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Estimating Fiscal Multipliers by Combining Statistical Identification with Potentially Endogenous Proxies
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The proxy Structural Vector Autoregression (SVAR) has become a popular tool to identify fiscal policy shocks. Prominent examples are mertens2014reconciliation who use a tax proxy and caldara2017analytics who rely on non-fiscal proxy variables like a TFP measure. Notably, both studies reach contradictory conclusions about the size of fiscal multipliers, angelini2020fiscal. A potential explanation for these conflicting results are endogenous proxies.
Alternatively, statistical identification methods can achieve point identification by imposing stronger assumptions on the stochastic properties of the shocks, see matteson2017independent, gourieroux2017statistical, keweloh2020generalized, lewis2021identifying, guay2021identification, jarocinski2024estimating, crucil2024monetary, pruser2024large, or kociecki2025non. However, even if the stronger stochastic assumptions are satisfied, inference based on statistical identification necessarily demands more from a finite sample than standard economic restriction- or proxy-based estimation approaches, see montiel2022svar.
In this paper, we propose a hybrid approach combining proxies and statistical identification which is robust to endogenous proxies. We leverage prior knowledge of an exogenous proxy to reduce estimation uncertainty while maintaining the flexibility to downweight or discard the proxy if the data provide evidence against its exogeneity. This mitigates the bias that would otherwise result from endogenous proxies. Allowing for potentially endogenous proxies, we find that increasing government spending is a more effective tool to stimulate the economy than lowering taxes, and previous contradictions in the literature seem to be caused by endogenous proxies.
Our econometric contribution consists of two parts. First, we combine proxies with a statistically identified SVAR using independent and non-Gaussian shocks. Crucially, in the statistically identified SVAR, we do not need to assume exogenous proxies. Instead, we estimate their correlation with non-target shocks and introduce a prior that shrinks towards exogeneity. We demonstrate that our shrinkage improves estimation precision when proxies are, in fact, exogenous. However, if the proxies are endogenous, the data can update the prior and reduce shrinkage, thereby mitigating the bias that endogenous proxies would otherwise introduce.
We build on a growing literature that combines statistical identification and economically motivated restrictions, see schlaak2023monetary, drautzburg2023refining, braun2021importance, keweloh2023monetary, HERWARTZ2023104630, or carriero2024blended. schlaak2023monetary combine identification by heteroskedasticity and proxy variables. They show that combining statistical identification with exogenous proxies improves efficiency, but performance deteriorates when the proxies are endogenous. Our approach advances this literature by shrinking the coefficients toward the exogenous proxy restriction instead of imposing these through a dogmatic prior. This makes sense if we believe the proxy is only weakly exogenous or are uncertain whether the proxy is exogenous at all. Importantly and in contrast to the hybrid literature cited above, our prior beliefs can still be updated by the data. This enables the model to inform us about endogenous proxies in a data-driven fashion, thereby mitigating the bias arising from their inclusion. But our aim extends beyond a mere test of proxy exogeneity. Instead, we try to leverage prior knowledge of an exogenous proxy (possible approximate) to improve the accuracy of the estimation and reduce estimation uncertainty. Overall, our framework preserves the key advantage of traditional hybrid approaches—namely, the ability to leverage exogenous proxies to improve estimation precision, while at the same time remaining robust to proxy endogeneity.
Our second econometric contribution is a novel approach to incorporate proxies into a Bayesian SVAR. In the frequentist setting, researchers can choose between two main strategies for estimating a proxy SVAR: (i) a moment-based approach, as in stock2012disentangling and mertens2013dynamic, which relies only on the assumption that the proxy is uncorrelated with non-target shocks; or (ii) the augmented proxy approach, as in angelini2019exogenous, which additionally imposes a specific data-generating process for the proxy. In contrast, Bayesian researchers currently lack this flexibility and must rely exclusively on the augmented proxy SVAR, as in caldara2019monetary. We fill this gap and introduce a new approach that incorporates moment conditions into the Bayesian SVAR, mirroring the frequentist moment-based proxy approach of stock2012disentangling and mertens2013dynamic, while preserving the advantages of Bayesian inference. Unlike the augmented proxy approach, and in line with moment-based frequentist proxy estimators, our proxy weighting method does not require specifying the functional form of the proxy. We show that this represents a key advantage in a non-Gaussian SVAR, where misspecifying the proxy's functional form can induce dependence among shocks and undermine the validity of non-Gaussian identification strategies.
We use our model to estimate the effects of fiscal policy shocks and evaluate whether commonly used proxies are exogenous. We provide evidence that the shocks identified using the fiscal proxy SVAR from mertens2013dynamic and using the non-fiscal proxy SVAR from caldara2017analytics are non-Gaussian. Specifically, all shocks show heavy tails and the tax and output shocks are left skewed. This data feature, which is used for identification, also makes sense from an economic point of view because tax reductions are generally larger (in absolute terms) than tax increases, and GDP usually falls stronger during recessions than it rises during expansions. It turns out that the government spending multiplier is larger than the tax multiplier. On impact, the tax multiplier is close to zero and it remains below unity for the entire impulse response horizon of five years. The government spending multiplier, however, is above unity for the first year after the shock and then slowly converges back to its pre-shock level. Evidently, our results are in some contrast to previous studies. In particular, mertens2014reconciliation estimate a tax multiplier around three. Our empirical findings indicate that the large differences between applying the mertens2014reconciliation approach and our model are due to the tax proxy not being exogenous. Specifically, we provide evidence that the narrative tax measure is negatively correlated with structural output shocks. Intuitively, not accounting for this correlation leads to identified tax cuts that also include exogenous increases in output, which increases the size of the estimated tax multiplier.
We emphasize that the reader should interpret our results with the understanding that they rely on strong assumptions about the error process, which may not necessarily hold. To address this concern, we carefully investigate the potential violation of these assumptions, but do not find any empirical evidence suggesting that they are invalid. Furthermore, we present historical evidence that validates our empirical findings based on the statistical identification approach. In particular, the narrative evidence indicates that the tax proxy shows a tendency to indicate an exogenous tax increase during periods of economic recessions and an exogenous tax decrease during episodes of economic expansion. Similarly, we also find evidence that the TFP proxy used by caldara2017analytics is negatively correlated with exogenous government spending shocks. Therefore, assuming an exogenous TFP proxy identifies output shocks which include spending shocks of opposite sign. As a consequence, the estimated spending multiplier is biased upward.
Several related papers use statistical identification to estimate the dynamic effects of fiscal shocks. For example, lewis2021identifying exploits time-varying shock variances, finding government spending and tax multipliers of similar size, both below one. guay2021identification relies on higher-order moments and shows that estimated multipliers depend heavily on additional assumptions. karamysheva2022we use non-Gaussian errors and also find similar-sized multipliers.
The remainder of the paper is organized as follows: Section (ref) describes our novel proxy weighting approach and the combination of proxy variables with non-Gaussianity. Section (ref) provides a summary of the results of a range of Monte Carlo simulations. Section (ref) uses the proposed model to analyze the effects of fiscal policy shocks. Section (ref) concludes.
This section presents the basic notation. Afterwards, we introduce a novel approach to incorporate proxy variables into a Bayesian SVAR using moment conditions. Subsequently, we combine the proxy moment conditions with independent non-Gaussian shocks and propose priors which shrink to proxy exogeneity. Finally, we discuss why it is important to use moment conditions to combine independent non-Gaussian shocks with proxy variables.
An SVAR with $n$ variables can be written as
with parameter matrices $\bm A_1,...,\bm A_p \in \mathbb{R}^{n \times n}$ and $\text{det}(\bm I-\bm A_1c-...-\bm A_pc^p)\neq0$ for $|c|\leq 1$, an intercept $\bm \nu$, an invertible matrix $\bm B_0 \in \mathbb{B} := \{\bm B \in \mathbb{R}^{n \times n} | \text{det}(\bm B)\neq 0 \}$, an $n$-dimensional vector of time series $\bm y_t=[\bm y_{1t} ,...,\bm y_{nt} ]'$, an $n$-dimensional vector of reduced form shocks $\bm u_t=[ u_{1t} ,..., u_{nt} ]'$, and an $n$-dimensional vector of serially uncorrelated structural shocks $ \bm \varepsilon_t=[ \varepsilon_{1t},..., \varepsilon_{nt}]'$ with mean zero and unit variance.
We define $\bm \pi_0 = \text{vec}([\bm v,\bm A_1',...,\bm A_p']')$ to collect the specific parameter matrices of the data-generating process and we use $\bm \pi $ to denote a vector of SVAR parameter matrices which can differ from the data-generating parameter matrices. Moreover, the innovation $e_{it}(\bm B,\bm \pi)$ denotes the $i$th component of $\bm e_t(\bm B,\bm \pi)=\bm B^{-1} (\bm y_t - \bm \nu - \bm A_1 \bm y_{t-1} - ... - \bm A_p \bm y_{t-p})$. Therefore, $\bm e_t(\bm B,\bm \pi)$ represents the shocks given $\bm \pi$ and $\bm B$. If $\bm \pi$ and $\bm B$ are equal to the corresponding values of the data-generating process, $\bm e_t(\bm B,\bm \pi)$ is equal to the structural shocks $\bm \epsilon_t$. Without additional restrictions, the model is not identified. We will use proxy variables and non-Gaussian shocks to archive identification.
A valid proxy is correlated with the target shock and uncorrelated with non-target shocks.
For simplicity, we assume that the first shock $ \varepsilon_{1t}$ is the target shock.
We propose a Bayesian proxy approach using moment conditions based on the proxy exogeneity assumption to reweight the likelihood of the SVAR. In contrast to the augmented proxy SVAR proposed by caldara2019monetary, our proposed method does not require specifying the functional form of the proxy. Section (ref), compares both methods and discusses the implications of misspecifying the functional form of the proxy.
Define the proxy exogeneity moment conditions measuring the correlation of proxy and non-target shocks $ e_{jt}(\bm B,\bm \pi)$ for $j=2,...,n$ with
and define $D(\bm z) = [D_2 (\bm z, \bm y, \bm B, \bm \pi),..., D_n (\bm z, \bm y, \bm B, \bm \pi)]'$. The joint density of the data $\bm y$ and the proxy exogeneity moment conditions is
The first term denotes the likelihood of the data $\bm y$ and the second term is the conditional density of the proxy exogeneity moment conditions given the data. For an exogenous proxy variable, we assume that
where $\bm \Sigma_z $ denotes the covariance matrix of $( z_t e_{2t} ,... ,z_t e_{nt} )'$, see section (ref).
The moment conditions do not constitute a generative model, and as such we view $p( D(\bm z) | \bm y , \bm B, \bm \pi)$ as a pseudo-likelihood following yin2009bayesian.\footnote{A pseudo-likelihood function is also used for Bayesian inference in narrative SVARs antolin2018narrative as well as in local projections ferreira2023bayesian. For a general discussion on pseudo-likelihoods for Bayesian inference, see ventura2016pseudo.} yin2009bayesian presents a Bayesian version of the Generalized Method of Moments (GMM) by defining a pseudo-likelihood function as follows
which is the kernel of a normal density. yin2009bayesian shows the validity of the posterior distribution resulting from this pseudo-likelihood. Specifically, yin2009bayesian demonstrate that the pseudo-likelihood defined by GMM criteria leads to valid Bayesian inferences in finite samples using probability coverage of posterior sets as suggested by monahan1992proper. kim2002limited justifies the incorporation of the GMM criterion in the Bayesian framework asymptotically. Furthermore, chernozhukov2003mcmc shows that a GMM criterion can be seen as the Laplace approximation of the negative true likelihood evaluated around the mode. Hence, we view our pseudo-likelihood as a convenient device that allows us to use the Bayesian toolkit when estimating our hyprid SVAR.
Throughout the paper, we refer to the model using a Gaussian likelihood $p(\bm y| \bm B, \bm \pi) $ and the conditional density based on an exogenous proxy variable in Equation ((ref)) as the Gaussian proxy weighting approach. If we assume a Gaussian likelihood, we need to assume that the proxy variables are exogenous to archive identification. Intuitively, the second term in Equation ((ref)) re-weights the likelihood of the VAR data $\bm y$ by giving more weight to structural parameters that result in non-target shocks that are uncorrelated with the proxy. The following proposition shows that for a valid proxy, the joint likelihood in Equation ((ref)) identifies the impact of the target shock.
The approach can be extended to the case of multiple proxies for multiple target shocks. A corresponding proposition can be found in the Online appendix.
The conditional density of the proxy exogeneity moment conditions downweights the likelihood of $\bm B$ and $\bm \pi$ values leading to non-target shocks which are correlated with the proxy, i.e. it downweights the solutions which lead to an endogenous proxy. However, the ability to downweight endogenous proxy solutions depends on the relevancy of the proxy. The inclusion of an exogenous but irrelevant proxy asymptotically only scales the joint likelihood in Equation ((ref)) but does not affect its shape or maxima.
This section explains how independent non-Gaussian structural shocks can be used to identify and estimate the simultaneous interaction in the SVAR. Technically, we impose the following assumptions:
lanne2017identification show that these assumptions are sufficient to identify the SVAR up to sign and permutation of the shocks. Assumption (ref) imposes independence and non-Gaussianity of the structural shocks. Both conditions have been relaxed in the literature: mesters2024non, keweloh2023uncertain, and herwartz2023identification explore relaxations of the independence assumption, while maxand2020identification addresses the relaxation of non-Gaussianity. For simplicity, we adopt the standard assumption of independent and non-Gaussian shocks in Assumption (ref).
Importantly, the identified shocks must be labeled manually by the researcher. A labeling strategy based on economic reasoning can attach an economic interpretation to the shocks and at the same time prevent permutation switches. We follow bertsche2022identification and label a shock as the target shock if it has the highest correlation with the corresponding proxy in absolute magnitude. We discuss labeling in more detail in the Online Appendix.
lanne2020identification, anttonen2021statistically, and braun2021importance propose Bayesian non-Gaussian SVAR models. We simplify anttonen2021statistically and assume that each shock follows a skewed t-distribution such that the density of the $i$th shock is given by
with $|\lambda_i|<1$, $q_i>2$ which implies that the fourth moment of the $i$th shock exists, and the normalization $m= \frac{2 v \lambda q_i^{0.5} \Gamma(q_i-0.5)}{\pi^{0.5}\Gamma(q_i+.5)}$, $v=q_i^{-0.5} \left[ (3 \lambda^2 + 1)(\frac{1}{2q_i-2}) - \frac{4\lambda^2}{\pi} (\frac{\Gamma(q_i-0.5)}{\Gamma(q_i)})^2 \right]^{-0.5}$ to mean zero and unit variance, which normalizes the size of the shocks. The likelihood of the data $\bm y$ given $\bm y_{-p+1},...,\bm y_0 $ follows from lanne2017identification and is equal to
Notably, by estimating $\bm \lambda$ and $\bm q$ our framework is flexible and the data can inform us over the degree of skewness and excess kurtosis, see anttonen2021statistically.
This section combines the re-weighting based on proxy exogeneity moment conditions with a non-Gaussian SVAR. Specifically, we use the non-Gaussian likelihood of Equation ((ref)) in Equation ((ref)), which ensures identification under Assumption (ref). Therefore, the proxy is not required for identification. Consequently, in contrast to the existing proxy approaches, we are able to identify shocks even if the proxy is endogenous. If the data do not support the exogeneity of the proxy, the model can effectively ignore its information. However, if the proxy is exogenous and relevant, incorporating it can lead to improved estimation accuracy and reduced estimation uncertainty in the non-Gaussian SVAR framework.
We generalize the proxy moment condition in Equation ((ref)) to
and allow for a non-zero mean $\bm \mu=(\mu_2,\dots,\mu_{n})$ of the moment conditions, which corresponds to endogenous proxies. Our aim is to estimate the exogeneity of the proxy variable $\bm \mu$ instead of restricting the proxy variable to an exogenous proxy by imposing $\bm \mu=0$.
This leads to the joint density
and for $\bm \mu = [ E[z_t e_{2t}],...,E[z_t e_{nt}]]'$ we have
For an exogenous proxy $\bm \mu$ contains only zeros. We propose a prior distribution for $\bm \mu$ that reflects the belief in an exogenous proxy. Specifically, we shrink the solution towards an exogenous proxy variable, i.e. we shrink all coefficients $\mu_{ j}$ in $\bm \mu$ to zero with the prior
We set $a=b=0$ which results in a flat prior, see tipping2001sparse. The prior variance $\sigma_{\mu j }^2$ controls shrinkage of $\mu_{ j}$ to zero. If we have strong beliefs in the validity of the proxy, we would set $\sigma_{\mu j }^2$ to be small. However, such prior beliefs can be controversial. To avoid fixing $\sigma_{\mu j }^2$ at an inappropriate value, we use the data and estimate $\sigma_{\mu {j}}^2$ hierarchically using an inverse Gamma prior. Hence, by shrinking $\mu_{ j}$ to zero, the model benefits from valid proxies, but if empirical warranted it can avoid such shrinkage and $\mu_{ j}$ can take on values different from zero to allow for endogenous proxy variables.
The joint posterior is proportional to
with $p(\bm \mu| \bm \Sigma_{\boldsymbol{\mu}})=N(\bm \mu; \bm 0,\bm \Sigma_{\boldsymbol{\mu}}) $ and $ \bm \Sigma_{\boldsymbol{\mu}} = \text{diag}( \sigma^2_{\mu_1},....,\sigma^2_{\mu_n})$.\footnote{We use flat priors for all model parameters. Alternatively, it would be possible to use a Minnesota type prior or a more flexible global local prior on $\bm \pi$, see e.g., huber2019adaptive, cross2020macroeconomic and pruser2023data.} The Markov Chain Monte Carlo (MCMC) algorithm to sample from the posterior is in the Online Appendix.\footnote{We rely on the algorithm proposed by ter2008differential to draw from the posterior distribution, which makes our approach more computationally intensive than alternative proxy methods, e.g. arias2021inference. However, we find that the sampler performs well across a wide range of Monte Carlo experiments and robustness checks. Our experience is consistent with the findings reported in anttonen2021statistically, who document similarly reliable performance using this algorithm.}
Throughout the paper, we refer to the model using the non-Gaussian likelihood $p(\bm y| \bm B, \bm \pi, \bm \lambda, \bm q) $, the conditional likelihood $p( D(\bm z) | \bm y , \bm B, \bm \pi, \bm \mu)$ based on the proxy moment conditions in Equation ((ref)), and the proxy exogeneity shrinkage prior in Equation ((ref)) as the non-Gaussian proxy weighting approach.
The conditional density $p( D(\bm z) | \bm y , \bm B, \bm \pi, \mu)$ still re-weights the SVAR likelihood and thereby utilizes the information of the proxy. However, by estimating $\bm \mu$ we allow for exogenous proxy ($\bm \mu=0$) and an endogenous proxy with ($\bm \mu \neq 0$). If we were to estimate $\bm \mu$ without shrinking it toward the exogeneity restriction $\bm \mu = 0$ via the shrinkage prior in Equation ((ref)), $D(z)$ would simply contribute an equal number of moment conditions and new parameters in $\bm \mu$. In that case, one could choose a $\bm \mu \neq 0$ to perfectly satisfy the moment conditions, resulting in no impact on the estimation of $\bm B$; that is, neither efficiency gain nor bias would arise. Conversely, if we were to fix $\bm \mu = 0$ to enforce an exogenous proxy, we would arrive at a typical blended identification approach, combining statistical identification with an economic restriction, as in schlaak2023monetary or carriero2024blended. In that case, the proxy moment conditions $D(z)$ can only be satisfied by choosing $\bm B$ such that the proxy is exogenous. This leads to efficiency gains in the estimation of $\bm B$ if the proxy is truly exogenous, but induces a bias if the proxy is endogenous.\footnote{ Analytically proving that combining statistical identification with economic restrictions yields efficiency gains over using either approach alone is challenging. Hybrid approaches typically demonstrate such gains via Monte Carlo simulations; we do so as well in our application. Intuitively, our non-Gaussian estimator identifies the model from higher-moment information, which can result in volatile estimates, whereas an exogenous proxy relies on a simple covariance restriction. Combining both sources of information stabilizes estimation and thereby improves efficiency. To our knowledge, the only analytical proof that combining statistical identification with economically motivated short-run restrictions improves efficiency over an estimator relying solely on the short-run restriction is given in keweloh2025higher. } Instead, we estimate $\bm \mu$ and impose the shrinkage prior in Equation ((ref)) to pull $\bm \mu$ toward zero. By estimating the variances $\sigma_{\mu_j}^2$ of $\bm \mu$, the data determine the cost of deviating from an exogenous proxy. If the proxy is exogenous, the estimated variances $\sigma_{\mu_j}^2$ will be small, making deviations of $\bm \mu$ from zero costly. In this case, the estimator will keep $\bm \mu$ close to zero and instead adjust $\bm B$ to satisfy the proxy moment conditions $D(z)$, thereby achieving efficiency gains. Conversely, if the proxy is endogenous, the estimated variances $\sigma_{\mu_j}^2$ will be large, making deviations from zero inexpensive. In that case, the estimator can use $\bm \mu$ to satisfy the proxy moment conditions $D(z)$ without inducing bias in $\bm B$. Our model therefore exploits information from proxy variables to improve the precision and reduce the uncertainty of the non-Gaussian SVAR estimation, while also mitigating bias when the proxies are endogenous.
Implementing the proxy weighting estimators in Sections (ref) and (ref) requires specifying or estimating $\bm \Sigma_z$, the covariance matrix of $( z_t e_{2t} ,... ,z_t e_{nt} )'$.
First, if the proxy is exogenous in a strong sense—meaning it is not only uncorrelated but also satisfies $E[z_t^2 \varepsilon_{jt} \varepsilon_{kt}]=E[z_t^2 ] E[\varepsilon_{jt} \varepsilon_{kt}] $ with non non-target shocks $\varepsilon_{jt} \varepsilon_{kt}$ which for example follows from independence—then $\bm \Sigma_z = \text{diag}(\sigma_z^2, \dots, \sigma_z^2)$, where $\sigma_z$ denotes the standard deviation of the proxy variable. Second, if the proxy is endogenous, the estimator stops shrinking $\mu_j$ toward zero, making the choice of $\bm \Sigma_z$ less critical. Specifically, when deviations of $\mu_j$ from zero entail little cost in Equation ((ref)), $\mu_j$ can be adjusted such that $D_j(\bm z, \bm y, \bm B, \bm \pi, \mu)$ in Equation ((ref)) equals zero. In this case, the proxy becomes irrelevant to the estimation, and so does the precise specification of $\bm \Sigma_z$.
In the remainder of the study, we use the simple assumption $\bm \Sigma_z=\text{diag}(\sigma_z^2,...,\sigma_z^2)$. However, we consider two alternatives approaches. For the first, we follow yin2009bayesian and set it to the empirical covariance matrix $ \bm \Sigma_z= \frac{1}{T}\sum_{t=1}^T (z_t e_{2t}, \dots,z_t e_{nt})'(z_t e_{2t}, \dots,z_t e_{nt})-D(\bm z)D(\bm z)' $. For the second we estimate $\bm \Sigma_z$ and shrink $\bm \Sigma_z$ to $\text{diag}(\sigma_z^2,...,\sigma_z^2)$. In particular, we assume that $\bm \Sigma_z (i,k)\sim N(m_{i,k},\sigma_{i,k}^2)$ with $m_{i,k}=\sigma_z^2$ if $i=k$ and otherwise we set $m_{i,k}=0$ and $\sigma_{i,k}^ 2\sim IG(c,d)$ with $c=d=0$. In the Online Appendix we show the all the alternatives lead to similar result in our empirical application as well as in our Monte Carlo simulations.
In the frequentist framework, there are two main approaches to incorporate proxy variables into an SVAR: (a) the moment-based proxy approach of stock2012disentangling and mertens2013dynamic, and (b) the augmented proxy SVAR approach of angelini2019exogenous. In contrast, the Bayesian proxy SVAR literature relies exclusively on the augmented proxy SVAR, see caldara2019monetary or arias2021inference.
This section compares the moment-based and augmented proxy SVAR approaches. We demonstrate that specifying the proxy’s data-generating process, as required in the augmented proxy SVAR, introduces a risk of misspecification, which can compromise identification—particularly in non-Gaussian models relying on independent shocks.
The augmented proxy SVAR approach adds an additional equation to the SVAR to model the proxy variable, such that the augmented system is equal to
with a measurement error $ \eta_t$ uncorrelated with the structural form shocks $\bm \varepsilon_{t}$. To simplify, let $\bm \nu_z=0$, $\bm \Sigma_{\eta}=1$, and $\bm \Gamma_{1} = \bm \Gamma_{2}=0$, which implies that the proxy is equal to a linear combination of structural shocks and measurement error, i.e., $ z_t = \bm \Phi \bm\varepsilon_{ t} + \eta_{t}$. Imposing a valid proxy implies zero restrictions on the $\bm \Phi$ matrix, such that the proxy variable is a linear combination of the target-shock and the measurement error, i.e. $z_t = \Phi_i \varepsilon_{it} + \eta_{t}$.
The augmented proxy SVAR requires to specify the data generating process of the proxy. Specifically, the proxy is defined as a linear function of the target shock and a measurement error. However, many proxy variables may not follow a linear process, like for instance the tax proxy in our empirical analysis, which may rather follow a process like $ z_t = \psi_t ( \Phi_i \varepsilon_{it} + \eta_{it}), $ where $\psi_t$ is a Bernoulli random variable, compare jentsch2019dynamic, bruns2022alternative, or budnikidentifying. If the proxy process is misspecified, it can lead to dependent shocks, which renders identification approaches based on independent shocks invalid.
In the misspecified augmented proxy SVAR, the measurement error $\eta_t$ is a function of the target shock $\varepsilon_{it}$. Consequently, measurement error and structural shocks are not independent in the misspecified augmented proxy SVAR. Therefore, estimators based on independent shocks should not be used to estimate augmented proxy SVARs with a misspecified proxy process. Of course, a correctly specified non-linear augmented proxy SVAR is a possible solution. Nevertheless, it again requires to correctly specify the non-linear proxy process and misspecifications again leads to similar problems.
One might argue that imposing Equations ((ref)) and ((ref)) amounts to specifying the proxy’s DGP, similar to the augmented proxy SVAR. However, the distribution of $D(z)$ does not depend on any distributional assumption about $z$; it follows asymptotically from the central limit theorem. In other words, regardless of the DGP of $z$, as long as the central limit theorem’s conditions hold, the moment conditions $D(z)$ will be asymptotically normal, thereby justifying Equations ((ref)) and ((ref)).
The consequences of misspecification in the augmented non-Gaussian proxy SVAR can be observed in the following Monte Carlo simulation. We generate data using a non-Gaussian SVAR with three variables and one exogenous proxy
In the first setup, the proxy is generated by a linear DGP, i.e. $z_t = \varepsilon_{\tau, t} + \eta_t$. In the second simulation, the proxy is generated by a truncated linear DGP, i.e. $z^{new}_t = \psi_t z_t$ where $\psi_t$ is Bernoulli random variable such that on average $80$% of the proxy observations are truncated to zero. The second case is relevant in our empirical work. We estimate the augmented proxy SVAR (without any zero restrictions implied by a valid proxy) using the non-Gaussian likelihood from Section (ref) . Table (ref) shows the estimated effect of the target shock $\varepsilon_{\tau, t}$. We observe that proxy misspecification in the augmented proxy SVAR leads to a bias of the non-Gaussian SVAR estimator and large estimation error (compare with results in Table (ref)).
In summary, the augmented proxy SVAR requires to specify the DGP of the proxy and misspecification of the DGP can lead to dependencies and render a non-Gaussian estimation based on the independence assumption invalid. In contrast, a moment-based proxy estimation approach never specifies the DGP of the proxy and hence, cannot suffer from misspecification consequences.
This section demonstrates the ability of our non-Gaussian proxy weighting approach in handling both exogenous and endogenous proxy variables. Our model is able to use the information of the proxy, which leads to a better performance compared to a purely non-Gaussian model. Furthermore, even when the proxy exhibits weak exogeneity, meaning it is only minimally affected by non-target shocks, our approach retains the ability to utilize the proxy to enhance efficiency, surpassing the performance of the non-Gaussian model alone. Moreover, our approach can detect whether the data provide evidence against the exogeneity of the proxy and can neglect information from endogenous proxies.
We simulate a system containing a government spending shock $ \varepsilon_{g, t}$, an output shock $ \varepsilon_{y, t}$, and a tax shock $\varepsilon_{\tau, t}$ with
The structural shocks are drawn independently and identically from a Pearson distribution with mean zero, variance one, skewness $0.68$ and excess kurtosis $2.33$ and simulate $1000$ data sets of length $T=250$ and $T=800$.
We construct a variable $z_{ t}$ as a proxy for the tax shock $\epsilon_{\tau, t}$. To demonstrate the flexibility of our model, we consider three different scenarios summarized in Table (ref). In the first scenario, the proxy is exogenous, in the second scenario, the proxy is weakly endogenous, and in the third scenario the proxy is endogenous. In all simulations, the proxy noise $\eta_{t}$ is i.i.d. and drawn from the same distribution as the structural shocks.
We consider four Bayesian estimation approaches.
Both non-Gaussian models use the true impact matrix $\bm B_0$ from the DGP in Equation ((ref)) to label the shocks in the MCMC. Specifically, for each proposal $\bm B$ we calculate $\bm B_0^{-1} \bm B$ and accept the proposal if each diagonal element of $\bm B_0^{-1} \bm B$ is larger in absolute value than the upper-right elements of $\bm B_0^{-1} \bm B$ in the same row, see keweloh2023uncertain.
Table (ref) shows the average point estimates and the mean squared error (MSE) of the estimated impact of $\varepsilon_{\tau ,t}$. The non-Gaussian model does not use the proxy variable and its performance is not affected by the different scenarios. Both Gaussian proxy approaches lead to very similar results and perform notably better than the non-Gaussian model if the proxy variable is exogenous, however, including an endogenous proxy leads to biased estimates. Combining statistical identification with proxy variables provides a balanced solution between these two extreme cases. If the proxy variable is exogenous, adding the proxy to the non-Gaussian model leads to an increase of the performance of the model, i.e. the MSE is twice as small compared to the non-Gaussian model. Therefore, the model is able to utilize the information of a valid proxy. However, in contrast to the Gaussian proxy models, our proposed combination approach is able to deal with endogenous proxy variables. If the proxy variable is only weakly endogenous, our non-Gaussian proxy weighting approach can still exploit the information of the proxy to deliver improved estimation accuracy in comparison to the non-Gaussian model. Moreover, if the proxy variable is endogenous, our non-Gaussian proxy weighting approach is less biased compared to the pure proxy approaches, and with more data and more evidence against the validity of the proxy variable, the prior gets updated and the bias decreases.
Next, we turn our attention to the finite sample properties of the $68$% credible bands of the models. Table (ref) shows the coverage rate (defined as the proportion in which the credible bands contain the true value) and the average length of the credible bands. The non-Gaussian model ignores the proxy variable, performs similarly throughout the three specifications, and has correct coverage rates (the coverage rate is close to the probability chosen for the credible bands). If the prior belief of an exogenous proxy is correct, adding the proxy also leads to correct coverage and more informative credible bands in the sense that the bands are up to $50$% smaller compared to the non-Gaussian model ignoring the proxy variable. Adding an endogenous proxy using to the non-Gaussian model via our weighting approach worsens the coverage rates. However, with an increasing sample size and more information against the prior belief of an exogenous proxy, the coverage rate improves and the difference of the error bands length between the model with and without prior vanishes. Importantly, adding a weakly endogenous proxy also helps in a small sample size to lower the lengths of the credible bands without distorting the coverage much. In the Online Appendix we present various additional Monte Carlo experiments to further demonstrate the flexibility of our approach.
This section applies our proposed non-Gaussian proxy weighting approach to estimate the effects of exogenous changes in tax revenues and government spending. First, we describe the data and show that the time series feature a sizable degree of non-Gaussianity. Then our main findings are discussed. We find a larger government spending than tax multiplier. We provide evidence indicating that the fiscal and non-fiscal proxies used by mertens2014reconciliation and caldara2017analytics, respectively, do not fulfill the crucial exogeneity assumption, which biases the results of Gaussian proxy approaches.
To achieve comparability, we use the same trivariate VAR as adopted by mertens2014reconciliation.\footnote{caldara2017analytics consider a slightly larger VAR consisting of five endogenous variables as baseline model but also report results for the three variables specification we rely on. We show below that our main findings are robust to extending the baseline trivariate VAR by inflation and the interest rate as done by caldara2017analytics.} The three endogenous variables are federal tax revenues $\tau_{t}$, federal government consumption and investment expenditures $g_{t}$, and output $y_{t}$, all in log real per capita terms and for the sample 1950Q2 to 2006Q4.\footnote{The data are downloaded from Karel Mertens' website.} The VAR has four lags and includes a constant, linear, and quadratic trends, and a dummy for 1975Q2 all contained in $\bm X_t$. The SVAR is given by
with tax shocks $\varepsilon_{\tau,t} $, government spending shocks $\varepsilon_{g,t} $, and output shocks $\varepsilon_{y,t}$. For comparison, we map the interaction into the notation used by mertens2014reconciliation
mertens2014reconciliation use a tax proxy $z_{\tau,t}$ and a zero restriction imposing that government spending does not respond contemporaneously to changes in economic activity. The tax proxy relies on a series of possibly unanticipated tax shocks, a subset of the Romer2010 tax shocks identified by studying narrative records of tax policy decisions. As argued, the tax proxy measures changes in the tax system that are not related to the state of the economy and thereby it should offer a valid proxy for tax shocks. Therefore, mertens2014reconciliation impose the following exogeneity assumptions
Instead of using a fiscal proxy, caldara2017analytics rely on the non-fiscal Fernald2012 TFP measure as a proxy $z_{y,t}$ for output shocks and additionally assume that government spending does not respond contemporaneously to tax shocks. The Fernald2012 technology series measures total factor productivity adjusted for changes in factor utilization. Fernald2012 carefully eliminates these sources of endogenous movements such that his resulting purified TFP series can be understood as solely reflecting exogenous technology variations which motivates the two exogeneity assumptions
used by caldara2017analytics. In what follows, we rely on the original TFP measure used by caldara2017analytics.
Both approaches start with carefully motivated identifying assumptions, yet both approaches lead to different conclusions regarding the effects of tax and spending shocks. In particular, the fiscal proxy leads to a large tax multiplier while the non-fiscal proxy leads to a large spending multiplier. This difference indicates that at least one of the identifying proxy exogeneity assumptions is invalid.\footnote{ In principal, the differences could be driven by the different zero restrictions. However, computing the tax shocks based on the tax proxy and the output shocks based on the TFP proxy leads to correlated shocks. This correlation is not affected by the zero restrictions and indicates that the different results are indeed driven by invalid proxies.} However, both approaches rely on the exogeneity assumptions to identify the SVAR and hence, cannot detect endogenous proxies. Our model fills this gap since it allows to evaluate the empirical support for the exogeneity assumptions and thus helps in understanding diverging findings regarding the size of fiscal multipliers between the fiscal and non-fiscal approach.
Besides the tax and TFP proxies, we use the growth rate of military spending per head of population as a proxy $z_{g,t}$ for government spending shocks. Hall2009,Barro2011,Miyamoto2019, amongst others, use military spending to identify exogenous government spending shocks. Changes in military spending are often large and regularly respond to foreign policy developments, suggesting that these changes are exogenous in the sense that they are less likely to be driven by domestic cyclical forces.
We estimate the SVAR in Equation ((ref)) using the non-Gaussian proxy weighting approach without any zero restrictions on the response of government spending. We normalize all proxies to mean zero and unit variance and use the proxy weighting moment conditions
with $D(z)=[D_1(\bm z_{\tau}, \mu_{\tau g} ), D_2(\bm z_{\tau}, \mu_{\tau y}),D_3(\bm z_{y},\mu_{y \tau} ),D_4(\bm z_{y},\mu_{y g}), D_5(\bm z_{g}, \mu_{g \tau} ),D_6(\bm z_{g},\mu_{g y} ) ]'$ and the proxy exogeneity shrinkage prior from Equation ((ref)). Therefore, we start with the prior that all proxies are valid and shrink towards exogenous proxies. However, the data can update the prior and deviate from a given exogeneity assumption. Moreover, we use the proxy variables to label the shocks similar to bertsche2022identification. Specifically, the shock exhibiting the highest correlation in absolute terms with the TFP proxy is labeled as the output shock. Among the two remaining shocks, the one displaying the strongest correlation in absolute terms with the tax proxy is identified as the tax shock, while the remaining shock is labeled as the government spending shock.\footnote{An alternative labeling approach relying on a first step estimator is shown in the Appendix and leads to similar results.}
For comparison, we also estimate the mertens2014reconciliation fiscal proxy SVAR with the Gaussian proxy weighting approach, the government spending restriction $b_{23}=0$, and the tax proxy weighting moment conditions $D_1(\bm z_{\tau},\mu_{\tau g} )$ and $D_2(\bm z_{\tau}, \mu_{\tau y})$ with $\mu_{\tau g}=\mu_{\tau y}=0$ and the caldara2017analytics non-fiscal proxy SVAR with the Gaussian proxy-weighting approach, the government spending restriction $b_{13}=0$, and the TFP proxy weighting moment conditions $D_3(\bm z_{y},\mu_{y \tau} )$and $D_4(\bm z_{y},\mu_{y g})$ with $\mu_{y \tau}=\mu_{y g}=0$. Therefore, both Gaussian models impose proxy exogeneity without the ability to deviate.\footnote{Note that the two Gaussian proxy weighting models yield impulse responses akin to those acquired through the conventional moment-based frequentist proxy estimator, as shown in the Appendix.}
A requirement for updating proxy exogeneity by the data is that we work with non-Gaussian structural shocks. Figure (ref) shows the posterior of the skewness and kurtosis of the estimated structural shocks in the non-Gaussian proxy weighting SVAR and the two Gaussian proxy weighting SVARs. All three models show a sizeable degree of non-Gaussianity in the estimated structural shocks. In particular, the skewness of the tax and output shock is centered around non-zero values and the kurtosis shows positive values above three. Across all models, the tax and output shocks are left skewed, which is economically reasonable because tax cuts tend to be larger than tax hikes and output falls stronger during recessions than it rises during expansions. In addition, there is some evidence that the government spending shock is right skewed, indicating that spending stimuli are larger than spending consolidations.
The second requirement for our non-Gaussian identification approach are independent structural shocks. The common critique to the independence assumption is a potentially shared volatility process, see montiel2022svar. Figure (ref) displays the posterior of the test statistic $S(E)=\sqrt{ \frac{1}{n(n-1)} (Corr(\varepsilon_{\tau,t}^2, \varepsilon_{g,t}^2)^2 +Corr(\varepsilon_{\tau,t}^2, \varepsilon_{y,t}^2)^2 +Corr(\varepsilon_{g,t}^2, \varepsilon_{y,t}^2)^2 )}$ proposed by montiel2022svar to measure the common volatility of all shocks and the posterior of the three possible cross-moments of squared shocks together with the corresponding distributions approximated by bootstrap under the null of independent shocks by randomly permutating the shocks similar to the approach in braun2021importance. In all cases, we find that the posterior with the posterior simulated under the null of independent shocks overlap to a large extend, suggesting no evidence against mutual independence. Moreover, the $90$% bands of the squared cross-moments contain zero, which is the value of the squared cross-moments $E[\varepsilon_{i,t}^2\varepsilon_{j,t}^2-1]$ of mutually independent shocks $\varepsilon_{i,t}$ and $\varepsilon_{j,t}$.
Figure (ref) plots results on the relevance of the proxies in our non-Gaussian proxy weighting model. The left column shows the posterior of the correlation between the tax proxy and tax shock, the middle column presents the posterior of the correlation between the TFP proxy and the output shock, and the right column shows the posterior of the correlation of the military spending proxy and the government spending shock. All distributions are centered around a positive mean with no support of values close to zero. Thus, we find that all proxies are strong in the sense that they satisfy the relevance condition.\footnote{Note that, we plot the correlation between the instrument and the structural shock, whereas mertens2014reconciliation and caldara2017analytics have to focus on the correlation between the instrument and the reduced form shock.}
Figure (ref) presents the estimated tax and spending multipliers, where the first column reports the tax multiplier, the second column shows the spending multiplier, and the third column presents the estimated difference between the tax and spending multiplier.\footnote{Similar to mertens2014reconciliation, we calculate tax multipliers by dividing the output response of a tax revenue shock of minus one percent by the average ratio of federal tax revenues to GDP in the sample of 17.5%. Government spending multipliers are calculated by dividing the output response of a public spending shock of one percent by the average ratio of federal spending to GDP in the sample of 9.1%. Equivalently, the numbers reflect the present response to a tax cut (government spending increase) that lowers (increases) tax revenues (government spending) by one percentage point of GDP.} The solid lines report the estimates of the non-Gaussian proxy weighting approach and shaded areas indicate 68% credible bands. As a comparison with the existing approaches, dashed lines show the estimates of the mertens2014reconciliation fiscal proxy SVAR and dashed-dotted lines report the estimates of the caldara2017analytics non-fiscal proxy SVAR.
We first discuss the estimates of our proposed non-Gaussian proxy weithing approach. The tax multiplier reported in the first column of Figure (ref) is estimated to be close to zero for the impact period. Only around one year after the shock, the multiplier increases. The multiplier peaks at a value of $0.7$ two years after the shock materialized. Thereafter, the response slowly converges back to its pre-shock level. The estimated government spending multiplier presented in the second column shows some stark differences. For the impact period, the spending multiplier is estimated to be different from zero taking on a value above but close to unity. The third column showing the difference between the government spending and tax multiplier clearly depicts the divergent dynamics of both multipliers. For the first year after the shock, the spending multiplier clearly exceeds the tax multiplier. Over the medium term, the difference becomes negative implying that the tax multiplier is larger than the government spending multiplier, although the point estimate for the difference is small. Thus, our baseline model suggests that positive government spending shocks have a larger stimulating impact on economic activity than exogenous tax cuts.
Figure (ref) compares our baseline estimates of the non-Gaussian proxy weighting SVAR with the estimates obtained when only relying on non-Gaussianity. The non-Gaussian model does not use any information included in the proxy variables. While the point estimates show only small differences to our baseline model, estimation uncertainty is much larger in the model without proxy variables. Thus, in line with our Monte Carlo simulation results presented above, the fiscal multiplier application shows that potentially endogenous proxies still provide important information that increases estimation precision. Identification through non-Gaussianity misses this information, such that error bands become wider.
As shown in Figure (ref), our estimates show strong differences to the fiscal proxy SVAR of mertens2014reconciliation. The fiscal proxy SVAR leads to a much larger tax multiplier compared to our baseline estimates. In particular, the fiscal proxy SVAR delivers an on-impact tax multiplier around two and the multiplier further increases in the subsequent periods reaching a peak value close to three. Given the much larger tax multiplier, the fiscal proxy SVAR implies that exogenous tax cuts are a more powerful tool to stimulate the economy compared to exogenous increases in government spending, which is the opposite of our baseline results. Additionally, the non-fiscal proxy SVAR of caldara2017analytics leads to a larger government spending multiplier compared to our baseline estimates. In particular, the spending multiplier peaks at a value above two, whereas our baseline estimate shows a maximum value above but close to unity. Similarly to our non-Gaussian proxy weighting model, applying the non-fiscal proxy SVAR also results in a positive difference between the government spending and tax multiplier. In the Appendix, we show that the differences in the estimated multipliers can be explained by different estimates on the cyclical elasticities of tax revenues ($\Theta_{y}$) and government spending ($\gamma_{y}$).
Our main finding that the government spending multiplier is larger than the tax multiplier is highly robust to modifications of the baseline empirical specification. We report a battery of robustness checks in the Appendix. Specifically, our results are robust to including additional endogenous variables in the VAR, controlling for fiscal foresight, splitting and extending the sample. Moreover, the results are not significantly affected when separately estimating the models with a single proxy compared to the baseline model that uses all proxies.
The key advantage of our non-Gaussian proxy weighting approach is that we can update the proxy exogeneity priors. That is we shrink towards exogenous proxies, however, if the data provide evidence against a given exogeneity assumption, our model can update the prior and stop to shrink towards exogenous proxies. In contrast, when applying the Gaussian fiscal proxy SVAR from mertens2014reconciliation or the Gaussian non-fiscal proxy SVAR from caldara2017analytics, it has to be assumed that the respective proxy is exogenous and this prior cannot be updated. In the following, we show that the data provide evidence against the exogeneity of both proxies, which further helps in understanding the different multiplier estimates across identification strategies.
Figure (ref) provides evidence on the exogeneity assumptions obtained from our non-Gaussian model. The two graphs in the first column show the posterior distributions of the correlation between the tax proxy and the structural government spending and output shock, respectively. The second column presents the posterior distributions of the correlation between the TFP proxy and the structural tax and government spending shock, respectively. The third column shows the posterior distributions of the correlation between the military spending proxy and the structural tax and output shock, respectively. With an exogenous proxy variable, meaning $\mu_j=0$, the proxy prior shrinks toward shocks without systematic correlation with the proxy variables. However, the data provide evidence against the exogeneity assumptions and the model prevents shrinkage towards some exogeneity assumptions. For example, the tax proxy has a clear negative correlation with the output shock. Put differently, positive (negative) output shocks coincide with negative (positive) values for the tax instrument. Intuitively, not accounting for this correlation leads to identified tax cuts that also include exogenous increases in output, and vice versa, which increases the size of the estimated tax multiplier. Concerning the TFP proxy, we find evidence that the instrument is negatively correlated with exogenous government spending shocks. Therefore, the TFP proxy identifies positive (negative) output shocks that also include negative (positive) government spending shocks, which reduces the fraction of GDP movements explained by the identified output shocks. As a result, the estimated government spending elasticity is reduced, which as shown in caldara2017analytics, increases the size of the estimated government spending multiplier. Finally, the military spending proxy seems to fulfill the exogeneity assumptions. The correlations with the structural tax shock and the output shock, respectively, are centered around zero.
Our findings on the endogeneity of the tax proxy align with bruns2024testing, who develop a proxy exogeneity test based on the idea that if a proxy $z_t$ is exogenous, any function of it, i.e. $z_t^2$, should also be exogenous. This yields an overidentified system, allowing for a J-test of proxy exogeneity. Applying this test, bruns2024testing find evidence of endogeneity in the tax proxy, but no such issues for the government spending proxy, consistent with our results. However, because the output proxy lacks skewness, $z_t^2$ becomes irrelevant, rendering the J-test powerless to detect its endogeneity. In contrast, our identification strategy based on the assumption of independent shocks does not suffer from this limitation. Indeed, we find indications of endogeneity in the output proxy as well, though the evidence is substantially weaker than for the tax proxy.
To further highlight the implied differences across identification strategies, in the Online Appendix we show the times series for the estimated structural shocks. Although the estimated shocks share a similar pattern for most periods, there are some discrepancies worth mentioning. In general, we find that the fiscal proxy VAR approach shows the tendency to interpret positive output shocks as negative tax shocks, and vice versa. As a consequence, by taking this shortcoming of the fiscal proxy into account, our non-Gaussian proxy weighting approach leads to a much smaller tax multiplier compared to the standard fiscal proxy identification strategy. We further highlight some episodes in which the non-fiscal proxy model and the non-Gaussian proxy weighting model differ in the sequence identified output and government shocks. We explain in detail why we think that the identified shock of our proposed non-Gaussian proxy weighting model fits better to the historical narrative.
Furthermore, we go one step further and construct new proxy measures that are orthogonal to the disturbances of the model. To get new proxy measures we proceed as follows. For the narrative tax proxy, we regress the proxy on a constant and the median structural government spending and output shocks obtained from applying the non-Gaussian proxy weighting approach. Similarly, for the TFP measure, we regress the proxy on a constant and the median structural tax and government spending shocks. The estimated residuals of these regressions capture movements in the original proxy measures that are not related to other disturbances of the model.\footnote{One should keep in mind that the construction of the new proxies is model-specific. In particular, for our application, we use the simple three variables baseline VAR which is a common model used in the fiscal policy literature. In addition, using the newly constructed proxies as observable data might suffer from a generated regressor problem.} The estimated fiscal multipliers using the new proxies using a Gaussian proxy approach are very similar to the ones of our baseline non-Gaussian proxy weighting approach. The new proxies and estimation results are discussed in the Appendix.
This paper discusses the challenge of measuring the effects of fiscal policy and the recent use of the proxy VAR approach as a tool to identify fiscal policy shocks. We propose a new non-Gaussian proxy weighting approach that combines non-Gaussian identification with proxy variables. The method shrinks towards exogenous proxy variables but allows for updating the prior and stopping the shrinkage if empirically warranted. We use our model to provide evidence that the contradicting results of mertens2014reconciliation and caldara2017analytics may be due to the use of invalid instruments. Finally, we find that increasing government spending is more effective in stimulating the economy than lowering taxes.
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