, (i.e., incorporating the same structural features). 24 To simplify the exposition, we abstract from nominal rigidities. 25 See Dupaigne and Fève (2015) for the analytics of the investment channel in small-scale DSGE models.  that Edgeworth complementarity is a better representation of the transmission mechanism of fiscal policy in the euro area.

CONCLUDING REMARKS
This paper uses the DSGE-VAR approach to assess the relative contributions to the size of estimated fiscal multipliers of two transmission mechanisms of government spending shocks, namely hand-to-mouth consumers and Edgeworth complementarity. Although a Bayesian prior predictive analysis highlights that the presence of hand-to-mouth consumers yields larger multipliers than the introduction of Edgeworth complementarity, our posterior estimates suggest the opposite. A model with Edgeworth complementarity provides a better fit and enriches the propagation mechanism of government spending shocks. In fact, a small change in the degree of Edgeworth complementarity substantially impacts the estimated share of hand-tomouth consumers. We also obtain that Edgeworth complementarity yields dynamic responses close to those obtained with the flexible DSGE-VAR model, i.e., a large output multiplier and a positive private consumption response. Conversely, the estimated share of hand-to-mouth consumers is too small to replicate the positive response of private consumption.
In our quantitative assessment, we deliberately abstracted from relevant details to concentrate on the two competing mechanisms. However, the relevant literature has emphasized other modeling and policy issues that might affect and enrich our findings. We mention two of them. First, we only concentrated our analysis on hand-to-mouth consumers and Edgeworth complementarity. There are other relevant mechanisms (externalities, deep habits, productive government investment), and a systematic evaluation of their relative merits may help to improve our understanding of the effects of government activity. Second, we assumed lump-sum taxes to finance the government deficit, but a more realistic representation would consider distortionary taxes with feedback rules. The way in which government expenditures are financed by distortionary taxes could impact the transmission mechanism of fiscal shocks in the euro area.

Online Appendix A. MEDIUM-SCALE DSGE MODELS
In this section we describe the DSGE models of the euro area economy with distinct transmission mechanisms for government spending shocks. All these models have a common core which is close to Smets and Wouters (2007) and Justiniano, Primiceri and Tambalotti (2010). In particular, each model includes features such as habit formation, investment adjustment costs, variable capital utilisation, monopolistic competition in goods and labor markets, and nominal price and wage rigidities. This setup is extended in two directions: (i) the introduction of households being hand-to-mouth consumers and (ii) the introduction of government spending in the household utility function in a non-separable way.

A.1. Baseline model
The economy is populated by five classes of agents: producers of a final good, intermediate goods producers, households, employment agencies and the public sector (government and monetary authorities).

A.1.1. Household sector
Employment agencies-. Each household indexed by j ∈ [0, 1] is a monopolistic supplier of specialised labor N j,t . At every point in time t, a large number of competitive "employment agencies" combine households' labor into a homogenous labor input N t sold to intermediate firms, . Profit maximization by the perfectly competitive employment agencies implies the labor demand function N j,t = W j,t W t − ε w,t ε w,t −1 N t , where W j,t is the wage paid by the employment agencies to the household supplying labor variety j, while is the wage paid by intermediate firms for the homogenous labor input sold to them by the agencies. The exogenous variable ε w,t measures the substitutability across labor varieties and its steady-state is the desired steady-state wage mark-up over the marginal rate of substitution between consumption and leisure. Household's preferences-. The preferences of the jth household are given by where E t denotes the mathematical expectation operator conditional upon information available at t, β ∈ (0, 1) is the subjective discount factor, h ∈ [0, 1] denotes the degree of habit formation, and ν > 0 is the inverse of the Frisch labor supply elasticity. C * t is a consumption measure (C * t = C t , where C t is real consumption, in the baseline version), N j,t is labor of type j, and ε b,t is a preference shock.
As we explain below, households are subject to idiosyncratic shocks about whether they are able to re-optimise their wage. Hence, the above described problem makes the choices of wealth accumulation contingent upon a particular history of wage rate decisions, thus leading to the heterogeneity of households. For the sake of tractability, we assume that the momentary utility function is separable across consumption, real balances and leisure. Combining this with the assumption of a complete set of contingent claims market, all the households will make the same choices regarding consumption and money holding, and will only differ by their wage rate and supply of labor. This is directly reflected in our notations. Finally, V (G t ) is a positive concave function, meaning that agents do not necessarily feel worse off when public expenditures increase. Notice that this term has no effect on the equilibrium.
Household j's period budget constraint is given by where I t is investment, T t denotes nominal lump-sum taxes (transfers if negative), B t is the one-period riskless bond, R t is the nominal interest rate on bonds, A j,t is the net cash flow from household's j portfolio of state contingent securities, D t is the equity payout received from the ownership of firms. The capital utilisation rate u t transforms physical capitalK t into the service flow of effective capital K t according to K t = u tKt−1 , and the effective capital is rented to intermediate firms at the nominal rental rate R k t . The costs of capital utilization per unit of capital is given by the convex function ϑ (u t ). We assume that u = 1, ϑ (1) = 0, and we define The physical capital accumulates according tō where δ ∈ [0, 1] is the depreciation rate of capital, and S (.) is an adjustment cost function which satisfies S (γ z ) = S (γ z ) = 0 and S (γ z ) = η k > 0, γ z is the steady-state (gross) growth rate of technology, and ε i,t is an investment shock. Households set nominal wages according to a staggering mechanism. In each period, a fraction θ w of households cannot choose its wage optimally, but adjusts it to keep up with the increase in the general wage level in the previous period according to the indexation rule W j,t = γ z π 1−γ w π γ w t−1 W j,t−1 , where π t ≡ P t /P t−1 represents the gross inflation rate, π is steady-state (or trend) inflation and the coefficient γ w ∈ [0, 1] is the degree of indexation to past wages. The remaining fraction of households chooses instead an optimal wage, subject to the labor demand function N j,t .

A.1.2. Business sector
Final good producers-. At every point in time t, a perfectly competitive sector produces a final good Y t by combining a continuum of intermediate goods Y t (ς), ς ∈ [0, 1], according to the . Final good producing firms take their output price, P t , and their input prices, P ς,t , as given and beyond their control. Profit maximization implies Y ς,t = P ς,t P t − ε p,t ε p,t −1 Y t from which we deduce the relationship between the final good and the prices of the intermediate goods . The exogenous variable ε p,t measures the substitutability across differentiated intermediate goods and its steady state is then the desired steady-state price markup over the marginal cost of intermediate firms.
Intermediate-goods firms-. Intermediate good ς is produced by a monopolist firm using the following production function where α ∈ (0, 1) denotes the capital share, K ς,t and N ς,t denote the amounts of capital and effective labor used by firm ς, F is a fixed cost of production that ensures that profits are zero in steady state, and Z t is an exogenous labor-augmenting productivity factor whose growth-rate is denoted by ε z,t ≡ Z t /Z t−1 . In addition, we assume that intermediate firms rent capital and labor in perfectly competitive factor markets.
Intermediate firms set prices according to a staggering mechanism. In each period, a fraction θ p of firms cannot choose its price optimally, but adjusts it to keep up with the increase in the 27 Later, we estimate η u rather than the elasticity ϑ (1) /ϑ (1) to avoid convergence issues. general price level in the previous period according to the indexation rule P ς,t = π 1−γ p π γ p t−1 P ς,t−1 ,where the coefficient γ p ∈ [0, 1] indicates the degree of indexation to past prices. The remaining fraction of firms chooses its price P ς,t optimally, by maximizing the present discounted value of future profits subject to the demand from final goods firms and the production function. Λ t+s is the marginal utility of consumption for the representative household that owns the firm.

A.1.3. Public sector
Real (unproductive) government purchases G t is set according to where g denotes the deterministic steady-state value of G t /Z t , ε g,t is a government spending shock, andG t is an endogenous component of the policy, assumed to follow the simple rulẽ The parameter ϕ g is the policy rule parameter linking the stationary component of government policy to demeaned output growth. If ϕ g > 0, the policy rule contains a procyclical component that triggers an increase in government expenditures whenever output growth is above its average value. In contrast, if ϕ g < 0, the policy rule features a countercyclical component, and thus reflects automatic stabilizers. When ϕ g = 0, the stationary component of government policy is exogenous.
The monetary authority follows a generalised-Taylor rule by gradually adjusting the nominal interest rate in response to inflation, the output gap and a change in the output gap: where R is the steady state of the gross nominal interest rate and ε r,t is a monetary policy shock. The output gap is defined as the ratio of actual to potential output Y f ,t (i.e. the level of output that would prevail under flexible prices and constant elasticity of substitution among intermediate goods and labor types). The parameter ϕ r captures the degree of interest-rate smoothing.

A.1.4. Market clearing and stochastic processes
Market clearing conditions on final goods market are given by ε p,t −1 dς is a measure of the price dispersion.

A.2. Introducing two transmission mechanisms of government spending shocks
We consider extended versions in order to introduce different transmission mechanisms of government spending shocks.
As in Galí, Lopez-Salido and Vallès (2007), we assume in a first specification (labelled 'HtM') (i) that a fraction ω of households, called hand-to-mouth consumers, do not have access to financial markets and simply consume their disposable income in each and every period, (ii) the employment agencies do not discriminate between household types in their labor demands, such that the number of hours worked N t is the same for all households. It follows that, in a symmetric equilibrium, all households have the same wage rate W t . Therefore, the hand-tomouth consumers set nominal consumption expenditure C r,t equal to their disposable wage income less lump-sum taxes T r,t . This results in the following period-by-period budget constraint: P t C r,t ≤ W t N t − T r,t The consumption of households who have access to financial markets is denoted C 0,t . Accordingly, total private consumption is then defined as C t = (1 − ω)C o,t + ωC r,t .
A second specification (labelled 'GiU') augments the baseline model with government spending in the utility function. As in Bouakez and Rebei (2007), we allow for complementarity/substitutability between private consumption and public expenditures. Formally, the consumption bundle C * t is now defined as where the parameter α g measures the degree of complementarity/substitutability between private consumption and public expenditures. The specification adopted here follows Christiano and Eichenbaum (1992), McGrattan (1994), Finn (1998), among others. If α g > 0, government spending substitutes for private consumption, with perfect substitution if α g = 1, as in Christiano and Eichenbaum (1992). In this case, a permanent increase in government spending has no effect on output and hours but reduces private consumption, through a perfect crowdingout effect. In the special case α g = 0, we recover the standard business cycle model, with government spending operating through a negative wealth effect on labor supply (see Aiyagari, Eichenbaum, 1992, Baxter andKing, 1993). When the parameter α g < 0, government spending complements private consumption. Then, it can be the case (depending on the labor supply elasticity) that private consumption will react positively to an unexpected increase in government spending.
A last specification (labelled 'Full') embeds both hand-to-mouth consumers and government spending in the utility function in the baseline model.

B.1. Equilibrium conditions
This section reports the first-order conditions for the agents' optimizing problems and the other relationships that define the equilibrium of the models.
Effective capital: Marginal utility of consumption: Tobin's Q: Capital utilization: where MC t is the nominal marginal cost. Capital renting: Aggregate price index: Aggregate wage index: Monetary policy rule: Resource constraint: Model with hand-to-mouth consumers: Model with government spending in the utility function:

B.2. Stationary equilibrium
To find the steady state, we express the model in stationary form. Thus, for the non-stationary variables, let lower-case denote their value relative to the technology process Z t : where we note that the marginal utility of consumption Λ t will shrink as the economy grows, and we express the wage in real terms. Also, we denote the real rental rate of capital and real marginal cost by r k t ≡ R k t /P t and mc t ≡ MC t /P t , and the optimal relative price as p t ≡ P t /P t . Then we can rewrite the model in terms of stationary variables as follows. Effective capital: Marginal utility of consumption: Tobin's Q: Capital utilization: Capital renting: βθ p s λ t+s λ t y t,t+s p t P t P t+s Π p t,t+s − ε p,t+s mc t+s = 0 Aggregate price index: Wage setting: Aggregate wage index: Government spending: Monetary policy rule: Resource constraint: Model with hand-to-mouth consumers: Model with government spending in the utility function:

B.3. Steady state
We use the stationary version of the model to find the steady state, and we let variables without a time subscript denote steady-state values. First, we have that R = (γ z π) /β and the expression for Tobin's Q implies that the rental rate of capital is and the price-setting equation gives marginal cost as The capital/labor ratio can then be retrieved using the capital renting equation: , and the wage is given by the labor demand equation as The production function gives the output/labor ratio as and the fixed cost F is set to obtain zero profits at the steady state, implying The output/labor ratio is then given by Finally, to determine the investment/output ratio, we use the expressions for effective capital and physical capital accumulation to get Given the government spending/output ratio g/y, the consumption/output ratio is then given by the resource constraint as Model with hand-to-mouth consumers: In what follows, t r is assumed to be zero.
Model with government spending in the utility function:

B.4. Log-linearized version
We log-linearize the stationary model around the steady state. Letχ t denote the log deviation of the variable χ t from its steady-state level χ:χ t ≡ log (χ t /χ). The log-linearized model is then given by the following system of equations for the endogenous variables. Effective capital:k Capital renting:r k t = mc t − (1 − α)k t + (1 − α)n t Phillips curve: Wage curve: Marginal rate of substitution: mrs t = νn t −λ t +ε b,t Government spending:ĝ C. PRIOR DISTRIBUTIONS     Table OA4 reports this contribution to the variance of observables for the four model versions. As it is clear from this table, the contribution of the government spending shock to output volatility is small for the baseline and HtM model versions (less or equal to 5%), while it is around 13% for the GiU specification. The discrepancy is even larger when it comes to the volatility of hours worked: 7% for the HtM specification and 42% for the GiU version. Introducing government expenditures in the utility directly affects the marginal rate of substitution between consumption and hours worked and thus acts as a labor wedge. This government spending based labor wedge then impacts output in the short-run.

I.1. The effects of data on estimation
One can legitimately wonder why the model with hand-to-mouth consumers differs so much from the model with Edgeworth complementarity at the estimation stage. The two propagation mechanisms can equally fit the data, as they both have the potential to yield a positive response of private consumption to a government spending shock (see Table 1 in the main text). As Guerron-Quintana (2010) has shown that the estimation of a structural model is sensitive to the set of observables, this section inspects the effect of data on the estimation of the share ω. We consider the following experiment. We calibrate the DSGE model according to the posterior estimates in the HtM, GiU and Full model versions, respectively. For the GiU and Full models, we set α g = 0, i.e. we eliminate the propagation mechanism related to Edgeworth complementarity. In other words, the HtM, GiU and Full versions only reflect a particular calibration of the remaining models' parameters. These three calibrations are considered as simple robustness check. Given a calibration, we only estimate the share of hand-to-mouth consumers for several sets of observables. We start with the smallest relevant set and progressively add observables. The results are reported in Table OA5. For comparison purpose, the table includes our benchmark results (i.e. with eight observables). When we consider private consumption and government spending, including or not investment, we obtain a larger estimated value of ω (whatever the calibration) compared to the benchmark estimates. The share ω is now close to 0.5 and the HtM model version can yield more likely an output multiplier larger than one. When we progressively extend the set of observables, the estimated value is reduced, especially if we include real wages, hours worked, inflation or the nominal interest rate.

I.2. Alternative specifications
In this section, we investigate the robustness of our findings to a number of perturbations: Sub-samples, news shocks in government spending, an alternative specification of technology shocks and non-separability between consumption and leisure in the utility function. All the results are reported in Table OA6. For all experiments, we use the same prior distributions for the parameters (see Table OA1), except special comments. To save space, we only report the parameter values for ω and α g and the marginal likelihood.
We first investigate whether our results still hold if we re-estimate the four model versions over different sub-samples. In the period between the mid-1990s and 2007, European countries enjoyed one of the greatest economic growth periods, known as the Great Moderation due to the low volatility of growth rates in those years. The mid-1990s also corresponds to the progressive realisation of Economic and Monetary Union. It seems then natural to split the overall sample in the following two parts: 1980Q1-1993Q4 and 1994Q1-2007Q4 (see Avouyi-Dovi and Sahuc, 2016. The results are reported in Panel (a) of Table OA6. All our previous findings are robust to this sub-sample analysis: The GiU model version outperforms the HtM one, the HtM specification adds very little to both the baseline model and the GiU model versions, the share of hand-to-mouth consumers decreases when this specification is considered together with Edgeworth complementarity.
Second, as emphasised by Ramey (2011b) and Schmitt-Grohé and Uribe (2012), the expected component in public expenditures constitutes an important element of government policy. We accordingly modify our benchmark specification to allow for news shocks in the government spending rule. The stationary component of government spending still follows an AR(1) but the innovation ζ g,t rewrites g,t−4 , and ζ (8) g,t−8 are three independent random variables that follow a normal distribution with zero mean and variance equals to σ 2 g,0 , σ 2 g,4 and σ 2 g,8 , respectively. All variances have the same prior distribution, i.e. an inverse gamma IG[1.00,2.00]. We obtain that government spending shocks explain around 20% of output variance, among which the expected components represent more than 30%. The estimation results are reported in Panel (b) of Table  OA6. As in the previous case, none of our main results are modified.
Third, we consider a stationary version for the productivity shocks. Indeed, one can argue that the presence of a random walk (with a positive drift) specification could affect our results as it implies that government spending growth is affected by technology shocks. 28 We relax this assumption and specify a stationary AR(1) process for the logarithm of total factor productivity (in deviation from a linear trend). Government expenditures are now only explained by their own shocks. We use the same prior as before for the autoregressive parameter and the variance of innovation. The results are reported in Panel (c) of Table OA6. We obtain a larger share of the hand-to-mouth consumers (with an output multiplier around 1), but this model version is still outperformed by the GiU specification (with an output multiplier around 1.80). Even if we obtain different numbers, we reach the same conclusions about the HtM and GiU specifications.
Finally, to lower the negative wealth effect of government spending shocks when agents are forward-looking, we introduce non-separability between consumption and leisure in the utility function as in Smets and Wouters (2007), where ψ is the elasticity of intertemporal substitution of consumption (for constant labor). Indeed, with non-separable preferences, an increase in hours worked has a positive effect on the marginal utility of consumption. The reason for this is that consumption and hours are complements in the utility function. Hence, unless monetary policy is very aggressive in increasing interest rates, the complementarity will work to drive up consumption with the increase in hours worked through the Euler equation. When the wealth effect on labor supply is reduced, there is no need for such a large share of HtM consumers or a large degree of Edgeworth complementarity (Panel (d) of Table OA6). For an elasticity of intertemporal substitution estimated around 1.60-1.80, we observe that ω and α g are reduced by more than 20%. However, this alteration has no effect on the models comparison.

J. DSGE-VAR METHODOLOGY
To setup the DSGE-VAR(λ, p), we follow Del Negro and Schorfheide (2004). The VAR representation of the n × 1 vector of endogenous variables x t can be written as with t = 1, ..., T and ε t ∼ N (0, Σ ε ). Let X be a T × n matrix with rows given by x t , S be the T × (np + 1) matrix with rows [1, x t−1 , ..., x t−p ] , Φ = [Φ 0 , Φ 1 , ..., Φ p ] , and ε be a T × n matrix with rows ε t . The VAR model can then be written as X = SΦ + ε, with likelihood function where tr [.] denotes the trace of a matrix.