Keywords: Fiscal policy, identification, vector autoregressions
Abstract:
Governments often use fiscal policy to stabilize economic fluctuations. For example, during the recent recession, the United States Congress approved the American Recovery and Reinvestment Act, which introduced increases in public spending and cuts in taxes by approximately 6% of GDP (CBO, 2010). The rationale for such fiscal stimulus rests on the assumption that fiscal interventions do affect economic activity. Yet, the size of fiscal multipliers, defined as the dollar response of output to an exogenous dollar spending increase or tax cut, is the subject of a longstanding debate in academia. As (Perotti, 2008) observes in his survey of the literature: "... perfectly reasonable economists can and do disagree on the basic theoretical effects of fiscal policy and on the interpretation of existing empirical evidence".
The presence of competing economic theories has motivated a large body of empirical investigations that measure the size of these fiscal multipliers. An important share of the literature relies on structural vector autoregressions (SVARs). Prominent examples include (Blanchard & Perotti, 2002), and (Mountford & Uhlig, 2009). The appeal of SVARs is that they control for endogenous movements in fiscal policies by imposing only a minimal set of assumptions, known as identification schemes. An alternative methodology identifies exogenous changes in taxation (Romer & Romer, 2010; Mertens & Ravn, 2011b) and public spending (Eichenbaum & Fisher, 2005; Ramey, 2011; Ramey & Shapiro, 1998) from narrative records, and studies their effects using VARs. Yet, despite their simple structure and the use of similar data, studies employing SVARs and narrative records report fiscal multipliers that are spread over a broad range of values. The lack of consensus prevents the profession from providing clear guidance on important policy choices, such as the size and composition of fiscal interventions.
Motivated by this knowledge gap, our paper asks two questions. Why do SVARs provide different measures of fiscal multipliers? Can we construct robust measures of fiscal multipliers using SVARs?
We answer the first question by deriving a unified analytical framework to compare competing identification schemes. We then apply this analysis to a fiscal VAR for the United States for the period 19472006. We show that existing identification schemes imply different restrictions on the output elasticity of tax revenue and government spending, which measure the endogenous response of tax and spending policies to economic activity.
We illustrate the framework for comparing different identification schemes with a tax policy example. Assume that only two shocks explain contemporaneous comovements between output and tax revenue: a tax shock and a nonpolicy shock. The object of interest is the response of output to the tax shock. The nonpolicy shock controls for comovements in the two variables due to automatic movements of tax revenue over the business cycle. In this setting, the identification of tax and nonpolicy shocks depends only on the restriction on one structural coefficient: the output elasticity of tax revenue.
The (Blanchard & Perotti, 2002) and the (Mountford & Uhlig, 2009) identification schemes imply output elasticities of tax revenue equal to 1.7 and 3.0, respectively. Standard sign restrictions on impulse response functions imply output elasticities of tax revenue between zero and infinity. Narrative identification of tax shocks imply an output elasticity of tax revenue above 3. Different restrictions on the output elasticity of tax revenue generate a large dispersion in the estimates of tax multipliers. For instance, we find that the impact tax multiplier is 0.17 dollars for an output elasticity of tax revenue equal to 1.7, whereas it is more than five times as large  at 0.93 dollars  for an output elasticity of tax revenue equal to . The impact tax multiplier is negative for all output elasticities of tax revenue smaller than 1.5. More in general, thanks to the analytical relations, we can readily map beliefs of policymakers and economists about plausible values of the output elasticity of tax revenue into tax multipliers.
These findings lead to the second question. We propose to measure fiscal multipliers more robustly by imposing restrictions on the output elasticities of fiscal variables in the form of probability distributions. In contrast to the existing literature, we impose distributions that encompass the existing empirical evidence on elasticities. The distribution of the output elasticity of tax revenue that we obtain ranges between 1.7 and 3. The distribution of the output elasticity of government spending ranges between 0.1 and 0.1. These restrictions are robust because they are generated by different approaches and empirical strategies and, hence, are less likely to be affected by particular assumptions or observations.
We apply this robust identification scheme to measure tax and spending multipliers associated with unexpected fiscal shocks.^{1} We document three findings. First, the median tax multiplier is 0.65 on impact, and it becomes larger than 1 five quarters after the policy intervention. Second, the median spending multiplier is larger than 1 at all horizons. Third, the probability that the tax multiplier is larger than the spending multiplier is below 0.5 at all horizons.
We also document a high probability that private consumption increases following a spending shock. Competing macroeconomic theories have different theoretical predictions regarding the effects of government spending shocks on private consumption. The standard neoclassical and New Keynesian models predict a decline in consumption (Linnemann & Schabert, 2003; Baxter & King, 1993). A recent branch of the literature (Ravn et al., 2006; Galí et al., 2007) proposes models that generate an increase in private consumption. The evidence is in line with the latter class of models.
The focus on the identification problem, as opposed to the estimation and specification of the reducedform VAR model, is based on evidence provided by (Chahrour et al., forthcoming) and (Caldara & Kamps, 2008). (Chahrour et al., forthcoming) employ a DSGEmodel approach to reject the hypothesis that the different tax multipliers estimated by the SVAR and narrative approaches are due to differences in the assumed reducedform transmission mechanisms. (Caldara & Kamps, 2008) find that, controlling for the specification of the reducedform VAR model, different identification schemes provide different estimates of tax and spending multipliers.
The remainder of the paper is organized as follows. Section I derives the analytical relation between output elasticities of fiscal variables and impulse response functions. It also characterizes theoretical properties of such relations. Section II reinterprets five alternative identification schemes used in the literature as restrictions on the output elasticities of fiscal variables. Section III analyses the implications of alternative priors on elasticities for fiscal multipliers. Section IV reviews the existing empirical evidence on elasticities and reports evidence on fiscal multipliers based on prior distributions that encompass the full range of elasticity estimates documented in the literature. Section V sheds light on two debates in the literature on the effects of government spending shocks: the response of private consumption and the role of fiscal foresight. Section VI concludes.
Consider the reducedform VAR model:
The reducedform disturbances will in general be correlated with each other and consequently do not have any economic interpretation. It is thus necessary to model the contemporaneous relation between reducedfrom disturbances to identify structural shocks with an economic interpretation:
Numerical results presented in the paper are based on the estimation of an 8equation VAR model in the logarithm of output, tax revenue, government spending (sum of government consumption and investment), private consumption, private nonresidential investment (all in real, percapita terms), CPI inflation, the 3month Tbill rate, and a measure of stock prices. We use quarterly data for the United States from 1947 to 2006.^{2} To illustrate our main results on identification uncertainty, we abstract from sampling uncertainty and present results based on OLS point estimates; the evidence on fiscal multipliers reported later in the paper accounting both for identification and sampling uncertainty instead relies on Bayesian estimates.^{3}
The estimation strategy addresses the wellknown misspecification problem of SVARs in the presence of anticipated fiscal shocks (Leeper et al., 2008). Variables such as consumption, investment, and prices react to signals about future policies. Lagged values of these variables predict future policy actions and, consequently, help to identify truly unexpected fiscal shocks (Forni & Gambetti, 2010; Giannone & Reichlin, 2006).^{4} By including these variables in our reducedform VAR model, we ensure that measures of anticipated fiscal shocks derived from narrative records (see (Ramey, 2011) for measures of anticipated government spending shocks, and (Mertens & Ravn, 2011b) for measures of anticipated tax shocks) do not Grangercause the fiscal shocks identified by our SVAR models (see Appendix A).
To understand how the choice of identifying restrictions affects inference, we consider a simplified example. We assume that the model consists of a nonpolicy variable and of a policy variable. The nonpolicy variable is output (). The policy variable, denoted by is either tax revenue (), or government spending ().
The relation between reducedform disturbances and structural shocks
can be written as:
Equation (3) states that unexpected movements in output are due to either unexpected movements in the policy variable ( ) or sources of business cycle fluctuations unrelated to the policy under investigation (). Equation (4) states that unexpected changes in the policy variable are either endogenous to the business cycle ( ) or exogenous to the business cycle and uncorrelated with nonpolicy sources of fluctuations (). Endogeneity of policy can arise either because policymakers react to contemporaneous developments in economic activity, or because of automatic feedback from activity to tax revenue and government spending. We follow (Blanchard & Perotti, 2002), B&P henceforth, and assume that the first channel is eliminated by the use of quarterly data. This is plausible due to information lags, legislative lags, and implementation lags faced by policy makers. Consequently, the coefficient captures the automatic response of fiscal variables to changes in economic activity, measured as the output elasticity of tax revenue ( ) and the output elasticity of government spending ( ), respectively.
In the bivariate case, we need to impose one identification restriction to identify the SVAR model: here this boils down to a restriction on .^{5} To highlight the restricted coefficient, we denote throughout the paper as . In the public finance literature, there is a long tradition of measurement of the output elasticity of fiscal variables in the context of the cyclical adjustment of budget balances. The output elasticity of tax revenue is the most familiar measure of sensitivity of taxes to income changes. This elasticity serves as an indicator of the tax system's overall progressivity. A proportional income tax has an elasticity of 1.0. Progressive tax systems, for which taxtoincome ratios all other things equal increase with income, have an elasticity larger than 1.0. As far as the output elasticity of government spending is concerned, most studies  including B&P  assume its value to be zero, based on the observation that government consumption and investment have at most weak cyclical components.
We view numerical restrictions as priors of the economist regarding a plausible value, or a set of plausible values, for the elasticities. As we describe in Section 2, in the literature economists have formed and implemented priors on using a variety of methods.
The system described by Equations (3) and (4) can be written in terms of impulse vectors as:
The object of interest is the contemporaneous response of output to a policy shock:^{6}
The denominator of this expression measures the strength of macroeconomic feedback. In the special cases in which either one of or is zero there is no feedback.
What we are interested in is to know how the output response to a policy shock depends on the prior of the econometrician about . In the bivariate model, there exists a simpleclosed form solution.^{7} The contemporaneous response of output to a policy shock can be rewritten as:^{8}
To facilitate the comparison between tax multipliers and spending multipliers, we compare shocks that are intended to stimulate output, i.e. we analyze the effects of structural spending increases but structural tax cuts. Figure 1 plots the impact fiscal multiplier as a function of the output elasticity of the respective fiscal variable, evaluating the covariance matrix at the OLS estimates of the tax and spending model, respectively.
Figure 1 highlights two important properties of expression (7). First, the set of output responses to a policy shock is bounded, and bounds have opposite signs. An important implication is that if the econometrician does not have any information to limit the set of plausible values for the output elasticity, the sign of the output response cannot be determined. Second, the output response to a policy shock is zero if and only if . Hence is the threshold elasticity to determine the sign of the output response to a policy shock.
The top panel of Figure 1 shows that, if the econometrician takes an agnostic view of plausible values for the output elasticity of taxes, the impact tax multiplier lies within a range between 1.07 and 1.07 dollars. However, typically at least some extramodel information may be available to narrow down the set of plausible assumptions about the output elasticity of taxes. For example, it appears implausible to assume that the business cycle has a negative effect on tax revenue. Yet excluding negatives values of would still be insufficient to pin down the sign  let alone the size  of the impact tax multiplier. Indeed, ruling out , the impact tax multiplier lies within a range between 0.92 and 1.07 dollars. Furthermore, even excluding , i.e. assuming that the tax system is at least proportional, is insufficient to pin down the sign of the impact tax multiplier. In this case, the impact tax multiplier lies within a range between 0.39 and dollars. To ensure that the impact tax (cut) multiplier is nonnegative the econometrician has to assume . In our application, the output elasticity of taxes has to be at least 1.5.
Turning to spending shocks, the bottom panel of Figure 1 shows that  again, if the econometrician takes an agnostic view of plausible values for the output elasticity of government spending  the impact spending multiplier lies within a range between 2.26 and 2.26 dollars. However, negative spending multipliers only occur if government spending is procyclical ( ), given that in empirical applications the correlation between output and government spending residuals is positive in general. In our application, for all values of the output elasticity of government spending smaller than 0.38 the impact spending multiplier is positive.
Summing up, we have shown that the sign and size of spending and tax multipliers depend on the choice of the output elasticity of tax revenue and government spending. We have also characterized analytically the identification problem. In the next section, we show how the alternative identification schemes used in the existing literature can be mapped into priors of economists regarding the output elasticities of fiscal variables.
In this section, we use analytical results to reinterpret identification schemes as priors on output elasticities of fiscal variables. We show that these priors are different enough to produce widely divergent fiscal multipliers. We examine five identification schemes used in the literature: the recursive approach, the traditional SVAR analysis implemented by B&P, the "pure" sign restriction approach, the penalty function approach to sign restrictions, and the narrative approach.
We first analyze the recursive approach, proposed by (Sims, 1980). The recursive approach imposes a dogmatic prior either on the impact multiplier (tax policy) or on the output elasticity of government spending (spending policy). In the recursive approach the ordering of the variables in the reducedform VAR model determines the contemporaneous effects of shocks: the variable ordered first in the VAR system is only affected contemporaneously by the first shock but not by the second shock, whereas the variable ordered second is contemporaneously affected by both shocks. The recursive VAR approach is applied via the lowertriangular Cholesky decomposition of the covariance matrix .
Tax Shocks. In our implementation of the recursive approach, we order output first and tax revenue second in the VAR system.^{10} On the one hand, assuming a zero contemporaneous response of output to a tax shock is restrictive. On the other hand, the alternative ordering, equivalent to assuming that tax revenue does not react at all contemporaneously to the business cycle, would be even more implausible. For the chosen order, the second property of the impact multiplier mentioned in Section 1 gives the result: the impact tax multiplier is zero if and only if the output elasticity of taxes is equal to .^{11} In our VAR, is 1.5, a value which, as we discuss in Section 4, lies within the range of empirically plausible elasticities. This is the point denoted `CHOL' in the top panel of Figure 1. This point is a useful reference point: the impact tax (cut) multiplier will be positive if and only if .
Spending Shocks. We order government spending first and output second. That is, we assume that government spending is acyclical, i.e. . As discussed in Section 4, this assumption is in line with the consensus view that in the U.S. the contemporaneous output elasticity of government spending is zero. The point denoted `BPCHOL' in the bottom panel of Figure 1 shows that the impact spending multiplier amounts to 1.25 dollars for this value of the elasticity. The label `BPCHOL' reveals that in the case of the spending model this recursive formulation is equivalent to the B&P approach, to which we turn next.
The B&P approach relies on institutional information about the tax and transfer systems and about the timing of tax collections in order to form a dogmatic prior about plausible out elasticities of fiscal variables. We provide a detailed analysis of the B&P methodology to calculate elasticities in Section 4.
Tax Shocks. The point denoted `BP' in the top panel of Figure 1 gives the value of the impact tax multiplier for the point estimate of the output elasticity of taxes constructed according to the B&P methodology ( ). For this value of the output elasticity of taxes the impact tax multiplier amounts to 0.17 dollars. Notice that in our sample, the B&P elasticity is only slightly larger than the elasticity implied by the recursive approach, with the implication that the B&P tax multiplier is only slightly larger than zero.
Spending Shocks. As discussed in the previous subsection, B&P assume that government spending is acyclical, i.e. , which is equivalent to the lowertriangular Cholesky decomposition with government spending ordered first. Accordingly, the B&P and recursive approaches provide identical estimates of spending multipliers ( 1.25 dollars on impact).
An alternative approach to identification is to impose sign restrictions on impulse responses. We base the discussion of this approach on the work by M&U. M&U impose sign restrictions on impulse responses in combination with a criterion function, discussed in the next subsection. The exercise in this subsection unveils what the inference on fiscal multipliers in M&U without penalty function would have been. Other studies identifying fiscal shocks using the pure signrestriction approach include (Canova & Pappa, 2007) and (Pappa, 2009). For the sake of simplicity, we only impose sign restrictions on impact responses.^{12} We continue to focus the theoretical discussion on a bivariate model, while presenting numerical results for a multivariate model. We provide intuition for the generalization of the theoretical results to multivariate models (see the appendix for details).
We follow Uhlig's suggestion to decompose the factor matrix into the lower triangular Cholesky factor of the reducedform covariance matrix, denoted , and an orthogonal matrix, denoted with . That is, for the pure sign restriction approach we have .
The system describing the relationship between reducedform disturbances and structural shocks can be written in compact form as . As shown in the appendix, this system  using the analytical solution for the Cholesky factorization  can be expressed as follows:
Contrary to the B&P and the recursive identification strategies, the pure signrestriction approach does not impose a dogmatic prior on the output elasticities or the impact multipliers. Instead, it places restrictions on the sign of impulse responses to the shock(s) of interest. These restrictions translate into restrictions on the set of admissible rotation angles ^{14} In general, there are (infinitely) many structural models satisfying the sign restrictions, each of which has the same likelihood.
To map the factorization of the covariance matrix described in (8) into the identification framework described in Section 1, we map restrictions on the rotation angle into restrictions on the output elasticity of the policy variable:
Tax Shocks. We apply the basic assumptions of M&U who identify a nonpolicy shock  labelled `business cycle shock'  and a tax shock. They assume that the business cycle shock drives up both output and tax revenue and that the tax shock is orthogonal to the business cycle shock. M&U leave the response of output to a tax shock (the object of interest) unrestricted. In our framework, these assumptions imply the following restrictions on the elements of the factor matrix :^{15}
Using the analytical expression for , Proposition 2 in the appendix characterizes the set of all output elasticities of tax revenue that satisfy this sign pattern. We show in particular that all elasticities between zero and plus infinity satisfy the above sign restrictions. Hence, in the top panel of Figure 1 all points on the line segment with nonnegative values of the output elasticity of tax revenue are elements of the set of pure sign restriction solutions.
Is this large set of pure sign restriction solutions an artefact of our dataset and reducedform VAR model? This is very unlikely. It is generally accepted that output can be viewed as a proxy for the tax base. When the tax base expands, so does tax revenue (for constant tax rates). This is why empirically the correlation coefficient between output and tax residuals is positive (very often, strongly positive). In this case, sign restrictions alone are insufficient to pin down the sign of the tax multiplier.
The sign restrictions given above pose a challenge in the bivariate context. For all values of the elasticity between 0 and , we obtain two shocks with identical sign pattern, which puts in question identification unless additional restrictions are imposed. The above result, however, is useful in that so far it has been hard to understand the implications of such sign restrictions for the sign of the response of output to a tax shock. For example, it would have been reasonable to assume that the sign restrictions on the business cycle shock and the orthogonality assumption are sufficient conditions to rule out negative impact tax cut multipliers (Mountford and Uhlig, 2009: 965).
To rule out the subset of pure sign restriction solutions with negative impact tax cut multipliers, at least one additional assumption is needed. In the next subsection, we discuss the M&U approach, which adds a criterion function to the pure signrestriction approach. In a second approach, discussed below, we impose an additional sign restriction on the response of output to a tax shock:
Proposition 4 in the appendix characterizes the set of all output elasticities of tax revenue that satisfy this sign pattern. With the additional sign restriction on the response of output to a tax shock all output elasticities of taxes between , i.e. the elasticity implied by the Cholesky factorization, and satisfy the sign restrictions.
Figure 2 plots the empirical distributions of the output elasticity of tax revenue and of the tax multiplier over the set of sign restriction solutions evaluated at the OLS estimate, assuming  as it is common practice in the literature  that the rotation angle is uniformly distributed over the range satisfying the sign restrictions. Imposing this additional sign restriction "reduces" the set of admissible elasticities to all elasticities between 1.5 and , and the set of admissible multipliers to dollars. Of course, a drawback of the additional assumption is that in principle we would like to leave open the sign of the response of output to a tax shock.
Finally, we can ask how the above results are affected if we move beyond the bivariate setting and impose restrictions on additional variables. For example, M&U identify the business cycle shock assuming that such shock increases not only output and taxes but also private consumption and nonresidential investment. Would these additional assumptions restrict the set of admissible elasticities? Would it make the additional assumption on the output response to a tax shock redundant?
In general, the answers to these questions depend on the correlation structure between the signrestricted variables (see the appendix for details). In our application, at the OLS estimates, all output elasticities of taxes between 0 and 73 remain admissible. In other words, the additional assumptions have only a minor effect on the set of pure sign restriction solutions identified in the bivariate case. The intuition is that consumption and investment are positively correlated with output and tax revenue. Hence, business cycle shocks that drive up output and tax revenue are very likely to also drive up consumption and investment.^{16} Turning to the second question, our application reveals that the lower bound of the set of admissible elasticities remains unaffected by the additional sign restrictions on the responses to a business cycle shock. These additional restrictions are, thus, by themselves insufficient to rule out negative tax cut multipliers.
Spending shocks. Similar to the identification of the tax shock, M&U identify the spending shock as a shock that increases spending and that is orthogonal to the business cycle shock. The M&U tax and spending models differ in one crucial dimension: there is no sign restriction on the response of government spending to the nonpolicy shock.^{17}
The implication of the lack of restriction on the spending response to a business cycle shock is that government spending can be procyclical, acyclical, or countercyclical. In fact, all output elasticities of government spending ranging between minus and plus infinity satisfy these loose restrictions.^{18} As can be seen in Figure 1, in our empirical application, the impact spending multiplier can range anywhere between 2.26 and 2.26 dollars, because all points on the line are elements of the set of pure signrestriction solutions.
This minimal set of assumptions therefore does not rule out solutions for which the responses to the two shocks follow the same sign pattern. To rule out those solutions implying the same sign pattern for the two shocks, it is necessary to restrict the set of admissible output elasticities to the range between minus infinity and zero. For this range, the impact spending multiplier is positive.
Summing up, the sets of pure sign restriction solutions are very large in fiscal VAR models. Standard sign restrictions applied in the literature are insufficient to pin down the sign, let alone the size, of impact tax and spending multipliers. To pin down the sign of the impact multiplier, it is necessary either to directly restrict the object of interest (the multiplier) or to augment the pure sign restriction approach with a selection criterion, such as the one embodied in the penalty function approach to which we turn next.
Pure sign restrictions alone are insufficient to pin down the sign of the multiplier. To address this limitation, M&U augment the pure sign restriction approach with a penalty function, as proposed by (Uhlig, 2005).
Tax Shocks. In a bivariate model, the M&U penalty function translates into the following objective function, maximized with respect to :
Proposition 4 in the appendix provides the analytical solution for this maximization problem. Importantly, we prove that the impact tax cut multiplier, evaluated at the penalty function solution, is positive for all admissible values of the correlation coefficient between output and tax residuals. An important implication is that the application of the M&U penalty function is equivalent to imposing the restriction that the output response to a tax increase is negative, as discussed in the previous subsection.
The penalty function solution in the bivariate setup maximizes the fraction of covariance between the output and tax disturbances explained by the business cycle shock. Such penalty function summarizes the belief of M&U, well grounded in the evidence provided by the DSGE literature, that fiscal shocks do not contribute substantially to business cycle fluctuations. Consequently, the role of the business cycle shock is to explain as much variability as possible in the restricted variables, letting fiscal shocks explain the residual variance. In comparison, the Cholesky factorization with output ordered first maximizes the fraction of output variance explained by the business cycle shock, explaining 100% of both the output variance and the covariance on impact. Recall that for this Cholesky factorization the impact tax multiplier is zero. For the impact multiplier to be positive, the business cycle shock has to explain more than 100% of the covariance. When this condition is fulfilled, the conditional covariance generated by the tax shock has to be negative, which is only possible if output declines in response to a tax shock meant to increase tax revenue. The penalty function in our setup selects a business cycle shock that explains 151% of the contemporaneous covariance between output and tax disturbances, while the tax shock explains 51%. Hence the penalty function, which maximizes the covariance between output and tax revenue explained by the business cycle shock, favors large positive tax cut multipliers.
In the top panel of Figure 1, the penalty function sign restriction solution is denoted `MU'. In our example, this point  compared to the B&P and recursive approaches  corresponds to a large value of the output elasticity of tax revenue ( ) and to a value of the impact tax cut multiplier of 0.93 dollars. Note that the penalty function solution satisfies any additional sign restrictions on the impact responses of private consumption and investment to a business cycle shock (see appendix for details).
Spending Shocks. To explain the identification of government spending shocks in the bivariate setting, we assume  consistent with the subsection on the pure sign restriction approach  that the business cycle shock drives up output only (leaving the response of government spending unrestricted) and that the government spending shock is orthogonal to the business cycle shock. Trivially, the solution to the penalty function associated to these restrictions is the Cholesky factorization with output ordered first and government spending ordered second. For this Cholesky factorization, government spending is procyclical; in our example, , and the impact spending multiplier is zero, compared to 1.25 dollars for the B&P approach.
What would happen if, following M&U, we identify a business cycle shock imposing restrictions on output and tax revenue, while keeping the response of government spending unrestricted? As we show in the appendix the penalty function solution under these assumptions implies a zero impact spending multiplier (while the penalty function solution picks the same  large  impact tax cut multiplier as in the bivariate tax model). In addition, the penalty function solution selects a positive value of the output elasticity of spending (in our application the output elasticity of government spending goes down slightly compared to the one obtained for the Cholesky decomposition, to 0.36).
Summing up, the penaltyfunction approach to sign restrictions can be interpreted as an additional identifying assumption beyond pure sign restrictions. Moreover, in fiscal VAR models, the penalty function as specified by M&U picks a solution favoring large tax multipliers and low spending multipliers. This explains the main result of M&U, namely that tax cuts are more effective than spending increases.
An alternative methodology for estimating the effects of fiscal policy shocks using VARs is the narrative approach. Prominent examples are (Romer & Romer, 2010), who identify tax shocks studying narrative records of tax policy decisions, and (Ramey, 2011), who identifies government spending shocks using changes in military spending associated with wars. Multipliers estimated using SVAR models are different from multipliers estimated using the narrative approach. Differences are in part due to the fact that most studies using the narrative approach identify anticipated fiscal shocks. Yet, (Mertens & Ravn, 2011b) construct a series of unanticipated tax shocks based on (Romer & Romer, 2010) narrative records, and find larger multipliers than SVARs. To understand what drives such differences, we conduct the following exercise:
where is the contemporaneous response of output to a narrative tax shock of size one standard deviation. This step follows the empirical specification for estimating tax multipliers using narrative measures of tax shocks proposed by (Favero & Giavazzi, forthcoming).^{19}
In our VAR model, , a value remarkably close to the elasticity associated to the M&U penalty function approach to sign restrictions.^{21} The associated impact tax multiplier is 0.95 dollars. In Figures 1 and 2 the results for our application of the narrative approach are denoted `NARR'.
The narrative tax multipliers presented in this paper are considerably smaller than the estimates reported by (Romer & Romer, 2010) and (Mertens & Ravn, 2011b). This is due to differences in the scaling of shocks. Narrative studies consider tax shocks that increase tax revenue by . SVARs instead consider tax shocks that increase tax revenue by i.e. SVARs account for macroeconomic feedback: if exogenous tax increases depress output, tax revenue will increase by less than oneforone. To compare the SVAR and narrative approaches, all shocks are rescaled following the SVAR convention.^{22}
In the previous section, we have concentrated on identification uncertainty to clarify the main properties of the alternative identification approaches. We now broaden the analysis, accounting also for sampling uncertainty and looking at dynamic multipliers beyond the impact period.
We start with the implications of sampling uncertainty on estimates of elasticities and impact multipliers. Figure 3 plots kernel densities of output elasticities and fiscal multipliers. The B&P and recursive approaches provide lower estimates of the output elasticity of tax revenue, and consequently of the impact tax multiplier, than the M&U and narrative approaches. The B&P and recursive approaches also provide lower estimates of the output elasticity of government spending than the M&U approach. This translates into a larger impact spending multiplier for the B&P and recursive approaches. All the identification approaches are dogmatic as regards structural uncertainty, i.e. for a given reducedform estimate, they imply a single elasticity as well as multiplier. Distributions of elasticities and impact multipliers are uniquely due to sampling uncertainty.
Figures 4 and 5 plot dynamic tax and spending multipliers for the B&P and M&U approaches.^{23} Focusing on the output multiplier, differences across approaches are substantial up to three years after the policy intervention, although differences across approaches diminish in the longrun, despite being very persistent. The topright and bottomleft panel of Figure 4 plot the response of taxes and spending to a tax shock. For up to the three years after the policy intervention, the B&P approach predicts a larger effect of a tax cut on the deficit than does the M&U approach. The reason is that tax cuts identified with the M&U approach are partly selffinancing, as they boost GDP and consequently the tax base.^{24} The opposite holds for the response of the deficit to spending shocks. As shown in Figure 5, the spending shock associated to the B&P approach is partly selffinancing, due to the boost in tax revenue associated to the increase in GDP.
The bottomright panel of Figure 5 plots the response of consumption to a spending shock. For the B&P approach, the response is positive at all horizons. For the M&U approach, the response is not significant for the first five quarters, and turns positive thereafter. We provide a detailed analysis of the response of consumption to a spending shock in Section 5.
Figure 6 plots the probability of the tax multiplier being larger than the spending multiplier for all four SVARbased approaches.^{25} The policy implication is very different: according to the B&P and the recursive approaches, tax multipliers are very likely to be smaller than spending multipliers for the entire forecast horizon. This probability reaches at most 0.2 two years after the policy intervention, and falls thereafter. The M&U approach instead finds that tax multipliers are very likely to be larger than spending multipliers. The probability is 1 for the first two years after the policy intervention. It declines to 0.45 after 5 years, and increases again thereafter. The pure sign restriction approach finds probabilities close to 0.5, reflecting the large structural uncertainty associated with this approach.
In the previous sections, we have shown that differences in priors on elasticities implicit in alternative identification schemes translate into large differences in fiscal multipliers. Some of the identification schemes appear very dogmatic, selecting a single value of the relevant output elasticity. Others appear quite loose, imposing almost no restriction on the relevant elasticity. In this section we strike a balance between these two extremes, surveying the existing literature on automatic stabilizers to derive distributions on elasticities that encompass the existing empirical evidence. Then we estimate fiscal multipliers based on these prior distributions.
Output elasticity of tax revenue. The size of automatic stabilizers is the subject of many empirical studies in the macro public finance literature. Several international organizations and national agencies estimate the output elasticity of tax revenue for different tax categories, using these elasticities to construct cyclicallyadjusted measures of the budget balance. Results for the B&P approach presented in this paper are based on elasticity estimates provided by (Follette & Lutz, 2010).^{26} These authors estimate the output elasticity of tax revenue using micro data for four different tax categories: personal income tax, social security contributions, corporate income tax, and indirect taxes. We aggregate these elasticities to obtain a point estimate for the output elasticity according to the following aggregator:
(11) 
where denotes the tax category, denotes the level of tax revenue, denotes total tax revenue, and denotes the output elasticity of tax category . Following B&P, we evaluate and at their sample mean. The point estimate for the period 19472006 is 1.71. (Caldara, 2011) shows that sampling uncertainty around the point estimate is small and can be safely neglected.
The NBER also estimates the output elasticity of personal income taxes and social security contributions using the TAXSIM model (Feenberg & Coutts, 1993). This model implements a microsimulation of the U.S. federal income tax system. The model is based on a large sample of actual tax returns prepared by the Statistics of Income Division of the Internal Revenue Service. The average elasticity of personal income taxes and social security contributions estimated using TAXSIM is 1.65, which would increase the estimate for the overall elasticity to 1.8. The OECD^{27} also estimates the output elasticity of tax revenue for the United States. Following the OECD methodology, the aggregate elasticity is 1.2.
All these estimates of the output elasticity of taxes are lower than the value of 2.08 reported by B&P for their sample period 19471997. This difference is mainly due to differences in the definition of tax aggregates considered. In line with the literature cited above we use total tax revenue as tax variable. B&P instead use a concept of net taxes, subtracting transfers and net interest payments from tax revenue. This procedure mechanically increases the output elasticity of (net) taxes. The reason is that subtracting transfers and net interest payments  whose elasticity B&P set to 0.1 and 0, respectively  increases the weight associated to the other subelasticities^{28}, in turn implying that the output elasticity of net taxes is much larger than the output elasticity of total tax revenue. Considering only genuine tax revenue, i.e. the four tax categories mentioned above, the B&P elasticity of taxes amounts to 1.5 for their sample period and their assumptions about subelasticities. This latter figure is in line with the evidence reported in the cyclicaladjustment literature cited above.
Elasticity estimates taken from this cyclicaladjustment literature are often used in DSGE modeling when authors want to move beyond the simplistic assumption of a proportional tax system (output elasticity of taxes equal to 1.0). A prominent example for the United States is (Leeper et al., 2010) who rely on elasticity estimates from B&P and the OECD to calibrate or set priors on the output elasticity of different tax categories.^{29}
In an independent study, (Mertens & Ravn, 2011a) argue in favor of values of the output elasticity of tax revenue around 3. Their estimates are based on narrative measures of tax shocks. They make three arguments to support their finding. First, elasticity estimates from public finance studies are based on regressions that, although based on micro data, might be subject to a simultaneity bias. The narrative measure of tax shocks is exogenous to the state of the economy, and hence is not subject to such bias. Second, conditional on observing output, an SVAR identified assuming an output elasticity of tax revenue of 3 has greater explanatory power for the dynamics of tax revenues than an SVAR identified assuming a smaller value of the elasticity. Third, an elasticity of 3 generates an endogenous drop in tax revenue in 20082009 consistent with the drop observed in the data.
All in all, the macro public finance literature consistently documents output elasticities of tax revenue ranging from 1.2 to 1.8. Studies based on narrative measures of tax shocks find elasticities around 3. To encompass this empirical evidence, we draw the output elasticity of tax revenue from two normal distributions centered at the B&P and M&U narrative elasticity estimates of 1.7 and 3. Both distributions have a standard deviation of 1 to ensure a wide coverage around their mean.^{30} The top panel of Figure 7 shows that for this prior distribution the median tax multiplier is 0.65 dollars on impact. It starts to exceed one dollar five quarters after the policy intervention.
Output elasticity of government spending. Most authors in the VAR literature assume that the output elasticity of government consumption and investment is zero. In a similar vein, the cyclicaladjustment literature  e.g. the OECD  shares this assumption and does not attempt to estimate this elasticity.
There are some studies in the politicaleconomy literature that estimate the output elasticity of government spending to assess whether fiscal policymakers behave procyclically. Examples include (Lane, 2003) and (Rodden & Wibbels, 2010). Aggregate elasticities are not statistically significant in general. Yet, these papers document that some components of government consumption, such as public wages or state and local spending, are mildly procyclical. Furthermore, international evidence on spending elasticities suggests that in some countries government spending is procyclical. Finally, (Leeper et al., 2010) model government consumption and investment as mildly countercyclical.
The existing evidence tends to support the B&P assumption that in the U.S. the output elasticity of government spending is zero. However, it can also not be ruled out that government spending is mildly cyclical. Therefore, we implement a prior on the output elasticity of government spending centered at zero. We set the standard deviation to 0.1 to allow for some uncertainty. The middle panel of Figure 7 shows that for this prior distribution the median spending multiplier is 1 dollar on impact and stays above 1 dollar over the entire horizon.
The bottom panel of Figure 7 shows that the probability of the tax multiplier being larger than the spending multiplier remains below 0.5 at all horizons. Hence, for these prior distributions on the output elasticity of fiscal variables, there is no evidence to support the view that tax policy provides a larger stimulus to output than spending policy.
The analytical results presented in the previous sections can help shed light on two ongoing debates in the literature on the effects of spending shocks: the response of private consumption and the role of fiscal foresight.
Standard RBC and New Keynesian models predict that, due to a negative wealth effect, consumption falls after a spending shock (Linnemann & Schabert, 2003; Baxter & King, 1993). Yet, SVARs model consistently find that consumption increases.^{31} Assuming that government spending is acyclical ( , the impact consumption multiplier is:
What is the effect of fiscal foresight on the estimated response of output and consumption to a spending shock? Similarly to equation (12), we can write the response of output to a spending shock as:
This prediction is satisfied for our VAR model. The impact spending multiplier estimated using a 3equation VAR in output, tax revenue, and government spending is 1.09 dollars, instead of 1.25 dollars in the full 8equation VAR model including additional variables likely to capture foresight. Similarly, the impact response of consumption in a 4equation VAR is dollars, compared to 0.09 dollars in the 8equation VAR model.^{32}
We provide comprehensive evidence on fiscal multipliers for the U.S. based on data for the period 19472006. Our novel analytical framework allows us to reveal the core properties of the alternative identification schemes used in the fiscal VAR literature. We show that differences in estimates of fiscal multipliers documented in the literature by (Blanchard & Perotti, 2002), (Mountford & Uhlig, 2009) and (Romer & Romer, 2010) are due mostly to different restrictions on the output elasticities of tax revenue and government spending. We use extramodel information to narrow the set of empirically plausible values of the elasticities, which in turn allows us to sharpen the inference on fiscal multipliers. Our results suggest that spending multipliers tend to be larger than tax multipliers.
The analytical framework developed in this paper can be applied to study identification problems in a large class of timeseries models, including VARs with timevarying reducedform coefficients, regimeswitching VARs and factor models. The use of such models can help unveil whether the transmission of fiscal policy shocks has changed over time^{33} or depends on the state of the economy (Auerbach & Gorodnichenko, forthcoming). Finally, the analytical framework developed here can be easily adapted to the study of other topics in empirical macroeconomics, such as the identification of monetary policy shocks.
We estimate the VAR model using Bayesian techniques. In particular, we impose prior distributions on the reducedform coefficients and following the methodology discussed in (Del Negro & Schorfheide, 2011). We implement this prior, which is a variant of the wellknown Minnesota prior, through dummy observations. The hyperparameters are chosen to impose a fairly loose prior, so that the comparison to the existing literature does not depend on the choice of the prior. Following the notation in (Del Negro & Schorfheide, 2011), the hyperparameters are:
All components of national income are taken from the NIPA Tables published by the U.S. Bureau of Economic Analysis. They are in real per capita terms and are transformed from the nominal values by dividing them by the GDP deflator (NIPA Table 1.1.4, Line 1) and the population measure (NIPA Table 2.1, Line 38). The remaining series are downloaded from Federal Reserve Bank of St. Louis FRED database. The table and row numbers given below refer to the organization of the data by the BEA. Data are at a quarterly frequency from 1947I to 2006IV. We use the logarithm of all national income variables.
Following (Ramey, 2011) we run the following Granger Causality tests:
We run the following regression:
where tax and spending shocks are identified by the B&P approach at the OLS estimates. The following table reports the Ftest for the null hypothesis .

The relation between reducedform residuals and structural shocks presented in (2) can also be written as:
Premultiplying equation (1) by matrix gives the structural form of the VAR model:
Finally, the relation between structural coefficients and reducedform coefficients is given by:
which describes a system of independent nonlinear equations.
In the bivariate model, we solve a system of three equations (as many as the distinct elements of ) in three unknowns :
The solution of this system is:
Substituting the analytical solution for in matrix ,we obtain the following analytical expression for the impact impulse responses:
The assumption that is positive definite ensures that the denominator of all impact responses is strictly larger than zero. This guarantees that impulse response functions are defined for all output elasticities .^{34}
In the bivariate model, we can rewrite the element i of the impulse vector associated with the policy shock as:
for .
Let us introduce a third variable into the system:
In a threeequation VAR model, we need three restrictions to identify the system (B.1). Without loss of generality, we assume that restrictions are imposed on and . In the interest of space, we do not report the solution to the threeequation system.^{35} The element of the impulse vector associated with the policy shock
can be written as:
If , the solution for the impulse vector (B.3) collapses to expression (B.2), the solution for the impulse vector found in the bivariate model. This result generalizes to VAR models of dimension under the following assumptions:
Under these assumptions, expression (B.2) describes the impact response of variable to a policy shock, for . The intuition is simple: what matters for the identification of the policy shock is the output elasticity of the policy variable and the reducedform residual How the remaining structural shocks generate fluctuations in is irrelevant for the identification of This is the same argument that explains why in a Cholesky decomposition, impulse responses associated to the shock ordered first do not depend on the identification of the remaining shocks, ie on the ordering of the remaining variables.
To derive analytical expressions for impulse responses at longer horizon we start from the Moving Average (MA) representation of the SVAR model:
where . The elements of matrices 's are functions of the autoregressive coefficients contained in the lag polynomial . ^{36} The matrix contains impulse responses quarters after the shock, which are linear combination of impact responses B.2. We do not inspect the analytical expression for impulse responses at horizon . The above assumptions ensures that, for given and , impulse responses to a policy shock only depend on the output elasticity of the policy variable at any horizon. Hence we can plot impulse responses as nonlinear functions of the elasticities for any value of the elasticities, as we did for the impact responses.^{37} For instance, Figure 8 plots the tax and spending multiplier 4, 8, and 12 quarters after the shock.
The output response to a policy shock (7) has the following properties:
Proof of Proposition 1. First, we prove existence of a global minimum and maximum of . Note that belongs to the family of rational functions, which are continuous and differentiable. So in order to find the global extrema of we have to investigate its first and second derivatives. With some abuse of notation, denote by . Equating the first derivative to zero we obtain two points that satisfy the necessary conditions for an extremum of :
Finally, the global minimum and maximum of
are:
This part of the Appendix provides formal derivations of the results for the pure sign restriction approach and the penalty function approach to sign restrictions cited in the main text of the paper. We start with the analytical solution to the Cholesky decomposition of the covariance matrix, which will prove useful in the derivation of the analytical results for the sign restriction approach.
We assume that the prediction error covariance matrix is a real symmetric positive definite matrix. The positive definiteness of implies for all and or, equivalently, for . This assumption guarantees that the Cholesky factorization of the covariance matrix exists, i.e. a unique lower triangular matrix with positive elements on the principal diagonal exists such that ((Golub & van Loan, 1996), Theorem 4.2.5, p. 143). We write the individual elements of the covariance matrix in terms of standard deviations and correlation coefficients, which is useful for the presentation of the analytical results in the paper:
The key to our analytical approach is the existence of an analytical expression for the lowertriangular Cholesky decomposition of the covariance matrix . The Cholesky decomposition has a recursive structure, which greatly simplifies the derivation of the individual elements of the Cholesky factor matrix. Denoting the Cholesky factor of by the algorithm for the computation of its individual elements can be expressed as follows:
The computation starts from the upper left corner of and proceeds to calculate the matrix either row by row (CholeskyBanachiewicz algorithm) or column by column (CholeskyCrout algorithm). The lowertriangular Cholesky decomposition of the covariance matrix yields the following expression for the Cholesky factor , where to save space we report only the elements of the first and second columns expressed as and , respectively:
We start with the derivation of Equation (8) in the main text, which gives the factor matrix for the pure sign restriction approach. Recall that the system can be written in compact form. For the bivariate system the Cholesky factor, with output ordered first and the policy variable ordered second, can be expressed as follows:
The orthogonal matrix can be expressed as follows in the bivariate case:
Using basic trigonometric identities this expression can be further simplified to yield Equation (8) in the main text^{39}:
We next characterize the set of pure sign restriction solutions under the baseline assumptions given in the main text (for the sake of brevity we concentrate on the more interesting case of the identification of tax shocks):
Proof. Abstracting from the sign restrictions, note that the interval
describes the set of all possible rotation angles, with angles measured in radians. The angle
corresponds to a clockwise rotation by
, while the angle
corresponds to a counterclockwise rotation by
. In other words, the interval describes the unit circle. Within these limits, the sign restrictions further restrict the set of admissible rotation angles:
First, note that the inclusion of reflections means that the sign restriction on can be disregarded as it will be satisfied, after signflipping where needed for all . Disregarding the sign restriction on implies that the set of pure sign restriction solutions grows to .
Second, consider the first shock for which the inclusion of reflections leads to the following conditions:
Note, however, that these conditions do not add solutions to the sign restriction set, once the signflipping is taken into consideration. The reason is that the signflipping can be implemented through a phase shift by (. Shifting the subsets satisfying by (adding to gives the subset or by (substracting from gives the subset , respectively, and taking the union of the two resulting subsets gives the same set as the one satisfying . The same holds for the signflipping of the solutions satisfying : shifting the subsets satisfying by (adding to gives the subset or by (substracting from gives the subset , respectively, and taking the union of the two resulting subsets gives the same set as the one satisfying .
Taken together the set of all pure sign restriction solutions is given by . This set is nonempty for all admissible values of the error correlation coefficient as for all . This completes the proof of Proposition 2.
It is worthwhile to quickly review the implications of Proposition 2 for our application. First, for the sign restrictions given by (C.4) the set will be the larger the smaller the value of , i.e. the larger the value of the correlation coefficient between reducedform output and policyvariable disturbances.^{40} In our application  but also in the VARs estimated by B&P and M&U  the correlation coefficient is positive and large: for our VAR  evaluated at the OLS estimate  (and ). This implies that the set is very large in terms of the range of admissible . However, regardless of the size of , it is always true that all output elasticities of taxes between zero and plus infinity will satisfy the sign restrictions given by (C.4). To see this recall the definition of given by Equation (9) in the main text:
For the lower bound of (
) the elasticity is zero because the numerator
while the denominator
for all
. For the upper bound of (
) the elasticity goes to plus infinity because the denominator
while the numerator
for all
. In addition,
is strictly increasing in over . To see this note that the expression for the elasticity can be transformed as follows (yielding the expression shown in Equation (9) in the main text):
To show that is strictly increasing in over it is sufficient to show that the first derivative of with respect to is positive for all , because in any case. Now, the first derivative of with respect to is , which is positive for all .
Note also that the expression for the impact tax cut multiplier for the sign restriction approach can be derived in analogy to Equation (6) in Section of the main text:
The main properties of the impact tax cut multiplier for the pure sign restriction approach and for the sign restrictions given by (C.4) can be summarized as follows:
We next turn to the set of pure sign restriction solutions under the alternative assumptions given in the main text (again we concentrate on the more interesting case of the identification of tax shocks) under which we restrict the object of interest (the output response to a tax shock) to be negative in order to rule out negative impact tax cut multipliers:
Proof. Considering rotations first, note that the sign restrictions restrict the set of admissible angles to:
The additional consideration of reflections does not affect this subset. To see this consider the following two subcases:
First, the subcase in which and but and . The individual restrictions are satisfied for the following angles:
Second, consider the subcase in which and but and . These restrictions individually are satisfied for the following angles:
Taken together the set of all pure sign restriction solutions is given by . For , i.e. for , the set is given by ; this set is nonempty for all admissible values of the error correlation coefficient as for all . For , i.e. for , the set is given by ; this set is also nonempty for all admissible values of the error correlation coefficient as for all . This completes the proof of Proposition 3.
It is worthwhile to quickly review the implications of Proposition 3 for our application. First, for the sign restrictions given by (C.5) the set is largest for , i.e. for , and shrinks as the (absolute) value of the correlation coefficient between output and policy disturbances increases. In our application , i.e. the set of pure sign restriction solutions satisfying the restrictions (C.5) is given by . The lower bound of this interval corresponds to the Cholesky factorization with output ordered first, for which as shown above the impact tax cut multiplier is zero and the output elasticity of taxes is equal to 1.5. At the upper bound of this interval the output elasticity of taxes is
As an alternative to the pure sign restriction approach, the literature has used the penalty function approach to select one particular solution out of the set of pure sign restriction solutions (see e.g. (Faust, 1998), (Uhlig, 2005) and (Mountford & Uhlig, 2009)). More generally, the penalty function approach is used to numerically solve constrained nonlinear optimization problems for which closedform analytical solutions may be hard to obtain or may not exist at all (see (Judd, 1998), pp. 12325). The idea is to replace the constraints  here: the sign restrictions  with a continuous penalty function that permits but (heavily) penalizes choices that violate the constraints. As a result, the constrained problem is replaced with an unconstrained one.
We show here that it is possible to analytically solve the nonlinear optimization problem underlying the signrestriction penalty function approach for the bivariate case. In particular, we show that the standard penalty function used in the literature implies that the optimum is an element of the subset ruling out negative impact tax cut multipliers that obtains when adding a sign restriction on the object of interest (the output response to a tax shock) as in (C.5). This shows what should be intuitively clear: the standard penalty function is an identifying assumption.
The standard penalty function has the sum of some or all impact impulse responses to a given shock as its arguments, where in general the impulse response of variable to shock is scaled by the standard deviation of this variable's onestepahead prediction error, .^{42} In our notation, the individual arguments of the penalty function can thus be written as . This expression is equivalent to the square root of the fraction of variable 's onestepahead forecast error variance explained by shock .
The literature, in general, proceeds as follows: first, numerically minimize a penalty function having some or all impulse responses to the first shock as its arguments. Second, if a second shock has to be identified, minimize a penalty function having some impulse responses to the second shock as arguments, imposing the further constraint that the second shock is orthogonal to the first shock. The identification of further shocks proceeds analogously.
In the bivariate case, the problem simplifies as there is a maximum of two shocks. There can be only one penalty function as the second shock imperatively has to explain all fluctuation not explained by the first shock. In line with M&U we assume that the penalty function has the impulse responses to the first shock of the signrestricted variables as its arguments. We here again focus on the more interesting case of the identification of the tax shock. In this case the penalty function has the impulse responses of output and taxes to the business cycle shock as its arguments. This yields the following objective function (10) given in the main text:
Expressing the problem in terms of trigonometric functions greatly facilitates the analytical solution to this optimization problem. First, the orthonormality restriction on the rotation matrix is automatically satisfied because holds for any . Second, since we can analytically characterize the set of all puresign restriction solutions, (see Proposition 2), we know that the domain of is a closed and bounded interval on . Third, as we show in the proof to the following proposition, since is a continuous and concave function on , we have by Weierstrass's theorem that achieves its global maximum on its domain. Furthermore, this global maximum is unique. In sum, we neither have to explicitly account for the equality constraint implied by the orthonormality assumption, nor for the inequality constraints associated with the sign restrictions. Both sets of constraints will be automatically satisfied.
The following proposition gives the analytical solution to the optimization problem underlying the penalty function approach:
Proof. First, we prove that the maximum is unique and global. Since we can analytically characterize the set of all puresign restriction solutions, (see Proposition 2), we know that the domain of is a closed and bounded interval on . Moreover, is a continuous and concave function on . The cosine function is continuous for all real numbers. To show that is concave on it is sufficient to show that and are both concave on (see (Simon & Blume, 1994), Theorem 21.8, p. 519). Using the secondderivative test it is straightforward to show that is concave on the interval and is concave on the interval . The set of all pure signrestriction solutions is a subset of both intervals for all . Then, by Weierstrass's theorem achieves its global maximum on its domain. To establish uniqueness we need to show that is strictly concave on its entire domain. Note, first, that and have continuous derivatives with respect to of every order, implying that is also infinitely continuously differentiable. Concavity implies that on and on for all . Furthermore, for identical angle of rotation if and only if (perfect negative correlation), which is ruled out by assumption. Thus, is strictly concave on for all , ensuring uniqueness of its global maximum.
Second, we prove that is the maximizer of on . The firstorder condition for a maximum is
0 for all 
Third, we prove that is the midpoint of both the set and the subset . First, recall that the set of all solutions satisfying the sign restrictions (C.4) is given by . The midpoint of this set is equal to half the sum of the lower and upper bound of , i.e. . Second, recall that the subset of all solutions satisfying the more restrictive sign restrictions (C.5) is given by . For , i.e. for nonnegative error correlation, this subset is given by . It is easy to see that the midpoint of this interval is . For , i.e. for nonpositive error correlation, this subset is given by , i.e. it is identical to , for which we already showed that its midpoint is equal to .
Finally, we prove that maximizes the fraction of onestep ahead forecast error covariance explained by the first shock if the error correlation is positive and minimizes it if the error correlation is negative. The fraction of covariance explained by the first shock is given by
It is worthwhile to quickly review the implications of Proposition 4 for our application.
At the penalty function solution we have and , which gives
We close the bivariate section with a summary of results for the spending model, which in the bivariate context is trivial. Recall that M&U do not sign restrict the response of government spending to a business cycle shock, which implies the following sign pattern
For these restrictions the set of pure sign restriction solutions is simply the entire interval of admissible rotation angles .
The objective function for the spending model is
It is easy to see that the solution is , i.e. the penalty function solution is nothing else than the Cholesky factorization with output ordered first. For this solution the impact spending multiplier is zero. The output elasticity of government spending implied by the penalty function solution is given by:
In our application, as is true also for other VAR studies in this literature, the output and government spending residuals are strongly positively correlated ( . For the penalty function solution  and the Cholesky factorization with output ordered first  100% of the correlation between one stepahead output and government spending errors is explained by the business cycle shock, implying strongly procyclical government spending ( ).^{43}
We extend the analysis to a threevariable system with output, taxes and a third variable. The third variable, denoted , will be private consumption, denoted , private investment, denoted , or government spending, , depending on the object of interest. We restrict attention to the set of pure sign restriction solutions derived for the bivariate tax model. In particular, we answer the following questions:
We should clarify that in this section we look at a subspace of the set of all solutions in the multivariate context. However, as we will see this is a very interesting subset, clarifying that the wide range of possible results that we derived for the bivariate context persists even if we don't look at the entire space of solutions in the multivariate context.
In the trivariate context, the orthogonal matrix can be written as the product of three Givens matrices , and , each rotating a different pair of columns of the matrix to be transformed:
In the following we focus attention on those sign restriction solutions that obtain when setting , in which case and .
We consider the following sign restrictions on the factor matrix :
These restrictions are comparable to the restrictions imposed by M&U: in the tax model we require the responses of output, taxes and the third variable (either or ) to be positive, whereas for the tax shock (second shock) we require the response of taxes to be positive and orthogonality to the business cycle shock.
We extend the bivariate system given by Equation (8) to a trivariate system, where denotes the residual of the third variable:
We are going to assume in the following that the crosscorrelations between the three residuals are all positive, i.e. . This assumption facilitates the exposition of the results but is not very restrictive as it is in line with the evidence for the fiscal VAR models used in the literature. In these models all candidate third variables (private consumption, private investment and government spending) are robustly positively correlated with output and taxes. It can easily be shown that for this assumption all elements of the Cholesky factor except for are nonnegative (see Equation C.3) and the sign of is equal to the sign of . Depending on the sign of the set of pure sign restriction solutions becomes
We can check the implications of this condition for our empirical application.
Consider first the case in which private consumption is the third variable. In this case,  with the covariance matrix evaluated at the OLS estimate  the element of the Cholesky factor is negative, i.e. the first case in (C.7) applies. This implies that the additional restriction on the sign of the consumption response shrinks the set of pure sign restriction solutions. How large is this effect? In our empirical application the set still covers all empirically plausible models: at the upper bound of the sign restriction set ( ) the output elasticity of taxes is equal to 73.2. In other words, the additional sign restriction on the consumption response has only a very minor effect on the set of pure sign restriction solutions derived for the bivariate model.
Consider next the case in which private investment is the fourth variable. In this case,  with the covariance matrix evaluated at the OLS estimate , the element of the Cholesky factor is positive, i.e. the third case in (C.7) applies. To know whether the additional restriction on the investment response shrinks the set we need to very whether . In our application this condition is not fulfilled, i.e. the sign restriction set is not affected at all by the additional sign restriction.
Note also that the above results do not depend on the analysis of a trivariate model. If instead we look at a fourvariable model with output, taxes, consumption and investment, the same results hold, with the sign restriction on the consumption response being the binding restriction in our empirical application. This is because the contemporaneous responses of consumption and investment to a business cycle shock do not depend on which one is ordered third or fourth in the system
Finally, we can ask whether the penalty function solution derived for the bivariate model, satisfies the additional sign restriction on the third variable:
Note that we have by assumption and and for all . If then the sign restriction solution for the bivariate model automatically satisfies the sign restriction on the third variable. If (which holds for ) the penalty function solution needs to satisfy the condition that
This condition is satisfied under the assumption of positive crosscorrelation between the reducedform errors. As we have argued above this assumption is empirically plausible. For all VAR models we are aware of output, taxes, private consumption and investment are robustly positively crosscorrelated.
We last turn to the triviate model with government spending, for which  in line with M&U  we do not sign restrict the response of government spending to a business cycle shock which gives the following sign restrictions on the factor matrix :
In this case all pure sign restrictions solutions derived for the bivariate model remain sign restrictions in the trivariate model. The sign restriction on the spending shock is automatically satisfied because postmultiplying the Cholesky factor by leaves the third column of unchanged and for positive definite . This latter property has an important implication: for all , i.e. the output response to a government spending shock is zero. Of course, this result would not hold if we broadened the analysis to include rotations beyond the outputtax subspace.
But this result is important because it also holds for the M&U penalty function if  in accordance with the assumptions made by M&U  only those responses to a business cycle shock that are sign restricted enter as arguments in the objective function to be maximized. Under this assumption the objective function to be maximized with respect to is still the one given by (10) and the maximum still obtains for .^{44}
The output elasticity of government spending implied by this solution is
The output elasticity of government spending, evaluated at , will be larger than for and smaller than for , recalling that for . In our application  evaluated at the OLS estimate  but the deviation from is small. The output elasticity for the penalty function solution is 0.36, compared to 0.38 for the Cholesky decomposition. But most importantly, the penalty function solution for standard sign restrictions implies a zero impact spending multiplier.