ASSESSING SIGN RESTRICTIONS

Transcription

1 ASSESSING SIGN RESTRICTIONS Matthias Paustian September 6, revised December 6 Bowling Green State University Abstract This paper assesses sign restrictions via a controlled experiment. A researcher estimates a VAR on an innite amount of data generated by a DSGE model. He or she then imposes sign restrictions on impulse responses to identify a structural shock while being agnostic about the response of a key variable of interest to this shock. Can such an agnostic identication procedure pin down the correct sign of this unconstrained response? The DSGE models of Erceg, Henderson, and Levin (), as well as of Smets and Wouters (3) are used as data generating processes. Two conditions must be met for the method to unambiguously deliver the correct sign of the unconstrained impulse response. First, a sufciently large number of restrictions must be imposed more than what is typically imposed in applied work. Second, the variance of the shock under study must be sufciently large larger than the values provided by Bayesian estimations of the DSGE models. Hence, sign restrictions can be a useful tool to recover structural shocks from VAR residuals. JEL classication: E3, C3. Keywords: Sign restrictions, structural VAR, monetary shocks, technology shocks. I would like to thank Fabio Canova, Charles Carlstrom, Lawrence Christiano, Martin Eichenbaum, Tim Fuerst, Mathias Trabandt, Harald Uhlig, as well as seminar participants at the Federal Reserve Bank of Cleveland and conference participants at the 7 Midwest Macro Meetings for comments and suggestions. The Editor and two anonymous referees have provided comments that greatly improved the paper. All errors are my own. This work was supported in part by a grant of computing time from the Ohio Supercomputer Center. Department of Economics, 3 Business Admin. Bldg, Bowling Green State University, Bowling Green, OH Phone: (419) , Fax: (419) , paustim@bgsu.edu.

2 1 Introduction Identifying restrictions are necessary in order to give a meaningful interpretation to residuals in vector-autoregressions (VARs). Without such restrictions impulse response functions typically do not trace out the effects of exogenous structural disturbances, such as government spending, technology, or monetary policy shocks. Instead, they typically pick up the effects of a linear combination of these structural shocks. It has recently become popular to identify a structural shock by requiring impulse responses to have certain signs across observable variables. 1 Such sign restrictions are attractive for several reasons. First, they arise naturally from dynamic stochastic general equilibrium (DSGE) models and can thus be rmly grounded in economic theory. In contrast, zero restrictions are seen by some researchers as inconsistent with a large class of general equilibrium models, see Canova and de Nicolo (, p. 1133). Second, sign restrictions can explicitly impose conventional views that are often only implicitly used when identifying VARs via zero restrictions, see Uhlig (5b). Third, they can be robust in the sense that they hold across several structural models or parameterizations of the same model. However, these advantages come at a cost: the structural shocks are not exactly identied. There are multiple matrices dening the linear mapping from orthogonal structural shocks to VAR residuals. All of these matrices satisfy the sign restrictions and imply the same reduced form covariance matrix of VAR residuals. In other words, they are observationally equivalent and equally consistent with theory. Should researchers be concerned about this multiplicity? This paper assesses sign restrictions via the following controlled experiment. A researcher observes an innite amount of data on several macro variables. The researcher is unaware of the structural DSGE model that generates the data and estimates a VAR of appropriate lag-length. He or she identies impulse responses to the structural shock of interest by imposing sign restrictions consistent with the underlying DSGE model. A natural but somewhat trivial question is whether the researcher recovers the true impulse response. Since sign restrictions do not achieve exact identication of the structural shock, the researcher will identify a set of impulse responses containing the true one. However, the following question is meaningful: If the researcher identies the shock by imposing some sign restrictions while leaving the response of a key variable of interest unconstrained, will the identied impulse responses in this set have a unique sign? This question is important as it relates to the ability of the sign restriction 1 See the recent papers by Faust (1998), Canova and de Nicolo (), Peersman (5), Pappa (5), Uhlig (5b), Farant and Peersman (6), Dedola and Neri (6), or Braun and Shioji (6) for an application of the sign restriction method.

3 method to discriminate among alternative economic theories. Competing economic models may predict different signs of certain impulse responses. For instance, real business cycle models often predict that hours worked rise in response to technology shocks, while sticky price models often predict that hours fall. Can sign restrictions be used to discriminate between such alternative theories? Clearly, a necessary condition is that the imposed restrictions uniquely identify the correct sign of the unconstrained response. This paper examines this necessary condition and focuses on two questions. What are the effects of monetary policy on output? How do hours worked respond to technology shocks? To this end the VAR representation of two DSGE models is derived: of the Erceg, Henderson, and Levin () model, as well as of the Smets and Wouters (3) model. Both models are calibrated using Bayesian estimation results, thus reasonable values for deep parameters as well as shock processes are used. The sign restriction method is then applied to the population VAR implied by the DSGE model. Hence, sampling uncertainty is ruled out in order to focus on the problem of non-exact identication. The overall result from this analysis is that sign restrictions can uniquely pin down the sign of unconstrained impulse responses as long as the imposed restrictions are sufciently numerous and the variance of the shock under inspection is sufciently large. Hence, sign restrictions can be a useful tool for identifying structural shocks. However, the variance of the shock under inspection must be larger than the values that are obtained from Bayesian estimations of the underlying DSGE models. In addition, the restrictions must be more numerous than what is typically imposed in the empirical literature. Imposing only a small number of sign restrictions in response to monetary policy shocks in the Erceg, Henderson, and Levin () model does not pin down the response of output to a contractionary policy shock. Output may fall or rise, and almost any contribution of monetary shocks to the variance decomposition of output is consistent with these restrictions. This result is independent of the number of periods for which the sign restrictions are imposed. It remains unchanged if the variance of the monetary shock is increased to tenfold the value estimated in Rabanal and Rubio-Ramirez (5). This model based nding provides an explanation for the similar empirical result in Uhlig (5b). Uhlig uses an agnostic identication procedure based on a small number of sign restrictions. He also nds that output may rise or fall in response to contractionary monetary shocks. Apparently, this result is driven by the method. The imposed restrictions are not informative enough to pin down the sign of the output response. However, when the response of all variables in the VAR to all shocks is constrained, except for the response of I am grateful to an anonymous referee for suggesting this interpretation of the paper.

4 output to monetary shocks, then the restrictions uniquely recover the correct sign of the output response. For this strict identication scheme to work, the variance of the monetary policy shock must be roughly ve times larger than its estimated value. The ndings are similar when analyzing the response of employment to technology shocks in the Smets and Wouters (3) model. Minimal identication schemes that only restrict the response of the real wage, consumption, and investment to technology shocks do not pin down whether employment rises or falls in response to this shock. This result holds for various reasonable values of the variance of this shock, and even when the restrictions are imposed for 3 quarters following the shock. It again becomes necessary to impose all sign restrictions except for the employment response to technology shocks in order to uniquely recover the correct sign of the impulse response under study. The variance of the technology shock must exceed a threshold value for this strict identication scheme to work. This threshold value is larger than the estimated variance of technology shocks in Smets and Wouters (3), but still well within a reasonable range. In fact, the contribution of technology shocks to the forecast error variance ranges from 5% to 4% at the threshold. This is good news as it shows that a sufciently large number of sign restrictions can be informative about technology shocks without requiring these shocks to be the overwhelming source of macroeconomic uctuations. The nal section of this paper briey analyzes small sample properties of the sign restriction method. A Monte Carlo experiment based on the Erceg, Henderson, and Levin () model suggests that the condence intervals for impulse responses identied via sign restrictions have reasonable properties. The mean bias is small. Actual coverage differs from nominal coverage, but the difference is mostly modest. The related literature is scarce, as most researchers are concerned with applying sign restrictions rather than testing their performance. Canova and Pina (5) compare contemporaneous and sign restrictions when the data generating process is a DSGE model. Fry and Pagan (5) highlight some conceptual problems resulting from the multiplicity of impulse vectors, but their analysis does not work within the framework of DSGE models. A conference discussion by Wouters (5) briey points out difculties in recovering structural shocks when applying sign restrictions to small samples of articial data. In contrast, this paper abstracts from small sample problems and focuses on non-exact identication. Finally, Uhlig (5a) looks at identication via sign restrictions in a standard demand and supply model.

5 The evaluation This section evaluates sign restrictions based on two log-linearized DSGE models. The methodology for evaluating the sign restriction approach is described rst..1 Methodology Recent work by Fernandez-Villaverde, Rubio-Ramirez, Sargent, and Watson (7), Ravenna (7), as well as Fernandez-Villaverde, Rubio-Ramirez, and Sargent (5), spells out the relationship between the solution to a DSGE model and the VAR representation of a vector y t of observables from the DSGE model. Two cases are of interest for most applied work. In the rst case, y t includes all endogenous state variables of the DSGE model. Under an invertibility condition, y t then possesses a nite order VAR representation, typically a VAR(). In the second case, some endogenous state variables are not included in y t. Under invertibility and stability conditions provided in Fernandez-Villaverde, Rubio-Ramirez, and Sargent (5), y t then possesses a VAR representation of innite or nite order. Ravenna (7) identies conditions for the existence of a nite order VAR. I only describe the methodology for the case of an innite order VAR. The procedure is essentially the same for the nite order case. Fernandez-Villaverde, Rubio-Ramirez, and Sargent (5) write the solution to the DSGE model in the form of a state space representation x t+1 = Ax t + Bw t, (1) y t = Cx t + Dw t. () x t is the vector of endogenous and exogenous state variables, y t is the vector of observable variables, and w t is a vector of white noise structural shocks. The matrices A, B, C, and D are nonlinear functions of the underlying structural parameters of the model. Throughout this paper, only cases where D is square and invertible are considered. Fernandez-Villaverde, Rubio-Ramirez, and Sargent (5) have shown that the DSGE model given by (1) and () implies a VAR( ) for the observable variables y t, if the eigenvalues of the square matrix Ω A BD 1 C are all inside the unit circle: y t = C Ω j BD 1 y t j 1 + Dw t. (3) j= Since the matrix D typically has nonzero elements everywhere, the innovations into the VAR are linear combinations of all structural shocks w t. Therefore, an

6 identication problem arises. Let the covariance matrix of the structural shocks be Σ w. The covariance matrix of the reduced form innovations v t Dw t in (3) is then given by Σ v = DΣ w D. This paper studies the following problem. A researcher has an innite amount of data on observables collected in the vector y t. All VAR coefcient matrices are therefore estimated without error. This allows the researcher to recover the reduced form innovations v t. The only unknown object is the matrix D, which relates the innovations v t to the structural shocks w t, whose effects are to be traced out. For this identication problem, the researcher uses a perfect estimate of the true covariance matrix Σ v as well as sign restrictions derived from economic theory. Note that this evaluation methodology is set up to isolate the issue of non-exact identication from small sample estimation problems by working entirely within population. For a given economic model, I start by computing the eigenvalue-eigenvector decomposition of Σ v. Next, the matrix M = Qλ 1 is formed, with Q being the matrix of eigenvectors corresponding to the eigenvalues that are collected on the main diagonal of the matrix λ. M has the property that MM = Σ v. Hence, the candidate matrix M is one way to back out uncorrelated structural shocks w t from the reduced form innovations v t. However, any other matrix M satisfying M M = Σ v would be an equally admissible decomposition that implies different impulse responses to the structural shocks. Finally, the impulse responses following a unit shock to an element of w t in the following system are computed: y t = Ξ j y t j 1 + MRw t. (4) j= I impose the true VAR coefcient matrices, i.e. Ξ j CΩ j BD 1. Here R is an orthonormal matrix that perturbs the impact response to structural shocks, but preserves the same reduced form covariance matrix of the VAR residuals. To generate the matrix R, random orthonormal matrices are drawn from the uniform distribution as in Rubio-Ramirez, Waggoner, and Zha (5). If a particular matrix R produces impulse responses compatible with economic theory to be outlined below, these impulse responses are stored. Otherwise, they are discarded. See Fry and Pagan (5) and Uhlig (5b) for a more detailed exposition of the sign restriction method. Below I dene the concept of identication of the sign of unconstrained impulse responses used throughout this paper. Denition 1 (Identication of the sign of unconstrained responses). A set of sign restrictions is said to identify the sign of some unconstrained impulse response at

7 horizon k, if for all rotational matrices R consistent with the imposed sign restrictions, the sign of the unconstrained impulse response is the same in period k after the shock. The rest of this paper focuses on characterizing the conditions under which a set of sign restrictions identies the sign of unconstrained impulse responses in the sense laid out above. Note that this evaluation framework is designed to work as much in favor of the sign restriction approach as possible. A number of problems that may arise in practice are ruled out here. First, the procedure abstracts from sampling uncertainty by working entirely within the population. Second, the imposed identifying assumptions are always chosen to be fully consistent with the DSGE model. Third, the shock under consideration produces impulse responses with a unique pattern of signs across the observable variables.. A simple generic example to build intuition It is instructive to rst consider the most simple generic model to which sign restrictions can be applied. This model has two observable variables which are driven by two shocks. Since I leave the structure of the model unspecied, the focus is on the contemporaneous response to shocks. Hence, it is assumed that the vector of endogenous variables y t is driven by structural shocks w t as y t = Dw t, or more explicitly [ y1t y t ] = d 11 + d 1 + d 1 d + [ w1t w t ] [ σ, with: E (w t w t) = Σ w = 1 σ ]. (5) Here, the subscripts + or denote one possible pattern of signs of the elements d i,j such that the structural shocks are potentially identiable. Clearly, if both shocks either raise or lower both variables, sign restrictions would be of little use for identication. A researcher observes an innite amount of data on y t and perfectly estimates the covariance matrix Σ y DΣ w D. The question at hand is whether imposing sign restrictions on three out of the four elements in D pins down the sign of the fourth unconstrained element. 3 To provide an example of such a kind of analysis, one can think of the variables as real wage and hours worked, and of the shocks as technology and government spending shocks. Hence, the signs of the elements in 3 The true model is such that opposite cross-correlations exists. Since the researcher leaves one sign unrestricted, in his view it possible that both shocks induce the same pattern of crosscorrelations. What I am asking is whether such a researcher would detect that opposite crosscorrelations exist and hence pin down the sign of the unconstrained response.

8 D can be taken to imply that technology shocks raise both real wages and hours. On the other hand, government spending shocks lower the real wage and increase hours worked. Such a pattern can be derived from DSGE models. A researcher who is agnostic about the response of hours to technology shocks imposes only three out of the four signs and leaves the central question under study unconstrained: the sign of the response of hours to technology shocks. This is similar in spirit to the agnostic identication of monetary shocks in Uhlig (5b). The following proposition establishes conditions under which imposing three sign restriction allows to identify the sign of the fourth unconstrained response at horizon k =. Proposition 1. Assume two variables are driven by two shocks such that one shock implies a positive cross-correlation of the variables' contemporaneous responses and the other a negative cross-correlation. The researcher imposes three out of four sign restrictions in accordance with the true impact matrix D and leaves the response of one variable to shock unconstrained. Dene σ σ/σ 1 as the variance of shock relative to shock 1. Then there exists a critical value σ = d 11 d 1 /( d 1 d ) such that 1. for σ > σ the sign of the fourth unconstrained impact response is uniquely determined by the three sign restrictions,. for σ < σ the sign of the fourth unconstrained impact response is not pinned down without further assumption on D. Proof of Proposition 1. In addition to three sign constraints, any candidate impact matrix M (whose elements are denoted by m i,j ) must satisfy MM = DΣ w D. These equations impose three constraints on M. Only one constraint restricts the sign of the elements of M. Without loss of generality, take the a priori unconstrained parameter to be m. The relevant constraint determining the sign of m reads m = α/m 1 m 11 m 1 /m 1, with α d 11 d 1 σ1 + d 1 d σ. When the true element d is positive, m 11 m 1 /m 1 must be positive since the three sign constraints must induce opposite cross-correlations. Hence, to obtain the correct sign α/m 1 must be positive. It can be easily shown that this requires σ > d 11 d 1 /( d 1 d ). The same condition is obtained when the true element d is negative. When σ < σ, m is given by the sum of two elements with opposite signs. Therefore, without further constraints on the magnitudes in D, the sign of m is not identied. A numerical example illustrates the above proposition. For simplicity, the impact matrix D is parameterized as d 11 = d 1 = d = 1 and d 1 = 1. Qualitatively, the results below do not depend on this particular calibration. With that

9 parameterization, a higher standard deviation of shock relative to shock 1 implies that the second shock drives the majority of the impact uctuations of each variable. In this case, it is convenient to use the Givens rotation matrix R(θ) to explore the space of all theory-consistent impact matrices, see Fry and Pagan (5). 4 Figure 1 displays the maximum and the minimum values of the unconstrained (,) element of M = DΣw 1/ R(θ) across those rotations consistent with the imposed signs for the remaining three constrained elements. Hence, the contemporaneous impact of a one standard deviation shock is identied. This minimum and maximum response is expressed as percentage deviations from the true (,) element of DΣ 1/ w and plotted against the relative standard deviation of the second shock. percentage deviation from truth relative standard deviation of shock Figure 1: Solid lines describe the minimum and maximum response of variable to a one standard deviation impulse to shock. Dashed horizontal lines at and at -1 are for reference only. Percentage deviation from the truth is dened as (identied response - true response)/(true response) 1. Figure 1 contains three interesting and intuitive results. First, as the variance of shock relative to shock 1 increases, the sign restrictions are increasingly precise in identifying the true unconstrained response. The interval of minimum and maximum percentage deviation from the true (,) element narrows towards zero. With increasing relative variance, more of the uctuations are driven by the shock under inspection. Consequently, the method has less difculties in pinning down the unconstrained response. 4 Note that vec(r(θ)) = [cos(θ), sin(θ), sin(θ), cos(θ)] is indexed by only one angle θ [, π]. Hence, one can efciently explore this space by partitioning [, π] in N points. N=1 is used in the analysis below. In addition, one needs to consider all possible decompositions where one or more columns have a switched sign, see Lütkepohl (5, p. 363).

10 Second, the three sign restrictions are able to uniquely pin down the sign of the contemporaneous response of variable to the second shock as long as the relative variance is bigger than σ = 1. To see this from Figure 1, recall that deviations are expressed as percentages. Therefore, the sign is not uniquely pinned down for negative deviations beyond -1. This is why a reference line is drawn at that value. Why does the cutoff occur at unity? Given the calibration of D, the unconditional correlation between y 1t and y t is zero at a relative variance of unity. Hence, imposing the sign restriction that shock 1 induces a positive conditional correlation implies that shock cannot induce a positive conditional correlation as well. Only a negative correlation conditional on shock is consistent with an unconditional correlation of zero. This pins down the unconstrained sign. Qualitatively, this nding suggests that agnostic identication schemes are not guaranteed to work without further assumptions on the relative variance of shocks. Apparently, the shock whose effect is partially unconstrained must be driving uctuations sufciently strong for the method to work. Third, a wide range of both positive and negative values for the (,) element are possible when the relative variance falls below unity. In this region, the sign restrictions are quite uninformative about the true response. When the second shock has a smaller variance than the rst one, the unconditional correlation between y 1t and y t becomes positive given the calibration of D. A positive correlation conditional on the second shock is clearly compatible with the a priori restriction and the positive unconditional correlation. In addition, a negative correlation conditional on the second shocks that is small enough to leave the unconditional correlation positive is also consistent with data and theory. Hence, the sign of d is not identied. Qualitatively, the ndings outlined above are not specic to this particular calibration. One would like to obtain analytical results for identiability in the general case with more than two shocks. However, in such a setting the sign restrictions implied by MM = DΣ w D are numerous and it is intractable to obtain general conditions for identiability. However, the following numerical example, where three variables are driven by three shocks, suggests that the results above extend beyond the bivariate case. I consider a simple numerical example where the model is again given by y t = Dw t, or more explicitly y 1t y t y 3t = w 1t w t w 3t, E ( w t w t) = Σw = σ 1 σ σ 3. (6) The imposed sign restrictions are consistent with the signs of eight elements in the matrix D and agnostic with respect to the response of y 3t to w 3t. Figure

11 displays the minimum of the unconstrained (3,3) element of M = DΣ 1/ w R(θ) across those rotations consistent with the imposed signs for the remaining eight constrained elements. 5 Figure : Percentage deviation of the minimum theory-consistent response from the true response. The x and y-axes depict the standard deviation of shock 3 relative to the standard deviations of shock 1 and shock, respectively. Figure shows that there exist threshold values for the relative standard deviation of the shock under inspection such that the unconstrained sign is uniquely pinned down. This is the area where the surface exceeds the value -1. Again, there is an apparent discontinuity. To summarize, this section argues that an important determinant for the usefulness of sign restrictions is the relative variance of shocks. If the shock under study is dominant, sufciently numerous sign restrictions can pin down the sign of an impulse response that the researcher leaves unrestricted. If the shock is not sufciently strong, this is not possible. Of course, sign restrictions are typically imposed for more than just one period in empirical work. The following subsections show that the basic nding from this section carries over to the case when sign restrictions are applied for longer than just on impact. 5 For computational reasons, the Givens rotation matrix is no longer used here. Rather, for each value on the grid of relative standard deviations, random orthonormal matrices are drawn until 1 theory-consistent matrices M are found. The minimum of the theory-consistent (3,3) elements is then expressed as a percentage deviation from the true value.

12 .3 Example 1: Response of output to monetary shocks The rst model used to test the sign restriction approach is the sticky price and sticky wage model of Erceg, Henderson, and Levin (). This model is used here, since it can serve as a rough theoretical framework behind the empirical studies that investigate the effects of monetary policy on output. The equilibrium conditions of this well known model are outlined below. All variables are expressed in log deviations from the steady state. y t = E t y t+1 σ (r t π t+1 + g t+1 g t ) (7) y t = a t + (1 α)n t (8) mc t = w t + n t y t (9) mrs t = 1 σ y t + γn t g t (1) w t = w t 1 + π w t π t (11) π w t = κ w [mrs t w t ] + βe t π w t+1 (1) π t = κ p [mc t + λ t ] + βe t π t+1, λ t N(, σ λ) (13) r t = ρ r r t 1 + (1 ρ r ) [γ π π t + γ y y t ] + m t, m t N(, σ m) (14) a t = ρ a a t 1 + u t, u t N(, σ a) (15) g t = ρ g g t 1 + v t, v t N(, σ g) (16) Equation (7) is the consumption Euler equation, where y t denotes output, r t nominal interest rate, π t price ination, and g t a preference shock. The production function is given by (8) with a t being an exogenous productivity process and n t being labor. Real marginal cost mc t is dened by (9), where w t is the real wage. Equation (1) denes the marginal rate of substitution mrs t. Equation (11) is an identity linking the real wage growth to the difference wage and price ination. The wage and price Philips curves arising from Calvo nominal rigidities are depicted in (1) and (13). λ t denotes a price markup shock that can be microfounded as reecting time variations in monopoly power. Here, πt w denotes wage ination. The slope of the price Phillips curve is given by κ p (1 θ)(1 βθ) 1 α. The θ (1 α+αɛ) slope of the wage Phillips curve is given by κ w (1 θw)(1 βθw) θ w. The central bank (1+φγ) adjusts the nominal interest rate r t according to a Taylor rule as in (14). The model is calibrated using the posterior mean of the estimation by Rabanal and Rubio-Ramirez (5, p. 1157). These authors explain the joint behavior of price ination, real wages, interest rates, and real output for the United States during the sample period 196:1 to 1:4. Table 1 summarizes the calibration.

13 β σ γ θ θ w ɛ φ α ρ r γ y γ π ρ a ρ g Table 1: Calibration of Erceg, Henderson, and Levin () model. The model has four exogenous stochastic processes driven by serially and mutually uncorrelated innovations with a mean of zero. The total factor productivity shock a t and the preference shock g t each follow a univariate AR(1) process with autocorrelation coefcients ρ a and ρ g. The monetary shock z t and the markup shock λ t have zero autocorrelation. The innovations into these exogenous processes have standard deviations 6 [σ a, σ g, σ m, σ λ ] = [.388,.1188,.33,.3167]. These values are again taken from the posterior mean of the estimation in Rabanal and Rubio- Ramirez (5, p. 1157), see their Table 1. For this particular calibration, the model implies signs of the impact responses to the four structural shocks as depicted in Table. Some studies employing the sign restriction framework are only interested in identifying one particular shock, rather than identifying the full set of shocks. For instance, Uhlig (5b) identies only monetary shocks based on the response of a subset of variables included in the VAR. I follow a similar approach here. The monetary shock is identied by imposing that the signs of impulse response for the nominal interest rate, the real wage, and ination are consistent with the corresponding signs in the DSGE model for the rst k = 4 quarters. Sensitivity analysis with respect to k is performed later on. No other shock produces the same pattern of signs, see Table. As in Uhlig (5b), the response of output is not constrained a priori, thereby mirroring his agnostic identication procedure for monetary shocks. observable monetary shock technology shock taste shock markup shock r t w t π t y t Table : Signs of impact responses from the DSGE model The vector of observables includes all endogenous state variables of the model. Therefore the VAR has lag length two. The results in Ravenna (7) are used to 6 For better visibility of the impulse response functions these standard deviations have been scaled by a factor of 1. This scaling does not affect the results.

14 compute the VAR coefcient matrices from the log-linear decision rules implied by the model. I use 1, draws of orthonormal matrices. % response % response NOMINAL RATE INFLATION.1..3 % response % response REAL WAGE OUTPUT quarters after shock quarters after shock Figure 3: Impulse response to monetary shock. Solid line: true response. Solid line with dots as markers: median identied response. Grey area: range of all theory-consistent responses. Sign restriction imposed for four quarters. Figure 3 plots the true impulse responses from the DSGE model together with the range of all theory-consistent impulse responses as indicated by the grey area. This range is dened by the minimum and the maximum pointwise at each horizon. It is important to note that all theory-consistent impulse responses are equally compatible with the reduced form VAR. Inspecting Figure 3 shows that the agnostic identication procedure is unable to identify the sign of the output response. Output may rise or fall in response to the monetary shock. Based on the mean response, output rises in response to the monetary shock, whereas the true response is negative. Note further how large the range of theory-consistent output responses is relative to the magnitude of the true response. Additionally, the range of theory-consistent responses is very wide even for those variables whose response is constrained a priori according to theory. Hence, the weak identication problem is severe. Would one be able to pin down the correct sign of the output response by increasing the horizon k for which the sign restrictions must hold? To answer this question, Figure 4 displays the histogram of the impact response of output to the

15 monetary shock for k equal to, 4, 8, and 1 quarters. The three imposed restrictions are fully consistent with the underlying DSGE model for up to 1 quarters. k = k = 4 percent of draws percent of draws percent of draws impact response of output k = impact response of output percent of draws impact response of output k = impact response of output Figure 4: Histograms for the impact response of output to a monetary shock. k denotes the number of periods during which the sign restrictions are imposed. n = 1, draws are used. Dashed vertical line denotes the true impact response from the DSGE model. Low variance case. As can be seen, the identication problem persists even when the sign restrictions are imposed for up to 1 quarters following the shock. Again, a wide range of initial responses is compatible with economic theory. As k increases, the distribution becomes centered farther away from the true impact response. A similar result was found in the empirical study of Uhlig (5b, p. 394), see his Figure 4. In that study, imposing the sign restrictions for the rst 3 months instead of the baseline value of ve months also shifted the distribution of impact responses of output largely into the positive region. This analysis shows that such a nding does not imply that one can be more condent that contractionary monetary shocks raise output. Rather, such results can be an artefact of the methodology. In light of the results from the previous subsection, one may suspect that the difculty in identifying the sign of the true output response is due to the fact that monetary shocks barely matter for output in the DSGE model. After all, the contribution of monetary shocks for the forecast error variance decomposition of output at horizon two years is extremely small, roughly one percent for the estimated parameter values of the model. To check for sensitivity along this dimension, the

16 standard deviation of the monetary shock is increased by a factor of 1. All other parameter values are at their baseline values. This change ascribes a much larger role for monetary shocks. Their contribution to the forecast error variance of output is now 5 percent. Figure 5 shows the histogram of impact responses to the k = k = 4 percent of draws percent of draws impact response of output k = impact response of output k = 1 percent of draws percent of draws impact response of output impact response of output Figure 5: Histograms for the impact response of output to a monetary shock. k denotes the number of periods during which the sign restrictions are imposed. n = 1, draws are used. Dashed vertical line denotes the true impact response from the DSGE model. High variance case. monetary shock in this case. Again, output may rise or fall after a monetary shock. Hence, the identication problem persists even when monetary shocks account for a large fraction of the variance in output. In this particular model, imposing a small number of sign restrictions does not allow to pin down the sign of the output response to monetary shocks, even when monetary shocks contribute considerably to driving output. However, identifying further shocks puts additional restrictions on the covariance matrix of reduced form residuals which should help in identifying the sign of the response of output to monetary shocks. This approach is pursued in the following paragraphs. I now impose 15 out of the 16 sign restrictions implied by the DSGE model and leave only the output response to monetary shocks unconstrained. These restrictions are required to hold for k = 4 quarters, with the exception of the response of ination to the markup shock. Since this sign restriction only holds on impact in the DSGE

17 model, it is imposed only for one period to ensure consistency with the model. 7 Figure 6 displays the distribution of theory consistent impact responses of output to monetary shocks as well as of the contribution of monetary shocks to the forecast error variance decomposition of output percent of draws 6 4 percent of draws impact response of output monetary contribution in % Figure 6: Histograms for the contribution of monetary shocks to forecast error variance decomposition of output at horizon eight quarters and for the impact response of output to a monetary shock. 15 sign restrictions are imposed for four quarters. n = 1 7 draws used. Dashed line denotes the true statistic from the DSGE model. Figure 6 shows that imposing 15 out of 16 sign restrictions for the rst four quarters still does not pin down the sign of the unconstrained output response to monetary shocks. The support of the impact responses ranges from roughly -.5 to 1. Therefore, almost anything is possible on impact. Similarly, the range of all theory consistent contributions of monetary shocks to the forecast error decompositions of output spans almost the entire interval [,1]. This nding is unchanged when the horizon k over which the sign restrictions are imposed is increased. For brevity, the corresponding graphs are not reported here. It is again instructive to increase the variance of the monetary shock. Table 3 reports how increasing the variance of this shock affects the variance decomposition. In the baseline calibration dictated by the estimates of Rabanal and Rubio-Ramirez (5), monetary shocks matter mainly for interest rates uctuations, but not for much else. When their standard deviation is increased by a factor of ve, monetary shocks contribute substantially to all uctuations except for ination. Increasing the standard deviation by a factor of 1 makes monetary shocks the dominant shock for most variables. 7 Due to the large number of sign restrictions, the number of draws of orthonormal matrices is increased to 1 million.

18 observable baseline vefold monetary stdv tenfold monetary stdv r t w t π t y t Table 3: Contribution of monetary shocks to forecast error variance decomposition at horizon when varying the standard deviation of monetary shocks. percent of draws impact response of output (a) percent of draws impact response of output (b) Figure 7: Histograms for the impact response of output to a monetary shock. 15 sign restrictions are imposed for four quarters. n = 1 8 draws used. Dashed line denotes the true response from the DSGE model. Standard deviation of monetary shocks is increased vefold relative to baseline in panel (a) and tenfold in panel (b). Figure 7 shows histograms for the case when the standard deviation of monetary shocks is increased vefold and tenfold relative to the benchmark. As before, 15 out of 16 sign restrictions are imposed for k = 4 quarters. When the standard deviation of monetary shocks is increased to vefold its baseline value, a very small number of draws still indicate a rise of output in response to contractionary monetary shocks. When the standard deviation is increased tenfold, the entire support of the distribution is negative. Therefore, identication of the sign of the output response is achieved. The numerical analysis in this subsection conrms the analytical results from the simple generic model. When the shock under inspection drives the uctuations strong enough, there exists a set of strict sign restrictions such that a researcher can uniquely pin down the sign of the unrestricted impulse response. However, under the model structure and estimated parameter values of Rabanal and Rubio-Ramirez

19 (5) it is not possible to identify the sign of the output response to monetary shocks unless these shocks have a standard deviation that is more than ve times larger than the estimated value. This nding provides one possible explanation for the results in Uhlig (5b). Using a small number of sign restrictions, Uhlig nds that contractionary monetary shocks may lower or raise output. This paper shows that such a result is to be expected when monetary shocks account for only small parts of the uctuations in the observable variables..4 Example : Response of employment to technology shocks Recently, a class of medium scale models has emerged in structural macro econometric research based on the work of Smets and Wouters (3), and of Christiano, Eichenbaum, and Evans (5). A variant of this model is employed by Dedola and Neri (6) to assess the impact of technology shocks on hours worked via sign restrictions. Therefore, the Smets and Wouters (3) model is used as a second example in order to test the sign restriction framework. A detailed description of this well known model can be found in Smets and Wouters (3). I simply present the log-linearized equilibrium conditions as used in this paper. 8 Ĉ t = hĉt h + EĈt h 1 h ( ) ˆRt Eˆπ t+1 + ˆε b t Eˆε b t+1 (1 + h)σ c (17) Î t = β Ît 1 + β 1 + β EÎt+1 + 1/S 1 + β ˆQ t β ( ) Eˆε I 1 + β t+1 ˆε I t (18) ( ) ˆQ t = ˆRt ˆπ t + 1 τ 1 τ + r E ˆQ r k k t τ + r k Eˆrk t+1 (19) ˆK t = (1 τ) ˆK t 1 + τît 1 () Ê t = βêt+1 + (1 βξ e)(1 ξ e ) (Êt ξ ˆL ) t e (1) ˆL t = ŵ t + (1 + ψ)ˆr t k + ˆK t 1 () Ŷ t = c y Ĉ t + τk y Î t + r k k y ẑ t (3) ) Ŷ t = φ (ˆε at + αψˆr kt + (1 α)ˆl t (4) 8 The equations in this paper differ slightly from the equations presented in Smets and Wouters (3) in the way the shocks enter. The reason is as follows. Smets and Wouters (3) note that their estimated model differs from the equations they present in their paper due to a normalization of the shocks in the estimation procedure. I tried to back out this normalization by inspecting their impulse responses.

20 ˆR t = ρ ˆR ( ) t 1 + (1 ρ) r πˆπ t 1 + r Y Ŷt G + r dπ (ˆπ t ˆπ t 1 ) + r dy Ŷ G t (5) ˆπ t = β γ p Eˆπ t βγ p (1 + βγ p ) ˆπ ( ) t 1 + κ p αˆr k t + (1 α)ŵ t ˆε a t (6) ŵ t = ŵt β + βeŵ t β + γ wˆπ t β + βeˆπ t β + (1 + βγ w)ˆπ t κ w (ŵ t mrs t ) 1 + β (7) with: 1 (1 βξ p )(1 ξ p ) κ p, 1 + βγ p ξ p (1 + βξ w )(1 + ξ w ) κ w, (1 + β) (1 + (1 + λ w )σ L /λ w ) ξ w mrs t ˆε l t + σ L ˆLt + σ c 1 h (Ĉt hĉt 1). Here, Ĉ t denotes consumption, ˆRt the nominal interest rate, ˆπ t price ination, ˆQt the relative price of capital, ˆr t k the rental rate of capital, and ŵ t the real wage. Ê t stands for employment, ˆL t for hours worked, ˆKt for capital, ẑ t for the capital utilization rate, and Ŷ t G for the deviation of output from its exible price and wage level. The exogenous stochastic processes evolve as: ˆɛ a t = ρ aˆɛ a t 1 + u a t, u a t N(, σ a), ˆɛ l t = ρ lˆɛ l t 1 + u l t, u l t N(, σ l ), (8) ˆɛ I t = ρ Iˆɛ I t 1 + u I t, u I t N(, σ I), ˆɛ b t = ρ bˆɛ b t 1 + u b t, u b t N(, σ b ). (9) The model's parameters displayed in Table 4 are taken from the posterior mean of the estimation results in Smets and Wouters (3). α β τ c y k y λ w S σ c h σ l ξ w ξ p γ w γ p φ ψ ξ L r π r dπ ρ r y r dy ρ a ρ b ρ I ρ l σ a σ b σ I σ l Table 4: Calibration of the Smets and Wouters (3) type model. The original model of Smets and Wouters (3) is driven by such a large number of shocks that it becomes too time consuming to thoroughly analyze the per-

21 formance of the sign restriction method. Therefore, only four main shocks are kept from the Smets and Wouters (3) model: a preference shock, ˆɛ b t; an investment shock, ˆɛ I t ; a labor supply shock, ˆɛ l t; and the technology shock, ˆɛ a t. The observables in a VAR generated by the DSGE model are taken to be employment, Êt; real wage, ŵ t ; investment, Ît; and consumption, Ĉ t. These shocks and observables were chosen because they explain a large amount of the uctuations in the full Smets and Wouters (3) model, give rise to a unique pattern of signs across the observables, and imply invertibility between the structural shocks and the VAR innovations as dened in Fernandez-Villaverde, Rubio-Ramirez, and Sargent (5). The invertibility requirement turned out to reject several of the a priori plausible selections of observables and shocks. Note that the implied VAR is of innite order. Since I work entirely within population, there is no need to estimate a nite order VAR. Instead the innite number of true VAR coefcients are assumed to be known to the researcher. The implied pattern of signs of impact responses to the structural shocks is as depicted in Table 5. observable preference investment labor supply technology Ê t ŵ t Î t Ĉ t Table 5: Signs of impact responses from the Smets and Wouters (3) type model. The question under study in this subsection is whether sign restrictions can identify the sign of the response of employment to technology shocks. In the model of Smets and Wouters (3), employment falls in response to that shock as in many other sticky price models. Since the seminal contribution of Gali (1999), the response of hours to technology shock is central in the debate between advocates of real business cycle theory and of new Keynesian sticky price models. I focus on the response of employment here, since the model of Smets and Wouters (3) is estimated on employment rather than on hours due to the absence of an Euro area wide series for hours. In the model, the responses of hours and employment to shocks are quite similar. Therefore, the focus on employment is not restrictive. The pattern of signs depicted in Table 5 allows to identify technology shocks without imposing restrictions on the response of employment. In fact, the technology shock is the only shock that induces positive bivariate correlations across consumption, investment, and the real wage. Hence, it is natural to identify the technology shock by imposing

22 that the real wage, investment, and consumption rise in response to this shock. In the DSGE model, these responses are very persistent, hence the sign restrictions are imposed for k = 3 quarters. The results below are not very sensitive to this choice of k. percentage deviation from truth standard deviation of technology shock increased by factor Figure 8: Grey area outlines the range of minimum and maximum impact response of employment to technology shocks expressed as percentage deviations from the true response. Sign restrictions imposed for k = 3 quarters. Solid horizontal lines at and at -1 are for reference only. Percentage deviation from truth is dened as (identied response - true response)/(true response) 1. Broken lines depict the centered 9th percentile and 68th percentile of all theory-consistent responses. In light of the results in the previous two subsections, it is expected that the technology shock must be sufciently strong for this identication scheme to work. Therefore, Figure 8 plots the range of the identied impact response of employment against a scalar that increases the standard deviation of the technology shock. This range is depicted via the minimum and the maximum impact response of employment and is expressed as a percentage deviation from the true response. Hence, when the range falls below -1, the restrictions are compatible with incorrectly concluding that employment may rise in response to the technology shock. I focus rst on the grey area of Figure 8, which outlines the range of responses given by the minimum and the maximum across 1, theory-consistent draws. Figure 8 shows that the imposed restrictions are compatible with employment rising in response to the technology shock even when this shock has a standard deviation that is 1 times larger than what is estimated in Smets and Wouters (3). Hence, the restrictions are compatible with the wrong sign of the employment re-

23 sponse even when the technology shock is the dominant source of uctuations. 9 Apparently, more restrictions are needed to unambiguously obtain the correct sign. How can this ambiguous result be reconciled with some clear cut ndings in the empirical literature? In contrast to this analysis, Dedola and Neri (6) nd that hours work rise in response to technology shocks while imposing only a similar small number of sign restrictions. This apparent discrepancy can be explained by two factors. First, it is always possible that technology shocks are much more important in the data than what is assumed in this paper. Secondly and more importantly, there is a subtle difference in the way identication uncertainty is treated in the empirical literature and in this paper. This paper focuses on the minimum and the maximum theory-consistent response. In contrast, much of the applied literature reports only certain percentiles of the distribution of theory-consistent responses, even for a xed vector of VAR parameters. The present analysis abstracts from sampling uncertainty. Hence, there are good reasons not to exclude the information in the tails of the distribution. All impulse responses satisfying the sign restrictions are equally compatible with the reduced form VAR and with economic theory. Therefore, the impulse responses in the tails are not any less likely given the data nor any less admissible given economic theory. Implicitly, Dedola and Neri (6) take a different stance on this issue. They document identication uncertainty in their analysis by keeping the VAR parameters xed at the ordinary least squares (OLS) estimates and report the range of theory-consistent responses via percentiles. According to the centered 9th percentile depicted in Figure 5 in their paper, none of the signs of unconstrained responses is pinned down at the OLS estimates - a similar nding as in this paper. According to the centered 68th percentile the sign of the hours response is pinned down except for a few initial periods. To be comparable with their approach, Figure 8 also reports these percentiles, which are denoted by the dashed lines. As can be seen from the gure, when the standard deviation of the technology shock exceeds roughly one and a half times its baseline value, the centered 68 percentile never falls below -1. Based on this percentile, the correct sign would be pinned down. This is the same nding as in Dedola and Neri (6). Therefore, the apparent conict between the nding in this paper and in Dedola and Neri can be traced down to a difference in treating identication uncertainty. They do not pay much attention to the information in the tails of the distribution of impulse responses even for xed parameter estimates. This stands in contrast to the approach chosen in this paper. It is clear that this different treatment of identication uncertainty extends to 9 The contribution of technology shocks to the forecast error variance decomposition is in excess of 78 percent for all variables at long horizons given this standard deviation.