Inference Based on SVARs Identified with Sign and Zero Restrictions: Theory and Applications

Transcription

1 FEDERAL RESERVE BANK of ATLANTA WORKING PAPER SERIES Inference Based on SVARs Identified wit Sign and Zero Restrictions: Teory and Applications Jonas E. Arias, Juan F. Rubio-Ramírez, and Daniel F. Waggoner Working Paper b Revised October 2017 Abstract: In tis paper, we develop algoritms to independently draw from a family of conjugate posterior distributions over te structural parameterization wen sign and zero restrictions are used to identify SVARs. We call tis family of conjugate posterior distributions normal-generalized-normal. Our algoritms draw from a conjugate uniform-normal-inverse-wisart posterior over te ortogonal reduced-form parameterization and transform te draws into te structural parameterization; tis transformation induces a normal-generalized-normal posterior distribution over te structural parameterization. Te uniform-normal-inverse-wisart posterior over te ortogonal reduced-form parameterization as been prominent after te work of Ulig (2005). We use Beaudry, Nam, and Wang's (2011) work on te relevance of optimism socks to sow te dangers of using alternative approaces to implement sign and zero restrictions to identify SVARs like te penalty function approac. In particular, we analytically sow tat te penalty function approac adds restrictions to te ones described in te identification sceme. JEL classification: C11, C32, E50 Key words: identification, sign restrictions, simulation Te autors tank Paul Beaudry, Andrew Mountford, Deokwoo Nam, and Jian Wang for saring supplementary material wit us and for elpful comments. Tey also tank Grátula Bedátula for er support and elp. Witout er, tis paper would ave been impossible. Tis paper as circulated under te title Algoritm for Inference wit Sign and Zero Restrictions." Juan F. Rubio-Ramírez also tanks te National Science Foundation, te Institute for Economic Analysis (IAE), te "Programa de Excelencia en Educación e Investigación" of te Bank of Spain, and te Spanis ministry of science and tecnology (Ref. ECO c03-01) for support. Te views expressed ere are te autors' and not necessarily tose of te Federal Reserve Bank of Atlanta, te Federal Reserve Bank of Piladelpia, or te Federal Reserve System. Any remaining errors are te autors responsibility. Please address questions regarding content to Jonas E. Arias, Researc Department, Federal Reserve Bank of Piladelpia, Ten Independence Mall, Piladelpia, PA , jonas.arias@pil.frb.org; Juan F. Rubio-Ramírez (corresponding autor), Economics Department, Emory University, Atlanta, GA 30322, jrubior@emory.edu; or Daniel F. Waggoner, Researc Department, Federal Reserve Bank of Atlanta, 1000 Peactree Street NE, Atlanta, GA , , daniel.f.waggoner@atl.frb.org. Federal Reserve Bank of Atlanta working papers, including revised versions, are available on te Atlanta Fed s website at Click Publications and ten Working Papers. To receive notifications about new papers, use frbatlanta.org/forms/subscribe.

2 1 Introduction Structural vector autoregressions (SVARs) identified wit sign and zero restrictions ave become prominent. Te fact tat identification generally comes from fewer restrictions tan in traditional identification scemes and tat any conclusions are robust across te set of SVARs consistent wit te restrictions as made te approac attractive to researcers. Most papers using tis approac work in te Bayesian paradigm. 1 In tis paper we develop algoritms to independently draw from a family of conjugate posterior distributions over te structural parameterization conditional on te sign and zero restrictions. We call tis family of conjugate posterior distributions normal-generalized-normal and we sow tat it is commonly used in te literature, mainly after te work of Sims and Za (1998). We focus on two different parameterizations of SVARs. In addition to te typical structural parameterization, SVARs can also be written as te product of te reduced-form parameters and te set of ortogonal matrices, wic we call te ortogonal reduced-form parameterization. Our algoritms will draw from a conjugate posterior distribution over te ortogonal reduced-form parameterization and ten transform te draws into te structural parameterization. We follow te literature in our coice of te family of conjugate posterior distributions over te reduced parameters and use te normalinverse-wisart density. 2 Tis coice is common because it is a conjugate family and it is extremely easy to independently draw from it. Our coice of conjugate posterior over te set of ortogonal matrices conditional on te reduced-form parameters is uniform. Tis uniform-normal-inverse-wisart density over te ortogonal reduced-form parameterization is a recurrent coice after te work of Ulig (2005). We ten develop a cange of variable teory tat allows us to caracterize te induced family of posterior densities over te structural parameterization. Tis teory sows tat a uniformnormal-inverse-wisart posterior density over te ortogonal reduced-form parameterization induces a normal-generalized-normal distributions posterior distribution over te structural parameterization. Te family of normal-generalized-normal densities over te structural parameterization is also conjugate and it is often used in te literature. In any case, our algoritms can be easily modified to consider a more general family of posterior distributions, bot conjugate and non-conjugate. Using our cange of variable teory we first sow tat current algoritms for SVARs identified only 1 Exceptions are Moon and Scorfeide (2012), Moon, Scorfeide and Granziera (2013), and Gafarov, Meier and Montiel Olea (2016a,b). Moon and Scorfeide (2012) analyze te differences between Bayesian probability bands and frequentist confidence sets in partially identified models. Moon, Scorfeide and Granziera (2013) and Gafarov, Meier and Montiel Olea (2016a,b) develop metods of constructing error bands for impulse response functions of sign-restricted SVARs tat are valid from a frequentist perspective. 2 Alternatively, one could use a normal-wisart density. 1

3 by sign restrictions, as described by Rubio-Ramírez, Waggoner and Za (2010), are in fact making independent draws from te normal-generalized-normal distribution over te structural parameterization conditional on te sign restrictions. Tese algoritms independently draw from te uniform-normalinverse-wisart distribution over te ortogonal reduced-form parameterization, only accepting draws suc tat te sign restrictions old and ten transforming te accepted draws into te structural parameterization. Next, we adapt tese algoritms to consider zero restrictions. Wile te set of all structural parameters satisfying te sign restrictions will be open in te set of all structural parameters, te set of all structural parameters satisfying te signs and zero restrictions is of measure zero in te set of all structural parameters. Tis invalidates te direct use of current algoritms wen zero restrictions are considered. But te set of all structural parameters satisfying bot te sign and zero restrictions is of positive measure in te set of all structural parameters satisfying te zero restrictions. Hence, we describe an algoritm tat makes independent draws from te set of all structural parameters satisfying te zero restrictions. Te key to tis algoritm is tat te class of zero restrictions on te structural parameters maps to linear restrictions on te ortogonal matrices, conditional on te reduced-form parameters. Tis algoritm independently draws from normal-inverse-wisart over te reduced-form parameters and from te set of ortogonal matrices suc tat te zero restrictions old. Because te zero restrictions define a lower dimensional smoot manifold in te set of all structural parameters, our cange of variable teory allows us to do two tings. First, we sow tat tis algoritm does not induce a posterior distribution over te structural parameterization from te family of normal-generalized-normal distributions conditional on te sign and zero restrictions. Second, we calculate te induced density and write an importance sampler tat independently draws from normalgeneralized-normal distributions over te structural parameterization conditional on te sign and zero restrictions. Wen using sign and zero restrictions, a commonly used algoritm is Mountford and Ulig s (2009) penalty function approac PFA encefort. We sow tat te PFA adds restrictions; ence, identification does not solely come from te sign and zero restrictions considered in te identification sceme. We sow te consequences of using te PFA by first replicating te results in Beaudry, Nam and Wang (2011), and by comparing tem wit te results tat a researcer would obtain if our importance sampler were to be used instead. Te aim of Beaudry, Nam and Wang (2011) is to provide new evidence on te relevance of optimism socks as an important driver of macroeconomic fluctuations by means of an SVAR identified by imposing a sign restriction on te impact response of stock prices to optimism socks and a zero restriction on te contemporaneous response of TFP to tese socks. Based on 2

4 te results obtained wit te PFA, Beaudry, Nam and Wang (2011) conclude tat optimism socks are clearly important for explaining standard business cycle type penomena because tey increase consumption and ours. Once our importance sampler is used, te identified optimism socks do not increase consumption and ours and, ence, tere is little evidence supporting te assertion tat optimism socks are important for business cycles. Te results reported in Beaudry, Nam and Wang (2011) are significantly affected by te additional restrictions imposed by te PFA. We are not te first ones to criticize te PFA. Tere is an existing literature tat already does exactly tat. For example, Caldara and Kamps (2012) and Binning (2013) sare some of our concerns, wile adding oters, about te PFA. In related and very original work, Giacomini and Kitawaga (2015) are also concerned wit te coice of te priors densities in SVARs identified using sign and zero restrictions. Tey also work on te ortogonal reduced-form parameterization and propose a metod for conducting posterior inference on IRFs tat is robust to te coice of priors densities. We see our paper as sympatetic to teir concern about te coice of priors densities. Finally, we ave to igligt Baumeister and Hamilton (2015). Tis paper directly draws in structural parameterization. Tis is a very interesting and novel approac since te rest of te literature (including us) works in te ortogonal reduced-form parameterization. Wile working in te structural parameterization as clear advantages, mainly being able to define priors densities directly on economically interpretable structural parameters, tis approac uses a Metropolis-Hastings algoritm to make te draws. Hence tis approac is inefficient compared wit ours and arder to implement in larger models. We wis to state tat te aim of tis paper is neiter to dispute nor to callenge SVARs identified wit sign and zero restrictions. In fact, our metodology preserves te virtues of te pure sign restriction approac developed in te work of Faust (1998), Canova and Nicoló (2002), Ulig (2005), and Rubio-Ramírez, Waggoner and Za (2010). 2 Pitfalls of te Penalty Function Approac Beaudry, Nam and Wang (2011) analyze te relevance of optimism socks as a driver of macroeconomic fluctuations using SVARs identified wit sign and zero restrictions. More details about teir work will be given in Section 8. At tis point it suffices to say tat in teir most basic SVAR, Beaudry, Nam and Wang (2011) use data on total factor productivity (TFP), stock prices, consumption, te real federal funds rate, and ours worked. Teir identification sceme defines optimism socks as 3

5 positively affecting stock prices but not affecting TFP contemporaneously and tey use te PFA to implement it. Beaudry, Nam and Wang (2011) also claim tat identification solely comes from tese two restrictions. Figure 1: IRFs to a one standard deviation optimism sock. Te solid curves represent te point-wise posterior medians and te saded areas represent te 68 percent point-wise probability bands. Te figure is based on 10,000 independent draws obtained using te PFA. Figure 1 replicates te main result in Beaudry, Nam and Wang (2011). As sown by te narrow 68 percent point-wise probability bands of teir IRFs, Beaudry, Nam and Wang (2011) obtain te result tat consumption and ours worked respond positively and strongly to optimism socks. If te IRFs sown in Figure 1 were te IRFs to optimism socks solely identified using te two restrictions described above, tey would clearly endorse te view of tose wo tink tat optimism socks are relevant for business cycle fluctuations. But tis is not te case. Wen te PFA is used, identification does not solely come from te identifying restrictions; as we sow below, te PFA introduces restrictions in addition to te ones specified in te identification sceme. In Section 8 we will sow tat if our importance sampler is used instead, te results do not back as strongly te view tat optimism socks are relevant for business cycle fluctuations. Hence, we can conclude tat te results reported in Figure 1 are mostly driven by te additional restrictions imposed by te PFA. 4

6 3 Our Metodology Tis section first describes te SVAR. It ten discusses te identification problem and te class of sign and zero restrictions considered in tis paper. Next, it introduces te structural and te ortogonal reduced-form parameterization. Our algoritms will draw from te ortogonal reduced-form parameterization and ten transform te draws to te structural parameterization. Hence, we must be able to transform te prior and posterior distributions from one parameterization to anoter. Te necessary teory to accomplis tis is also described in tis section. Te usual cange of variable teorem is not sufficient because we are not transforming between open subsets of Euclidean spaces of te same dimension, but instead are transforming between smoot manifolds of te same dimension. We will briefly outline te volume measure on smoot manifolds and state te appropriate generalizations of te cange of variable teorem. Finally, we explicitly specify te conjugate prior distributions tat will be used. Wile te algoritms developed ere can be applied to oter non-conjugate prior distributions, tey are most efficient in te conjugate case. 3.1 Te Model Consider te SVAR wit te general form, as in Rubio-Ramírez, Waggoner and Za (2010) y ta 0 = p y t la l + c + ε t for 1 t T, (1) l=1 were y t is an n 1 vector of endogenous variables, ε t is an n 1 vector of exogenous structural socks, A l is an n n matrix of parameters for 0 l p wit A 0 invertible, c is a 1 n vector of parameters, p is te lag lengt, and T is te sample size. Te vector ε t, conditional on past information and te initial conditions y 0,..., y 1 p, is Gaussian wit mean zero and covariance matrix I n, te n n identity matrix. Te model described in Equation (1) can be compactly written as y ta 0 = x ta + + ε t for 1 t T, (2) were A + = [ A 1 A p c ] and x t = [ y t 1 y t p 1 ] for 1 t T. Te dimension of A + is m n, were m = np + 1. Te reduced-form representation implied by Equation (2) is y t = x tb + u t for 1 t T, (3) 5

7 were B = A + A 1 0, u t = ε ta 1 0, and E [u t u t] = Σ = (A 0 A 0) 1. Te matrices B and Σ are te reduced-form parameters, wile A 0 and A + are te structural parameters. 3.2 Te Identification Problem and Sign and Zero Restrictions Following Rotenberg (1971), te parameters (A 0, A + ) and (Ã0, Ã+) are observationally equivalent if and only if tey imply te same distribution of y t for all t. For te linear Gaussian models of te type studied in tis paper, tis statement is equivalent to saying tat (A 0, A + ) and (Ã0, Ã+) are observationally equivalent if and only if tey ave te same reduced-form representation. Tis implies tat te structural parameters (A 0, A + ) and (Ã0, Ã+) are observationally equivalent if and only if A 0 = Ã0Q and A + = Ã+Q for some Q O(n), wic is te set of all n n ortogonal matrices. To solve te identification problem, one often imposes sign and/or zero restrictions on eiter te structural parameters or some function of te structural parameters, like te IRFs. For instance, te element in row i and column j of (A 1 0 ) is te contemporaneous response of te i t variable to te j t sock. 3 Restricting tis element to be zero would imply tat te i t variable does not respond contemporaneously to te j t sock. Restricting tis element to be positive would imply tat te initial response of te i t variable to te j t sock is positive. Te teory and simulation tecniques tat we develop apply to sign and zero restrictions on any function F(A 0, A + ) from te structural parameters to te space of r n matrices tat satisfies te condition F(A 0 Q, A + Q) = F(A 0, A + )Q, for every Q O(n), wic is true for IRFs. 4 To set te notation, let S j be a s j r matrix of full row rank, were 0 s j, and let Z j be a z j r matrix of full row rank, were 0 z j n j for 1 j n. Te S j will define te sign restrictions on te j t structural sock and te Z j will define te zero restrictions on te j t structural sock for 1 j n. In particular, we assume tat S j F(A 0, A + )e j > 0 and Z j F(A 0, A + )e j = 0 for 1 j n, were e j is te j t column of I n. In Rubio-Ramírez, Waggoner and Za (2010), sufficient conditions for identification are establised. Te sufficient condition for identification is tat tere must be an ordering of te structural socks so tat tere are at least n j zero restrictions on te j t structural sock, for 1 j n. In addition, 3 More generally, te IRF of te i t variable to te j t structural sock at orizon k is te element in row i and column j of te matrix L k (A 0, A + ), were L 0 (A 0, A + ) = ( A 1 ) 0 and Lk (A 0, A + ) = min{k,p} ( l=1 Al A 1 ) 0 Lk l (A 0, A + ), for k > 0. An induction argument on k sows tat L k (A 0 Q, A + Q) = L k (A 0, A + ) Q, for all k and Q O(n). 4 In addition, a regularity condition on F is needed. For instance, it suffices to assume tat F is differentiable and tat its derivative is of full row rank. 6

8 tere must be at least one sign restriction on te impulse responses to eac structural sock. 5 Te necessary order condition for identification, Rotenberg (1971), is tat te number of zero restrictions is greater tan or equal to n(n 1)/2. In tis paper, we will ave fewer tan n j zero restrictions on te j t structural sock. Tis means tat no matter ow many sign restrictions are imposed, identification will only be a set identification. However, if tere are enoug sign restrictions, ten te identified sets will be small and it will be possible to draw meaningful economic conclusions. 3.3 Te Ortogonal Reduced-Form Parameterization Equation (2) represents te SVAR in terms of te structural parameterization, wic is caracterized by A 0 and A +. Given te discussion in Section 3.2, te SVAR can alternatively be written in wat we call te ortogonal reduced-form parameterization. Tis parameterization is caracterized by te reduced-form parameters B and Σ togeter wit an ortogonal matrix Q and is given by te following equation y t = x tb + ε tq (Σ) for 1 t T, (4) were te n n matrix (Σ) is any decomposition of te covariance matrix Σ satisfying (Σ) (Σ) = Σ. We will take to be te Colesky decomposition, toug any differentiable decomposition would do. As we will see, te ortogonal reduced-form parameterization is convenient for drawing. However, te researcer will be interested in making draws from te structural parameterization; tus, we will need to transform (B, Σ, Q) into (A 0, A + ). Given Equation (2), Equation (4), and te decomposition, we can define a mapping between (A 0, A + ) and (B, Σ, Q) by f (A 0, A + ) = (A + A 1 0, (A 0 A }{{} 0) 1, ((A 0 A }{{} 0) 1 )A 0 ). }{{} B Σ Q By a direct computation, it is easy to see tat ((A 0 A 0) 1 )A 0 is an ortogonal matrix. Te function f is invertible, wit inverse defined by f 1 (B, Σ, Q) = ((Σ) 1 Q, B(Σ) 1 Q }{{} A 0 ). }{{} A + Te ortogonal reduced-form parameterization makes clear ow te structural parameters depend 5 Often, one is only interested in partial identification. If tere is an ordering suc tat tere are at least n j zero restrictions and at least one sign restriction on te impulse responses to te j t structural sock for 1 j k, ten te first k structural socks under tis ordering will be identified. 7

9 on te reduced-form parameters and ortogonal matrices. Given te reduced-form parameters and a decomposition, one can consider eac value of Q O(n) as a particular coice of structural parameters. 3.4 Cange of Variable Teorems As mentioned above, te researcer will be interested in making draws from te structural parameterization but it is simpler to make draws from te reduced-form parameterization and ten transform tem into te structural parameterization using f 1. Hence, it is crucial to understand ow to transform densities between tese two representations. In tis section, we discuss te cange of variable teorems tat will allow us do exactly tat. Wile we will apply tese teorems to te structural parameterization, tey can be used wit any parameterization as long as te mapping between te ortogonal reduced-form parameterization and te desired parameterization can be explicitly computed. 6 As we will see, tere are differences between transforming densities wen tere are only sign restrictions as opposed to wen tere are also zero restrictions. Details and proofs, or references to proofs, will be relegated to Appendix A. Te usual cange of variable teorem can be stated as follows. Teorem 1. Let U R b be an open set and let γ : U R b be a one-to-one and continuously differentiable function. If A γ(u) and λ : A R is an integrable function, ten Proof. See Appendix A.1. A λ(v)dv = γ 1 (A) λ(γ(u)) det(dγ(u)) du. (5) Note tat te integral on eac side of Equation (5) is wit respect to Lebesgue measure over R b. Te term v γ (u) = det(dγ(u)) is te volume element of γ at u, were Dγ(u) denotes te derivative of γ evaluated at u. Note tat Dγ(u) is a b b matrix. Wen te range of γ is not R b, but is instead a b-dimensional smoot manifold in R a, one as te following cange of variable teorem. 7 Teorem 2. Let U R b be an open set, let V R a be a b-dimensional smoot manifold, and let γ : U V be a one-to-one and continuously differentiable function. If A γ(u) and λ : A R is an 6 In particular, it is easy to replicate te teory and algoritms of tis paper for te IRF parameterization. Tis parameterization is caracterized by te IRFs of te SVAR and te details are in Appendix B. 7 A b-dimensional smoot manifold in R a is a subset V of R a tat admits a local b-dimensional coordinate system in V. Tis means tat for eac v V, tere is an open set U R b and a continuously differentiable function γ : U V suc tat γ(u) is open in V, Dγ(u) is of rank b for every u U, te inverse of γ exists and is continuous, and v γ(u). Te function γ : U V is a coordinate system in V about v and γ(u) is a coordinate patc in V about v. 8

10 integrable function, ten λ(v)dv = λ(γ(u)) det(dγ(u) Dγ(u)) 1 2 du. (6) A γ 1 (A) Proof. See Appendix A.1. Te integral on te rigt and side of Equation (6) is wit respect to Lebesgue measure over R b, but te integral on te left and side cannot be wit respect to Lebesgue measure in R a if a > b because V is of measure zero in R a. However, te smoot manifold structure of V, togeter wit Lebesgue measure over R b, uniquely defines te volume of a set in V, wic determines a well-defined measure over V tat we call te volume measure. More formally, if γ : U V is a coordinate system, A γ(u), γ 1 (A) is Lebesgue measurable over R b, and λ(v) = 1 for every v A, ten Equation (6) can be taken as te definition of te volume of A. It can be sown tat tis definition is independent of te coice of coordinate system and can be extended to sets not contained in a single coordinate patc. Te integral on te left and side of Equation (6) is wit respect to te volume measure over V. Te matrix Dγ(u) is a b, so tat Dγ(u) Dγ(u) is a b b matrix. As before, te term v γ (u) = det(dγ(u) Dγ(u)) 1 2 is te volume element of γ at u. Wen a = b, Teorem 2 reduces to Teorem 1. As we will see below, Teorem 2 will be used to transform densities wen only sign restrictions are considered. Te final generalization of te cange of variable teorem is given below, wic will be of use wen tere are zero restrictions. Teorem 3. Let U R b be an open set, let V R a be a d-dimensional smoot manifold, and let te functions γ : U R a and β : U R b d be continuously differentiable wit Dβ(u) of rank b-d wenever β(u) = 0. Define U = β 1 ({0}) and suppose tat γ(u) V and γ is one-to-one on U. If A γ(u) and λ : A R is an integrable function, ten λ(v)dv = λ(γ(u)) det(n u Dγ(u) Dγ(u) N u ) 1 2 du, (7) A γ 1 (A) U were N u is any b d matrix wose columns form an ortonormal basis for te null space of Dβ(u). Proof. See Appendix A.1. Te conditions on te function β, wic will be used to describe te zero restrictions, imply tat U is a d-dimensional smoot manifold in R b and te integral on te rigt and side of Equation (7) is wit 9

11 respect to te volume measure over U. By assumption, γ(u) is contained in te d-dimensional smoot manifold V and te integral on te left and side of Equation (7) is wit respect to te volume measure over V. Te matrix Dγ(u) is a b, so tat N u Dγ(u) Dγ(u) N u is d d. Te matrix N u is not unique. If N u and Ñu are two matrices wose columns form an ortonormal basis for te null space of Dβ(u), ten tere exists a d d ortogonal matrix X suc tat N u = ÑuX. Because te determinant of a product of square matrices is equal to te product of te determinants and te determinant of a ortogonal matrix is plus or minus one, te value of te expression det(n u Dγ(u) Dγ(u) N u ) is independent of te coice of N u. As before, te term det(n u Dγ(u) Dγ(u) N u ) 1 2 is te volume element of γ restricted to U at u. To empasize te importance of te restriction, we denote tis volume element by v γ U (u). Bot Teorems 2 and 3 will be used in subsequent sections in te following way. We will ave a distribution over V wose density wit respect to te volume measure over V evaluated at v is p(v). A draw v from tis distribution can be uniquely transformed to u = γ 1 (v) and we will want to compute te density over te u. 8 Wen tere are only sign restrictions, Teorem 2 will be applicable and te density of u will be p(γ(u))v γ (u) wit respect to Lebesgue measure over R b. Wen tere are zero restrictions given by β, Teorem 3 will be applicable and te density of u will be p(γ(u))v γ U (u) wit respect to te volume measure over U. Wen applying tese teorems, v will be an ortogonal reducedform parameter and u will be a structural parameter. If te researcer is using a parameterization oter tan te structural parameterization, u will belong to it instead. In tis section we ave discussed cange of variable formulas for integration over smoot manifolds wit respect to te volume measure. In order to fix ideas, it is useful to relate te volume measure over commonly used smoot manifolds to oter measures tat could be defined over te same smoot manifolds. Some of tese examples will be used later in te paper. First, an open subset U R b is a b-dimensional smoot manifold in R b. Te volume measure over U is identical to Lebesgue measure over R b in tis case. Second, an open subset of a b-dimensional linear subspace of R a is a b-dimensional smoot manifold in R a and so tere is a well-defined volume measure over tese sets. For instance, te set of all n n symmetric and positive definite matrices is a n(n+1) 2 -dimensional smoot manifold in R n2. If V R a is a b-dimensional linear subspace, we know tat tere exists a linear mapping γ from R b onto V. Because γ is linear, te volume element is constant. Tus, by Teorem 2, te volume of A V will be te 8 It must be te case tat te support of p(v) is contained in γ(u) wen applying Teorem 2 and contained in γ(u) wen applying Teorem 3. Tis ensures tat u = γ 1 (v) exists and is unique. 10

12 Lebesgue measure of γ 1 (A) R b times te constant value of te volume element. Often te constant volume element is ignored, in wic case te measure will not be te volume measure. Because of te simple nature of linear subspaces tis generally causes no problems; owever, in tis paper we will always use te volume measure and tus te constant volume element will always be explicitly taken into account. Tird, te set of all n n ortogonal matrices, O(n), is a n(n 1) 2 -dimensional smoot manifold in R n2. In addition to te volume measure over O(n), tere are Haar measures defined over O(n), wic is any measure tat is invariant to multiplication by ortogonal matrices. Any two Haar measures differ only by a constant scale factor. Because volume is invariant to rigid transformations, wic multiplication by an ortogonal matrix is, te volume measure over O(n) is a Haar measure. Trougout te rest of te paper all densities will be wit respect to te volume measure, even toug we will not explicitly state it. Sometimes, for instance wen tere are no zero restrictions and we are working wit A 0, A +, or B, te volume measure will be Lebesgue measure. However, wen we are working wit symmetric and positive definite matrices, ortogonal matrices, or wen tere are zero restrictions, te volume measure will not be Lebesgue. 3.5 Conjugate Priors and Posteriors Wile te tecniques developed ere will work wit any prior distribution, tey are most efficient wen used wit prior distributions tat belong to a certain family of conjugate distributions. 9 For te reduced-form representation in Equation (3), te normal-inverse-wisart family of distributions is conjugate. 10 If te prior distribution over te reduced-form parameters is NIW ( ν, Φ, Ψ, Ω), ten 9 A family of distributions is conjugate if te prior distribution being a member of tis family implies tat te posterior distribution is a member of te family. Some autors also require te likeliood to be a member of te family. 10 A normal-inverse-wisart distribution over te reduced-form parameters is caracterized by four parameters: a scalar ν n, an n n symmetric and positive definite matrix Φ, an m n matrix Ψ, and an m m symmetric and positive definite matrix Ω. We denote tis distribution by NIW (ν, Φ, Ψ, Ω) and its density by NIW (ν,φ,ψ,ω) (B, Σ). Furtermore, NIW (ν,φ,ψ,ω) (B, Σ) det(σ) ν+n+1 2 e 1 2 tr(φσ 1 ) det(σ) m 2 e 1 2 vec(b Ψ) (Σ Ω) 1 vec(b Ψ). }{{}}{{} inverse-wisart conditionally normal 11

13 te posterior distribution over te reduced-form parameters is NIW ( ν, Φ, Ψ, Ω), were ν = T + ν, Ω = (X X + Ω 1 ) 1, Ψ = Ω(X Y + Ω 1 Ψ), Φ = Y Y + Φ + Ψ Ω 1 Ψ Ψ Ω 1 Ψ, for Y = [y 1 y T ] and X = [x 1 x T ]. If π(q B, Σ) is any conditional density over O(n), ten prior densities of te form NIW ( ν, Φ, Ψ, Ω) (B, Σ)π(Q B, Σ) over te ortogonal reduced-form parameterization will be conjugate. We will take π(q B, Σ) to be te uniform density. We make tis coice for tree reasons. First, as we will see below priors densities over te ortogonal reduced-form parameterization of tis form induce standard prior densities over te structural parameterization. Second, prior and posterior densities over te ortogonal reduced-form parameterization of tis form will be very easy to independently draw from. Tird, te likeliood is of tis form and so tis family of densities will be conjugate in even te stronger sense. We call tis te uniform-normal-inverse-wisart distribution over te ortogonal reduced-form parameterization; denote it by U N IW (ν, Φ, Ψ, Ω), and denote its density over te ortogonal reduced-form parameterization by UNIW (ν,φ,ψ,ω) (B, Σ, Q). 11 Densities over te ortogonal reduced-form parameterization induce densities over te structural parameterization via te function f. If π(b, Σ, Q) is any density over te ortogonal reduced-form parameterization, ten by Teorem 2, te induced density over te structural parameterization will be π(f (A 0, A + ))v f (A 0, A + ). It is easy to verify tat te ypoteses of Teorem 2 are satisfied and so Teorem 2 is applicable. Te volume element could be computed numerically, but for any function f it can be computed analytically using Proposition 1, described below. Te reader sould notice tat te volume element does not depend on te coice of. 11 It is te case tat UNIW (ν,φ,ψ,ω) (B, Σ, Q) = NIW (ν,φ,ψ,ω) (B, Σ)/ 1dQ. Because O(n) is compact, O(n) 1dQ is finite. O(n) 12

14 Proposition 1. Te volume element of f at (A 0, A + ) is v f (A 0, A + ) = 2 n(n+1) 2 det(a 0 ) (2n+m+1). Proof. See Appendix A.2. Using Teorem 2, Proposition 1, and te definition of te normal-inverse-wisart distribution, te density over te structural parameterization induced by te uniform-normal-inverse-wisart density over te ortogonal reduced-form parameterization is NGN (ν,φ,ψ,ω) (A 0, A + ) = UNIW (ν,φ,ψ,ω) (f (A 0, A + ))v f (A 0, A + ) det(a 0 ) ν n e 1 2 vec(a 0) (I n Φ) vec(a 0 ) e 1 2 vec(a + ΨA 0 ) (I n Ω) 1 vec(a + ΨA 0 ) }{{}}{{}. (8) generalized-normal conditionally normal Tus, if we independently draw (B, Σ, Q) from a uniform-normal-inverse-wisart distribution over te ortogonal reduced-form parameterization wit parameters (ν, Φ, Ψ, Ω) and ten transform te draws to (A 0, A + ) using f 1 we are in fact independently drawing from te density over te structural parameterization represented in Equation (8). We call tis a normal-generalized-normal distribution over te structural parameterization; denote it by NGN(ν, Φ, Ψ, Ω), and denote its density over te structural parameterization by NGN (ν,φ,ψ,ω) (A 0, A + ). Wen ν = n te marginal distribution of vec(a 0 ) is normal wit mean zero and variance I n Φ 1. In general we call it a generalized-normal distribution. Te distribution of vec(a + ), conditional on A 0, is normal wit mean vec(ψa 0 ) and variance I n Ω. Because te uniform-normal-inverse-wisart family of distributions is conjugate over te ortogonal reduced-form parameterization, te normal-generalized-normal family of distributions over te structural parameterization is conjugate. Tis is because if te prior and posterior densities ave te same functional form in one parameterization, ten, because te volume element will be te same for te prior and posterior densities, te induced prior and posterior densities in te oter parameterization will also ave te same functional form. Normal-generalized-normal prior distributions over te structural parameterization are often used in te literature, particularly wit ν = n. For instance, any prior distribution over te structural parameterization tat can be implemented troug dummy observations will be of tis form. Tus, te Sims-Za prior distribution over te structural parameterization is also of tis form. If te researcer needs to work wit a parameterization oter tan te structural parameteriza- 13

15 tion, an analog to Equation (8) can easily be obtained as long as te mapping between te ortogonal reduced-form parameterization and te desired parameterization can be explicitly computed. However, it may not be possible to derive an analytical expression for te volume element as in Proposition 1, but Teorem 2 can always be used to numerically compute te density over te desired parameterization induced by te uniform-normal-inverse-wisart density over te ortogonal reduced-form parameterization. It is very easy to independently draw from te uniform-normal-inverse-wisart distribution. Matlab, Matematica, and R ave routines for making independent draws from bot te inverse-wisart distribution and te normal distribution. Tere are efficient algoritms for making independent draws from te uniform distribution over O(n). Faust (1998), Canova and Nicoló (2002), Ulig (2005), and Rubio-Ramírez, Waggoner and Za (2010) all propose algoritms to do tis. Te algoritm of Rubio- Ramírez, Waggoner and Za (2010) is te most efficient, particularly for larger SVAR systems (e.g., n > 4). 12 Rubio-Ramírez, Waggoner and Za s (2010) results are based on te following teorem. Teorem 4. Let X be an n n random matrix wit eac element aving an independent standard normal distribution. Let X = QR be te QR decomposition of X wit te diagonal of R normalized to be positive. Te random matrix Q is ortogonal and is a draw from te uniform distribution over O(n). Proof. Te proof follows directly from Stewart (1980). In tis section we ave explicitly derived expressions for prior densities over te structural parameterization tat are conjugate and are induced by a uniform-normal-inverse-wisart prior density over te ortogonal reduced-form parameterization. Furtermore, tey are a family of prior densities often used in te literature. 4 Sign Restrictions Because F is continuous, te set of all structural parameters satisfying te sign restrictions will be open in te set of all structural parameters. An important point to make ere is tat te condition F(A 0 Q, A + Q) = F(A 0, A + )Q for every Q O(n) and te regularity condition are only needed to implement te algoritms for zero restrictions to be presented later. Wen only sign restrictions 12 See Rubio-Ramírez, Waggoner and Za (2010) for details. 14

16 are considered, it is enoug to assume tat F is continuous. So, if te sign restrictions are nondegenerate, so tat tere is at least one parameter value satisfying te sign restrictions, ten te set of all structural parameters satisfying te sign restrictions will be of positive measure in te set of all structural parameters. Tis justifies algoritms of te following type. Algoritm 1. Te following algoritm independently draws from te N GN(ν, Φ, Ψ, Ω) distribution over te structural parameterization conditional on te sign restrictions. 1. Draw (B, Σ) independently from te NIW (ν, Φ, Ψ, Ω) distribution. 2. Draw Q independently from te uniform distribution over O(n) using Teorem Keep (A 0, A + ) = f 1 (B, Σ, Q) if te sign restrictions are satisfied. 4. Return to Step 1 until te required number of draws as been obtained. Algoritm 1 follows te steps igligted in Section 3.4: it draws from a distribution over te ortogonal reduced-form parameterization conditional on te sign restrictions and ten transforms te draws into te structural parameterization using f 1. It follows from te discussion in Section 3.5 tat te independent draws of (A 0, A + ) produced by Algoritm 1 will be from te NGN(ν, Φ, Ψ, Ω) distribution over te structural parameterization conditional on te sign restrictions. As argued by Baumeister and Hamilton (2015), one sould coose te parameterization so tat at least some, if not all, of te parameters ave an economic interpretation and te prior distribution over te selected parameterization sould be cosen to reflect wat economic teory as to say about tose parameters. Usually, tis parameterization will not be te ortogonal reduced-form parameterization since tose parameters, particularly te ortogonal matrix, Q, are ard to interpret from an economic point of view. Altoug we agree wit Baumeister and Hamilton (2015), we will draw from te ortogonal reduced-form parameterization because Algoritm 1 is relatively efficient. We will ten transform te ortogonal reduced-form draws back to te desired parameterization. Wile Algoritm 1 is stated in terms of te structural parameterization, it will work for any parameterization as long as one can explicitly compute te transformation between te ortogonal reduced-form and te desired parameterization and te draws produced by tis algoritm will be from a conjugate distribution over te desired parameterization. Clearly, if te desired parameterization is te ortogonal reducedform parameterization, te transformation is te identity and Algoritm 1 can be used to produce independent draws of (B, Σ, Q) from te UNIW (ν, Φ, Ψ, Ω) distribution over te ortogonal reducedform parameterization conditional on te sign restrictions. 15

17 Given te desired parameterization, conjugate prior distributions ave many useful properties and if one wants to use a conjugate prior distribution tat is induced by a uniform-normal-inverse-wisart distribution over te ortogonal reduced-form parameterization, ten Algoritm 1 is an efficient tecnique for making independent draws from te associated posterior distribution. If one wants to use a different prior distribution, ten Algoritm 1 can still be used as te proposal density in an importance sampler. Of course te efficiency of tis algoritm depends eavily on ow close suc a prior distribution is to one tat can be induced by a uniform-normal-inverse-wisart distribution over te ortogonal reduced-form parameterization. In tis paper we work only wit conjugate priors distributions over te structural parameterization wose density can be described by Equation (8) as tey are commonly used in te literature, as mentioned in Section Zero Restrictions We now adapt Algoritm 1 to andle te case of sign and zero restrictions. Wen tere are only sign restrictions, te set of all structural parameters satisfying te restrictions is of positive measure in te set of all structural parameters. However, wen tere are bot sign and zero restrictions, te set of all structural parameters satisfying te restrictions is of measure zero in te set of all structural parameters. Tis invalidates te direct use of Algoritm 1. But since te set of all structural parameters satisfying bot te sign and zero restrictions is of positive measure in te set of all structural parameters satisfying te zero restrictions, if we could make independent draws from te set of all structural parameters satisfying te zero restrictions, ten we could apply a variant of Algoritm 1 to obtain independent draws from te set of all structural parameters satisfying bot te sign and zero restrictions. Algoritm 3, described in tis section, does precisely tat. 5.1 Zero Restrictions in te Ortogonal Reduced-Form Parameterization Te zero restrictions in te structural parameterization are Z j F(A 0, A + )e j = 0 for 1 j n. From te definition of f and te fact tat F(A 0 Q, A + Q) = F(A 0, A + )Q, te zero restrictions in te ortogonal reduced-form parameterization are Z j F(f 1 (B, Σ, Q))e j = Z j F(f 1 (B, Σ, I n))qe j = 0 for 1 j n. 16

18 Tis means tat te zero restrictions in te ortogonal reduced-form parameterization are really just linear restrictions on eac column of te ortogonal matrix Q, conditional on te reduced-form parameters (B, Σ). It is tis observation tat is key to being able to make independent draws from te set of all structural parameters satisfying te zero restrictions. Algoritm 2. Te following makes independent draws from a distribution over te structural parameterization conditional on te zero restrictions. 1. Draw (B, Σ) independently from te NIW (ν, Φ, Ψ, Ω) distribution. 2. For 1 j n, draw x j R n+1 j z j independently from a standard normal distribution and set w j = x j / x j. 3. Define Q = [q 1 q n ] recursively by q j = K j w j for any matrix K j wose columns form an ortonormal basis for te null space of te (j 1 + z j ) n matrix M j = [ q 1 q j 1 (Z j F(f 1 (B, Σ, I n))) ]. 4. Set (A 0, A + ) = f 1 (B, Σ, Q). 5. Return to Step 1 until te required number of draws as been obtained. Te null space of M j will be of dimension n + 1 j z j if and only if M j is of full row rank. Because of te regularity condition on te function F, tis will always be case. Te details of tis argument appear in Appendix A.3. Tis is crucial because oterwise te product K j w j is not defined. It is also te case tat matrix K j is not unique. If te columns of K j form an ortonormal basis for te null space of M j, ten so will te columns of K j X for any X O(n + 1 j z j ). Te particular coice of K j does not make a material difference in te output of Algoritm 2, but in te next section, wen we compute te density over te structural parameterization conditional on te zero restrictions implied by Algoritm 2, we will need te function K j = K j (B, Σ, q 1,, q j 1 ) to be differentiable almost everywere. 13 In Appendix A.3 we define K j so tat it is differentiable almost everywere. 13 Te function K j depends on f 1 (B, Σ, I n), and so implicitly requires Σ to be symmetric and positive definite. Tus te domain of K j is not an open set in R n(m+n+j 1). In Appendix A.2, we extend te definition of f 1 so tat te domain of K j is an open set and te derivative can be defined. Also, in general, it is not possible to define K j so tat it is differentiable everywere. For instance, if tere are no restrictions, ten te existence of everywere continuous K j would imply tat O(n) is topologically equivalent to a product of speres, wic is not true if n 3. 17

19 Our implementation is quite straigt forward and te computation of eac K j requires only a single QR-decomposition of an n n invertible matrix. 14 By construction, te vector q j is perpendicular to te rows of M j and q j = w j = 1. Tus te matrix Q obtained in Steps 2 and 3 is ortogonal and Z j F(f 1 (B, Σ, I n))qe j = 0 for 1 j n. So, te algoritm produces independent draws from a distribution over te structural parameterization conditional on te zero restrictions. In te next subsection, we sow ow to numerically compute te density of tis distribution using Teorem 3. Unlike te sign restriction only case, te distribution over te structural parameterization conditional on te zero restrictions implied by Algoritm 2 is not equal to te NGN(ν, Φ, Ψ, Ω) distribution conditional on te zero restrictions. However, once we know ow to numerically compute its density, we can use Algoritm 2 as a proposal distribution for an importance sampler to draw from te N GN(ν, Φ, Ψ, Ω) distribution over te structural parameterization conditional on te zero restrictions. Because Algoritm 2 will be used as a proposal distribution for an importance sampler for a distribution wose support is te set of all structural parameters satisfying te zero restrictions, it must be te case tat te support of te distribution implied by Algoritm 2 is also te set of all structural parameters tat satisfy te zero restrictions. To see tat tis is te case, suppose tat (A 0, A + ) = f 1 (B, Σ, Q) satisfies te zero restrictions. To sow tat tese parameters are in te support of te distribution implied by Algoritm 2, it suffices to sow tat tere exist w j R n+1 j z j, for 1 j n, suc tat Step 3 of Algoritm 2 maps to te ortogonal matrix Q. Let w j = K jq j. Since f 1 (B, Σ, Q) satisfies te zero restrictions and te matrix Q is ortogonal, te jt column of Q is in te null space of M j. Tus K j w j = K j K jq j = q j, because multiplication by K j K j is projection onto te null space of M j. Algoritm 2 also follows te steps igligted in Section 3.4: it draws from a distribution over te ortogonal reduced-form parameterization conditional on te zero restrictions and ten transforms te draws into te structural parameterization using f 1. As mentioned, in tis case te independent draws of (A 0, A + ) produced by Algoritm 2 will not be from te NGN(ν, Φ, Ψ, Ω) distribution over te structural parameterization conditional on te zero restrictions. Te density implied by Algoritm 2 will be analyzed below. 14 In Matlab, an obvious coice would be to define K j = null(m j ). Wile it is surely te case tat tis coice would be differentiable almost everywere, to prove tis would require details of te Matlab implementation of tis function. 18

20 5.2 Te Density Implied by Algoritm 2 In tis section we use Teorem 3 to sow ow to numerically compute te distribution over te structural parameterization conditional on te zero restrictions implied by Algoritm 2. In order to do tat we need to carefully caracterize te mapping implied by te steps in te algoritm. Step 1 of Algoritm 2 independently draws B and Σ from te NIW (ν, Φ, Ψ, Ω) distribution. Step 2 draws w j from te uniform distribution on te unit spere in R n+1 j z j. Tis implies tat te density over (B, Σ, w 1,, w n ) will be proportional to NIW (ν,φ,ψ,ω) (B, Σ). Step 3 maps (B, Σ, w 1,, w n ) to (B, Σ, Q) and Step 4 maps (B, Σ, Q) to (A 0, A + ) = f 1 (B, Σ, Q). It is tis composite mapping, togeter wit Teorem 3, tat we will use to compute te density. It will be sown in Appendix A.3 tat tere exists an open set V R nm+n2 +n 2 suc tat te functions K j = K j (B, Σ, q 1,, q j 1 ) can be defined so tat tey are differentiable for all (B, Σ, Q) V. 15 Tus, we can define a differentiable function g : V R nm+n2 + n j=1 (n+1 j z j) by g(b, Σ, Q) = (B, Σ, (K 1 (B, Σ) q 1,, K n (B, Σ, q 1,, q n 1 ) q n )). On te set of all (B, Σ, Q) V suc tat Σ is symmetric and positive definite, Q is ortogonal, and (A 0, A + ) = f 1 (B, Σ, Q) satisfies te zero restrictions, te function g will be one-to-one. Te easiest way to see tis is tat te function defined by Step 3 of Algoritm 2 is te inverse of g on tis restricted set. Te argument is identical to tat used to sow tat te support of te distribution implied by Algoritm 2 is te set of all structural parameters satisfying te zero restrictions. Let U = f 1 (V ), wic will be an open set in te set of all structural parameters. Te composite function g f, wen restricted to te (A 0, A + ) U tat satisfy te zero restrictions, will be te inverse of te function defined by Steps 3 and 4 of Algoritm 2. If Z denotes te set of all structural parameters tat satisfy te zero restrictions, ten by Teorem 3 te density over te structural parameterization conditional on te zero restrictions implied by Algoritm 2 is proportional to NIW (ν,φ,ψ,ω) (B, Σ)v (g f ) Z (A 0, A + ), were (B, Σ, Q) = f (A 0, A + ). It is easy to verify tat te ypoteses of Teorem 3 are satisfied. Te only difficulty is to ceck weter te derivative of te function describing te zero restrictions, wic is given by β(a 0, A + ) = (Z j F(A 0, A + )e j ) n j=1, as full row rank. Tis will follow from te regularity conditions on te function F, te details of wic are in Appendix A.3. Finally, it must be te case tat any structural parameter satisfying te zero restrictions must 15 Here, we are implicitly considering K j to be a function of Q, even toug it actually only depends on te first j 1 columns of Q. 19

21 almost surely be in te set U. If tis were not te case, ten tere would be a set of positive measure for wic te tecniques of te section would not apply. As wit te oter details in tis section, tis will be sown in Appendix A An Importance Sampler Te results of Sections 5.1 and 5.2 sow tat, first, Algoritm 2 generates independent draws from a distribution over te structural parameterization conditional on te zero restrictions tat is not equal to te NGN(ν, Φ, Ψ, Ω) distribution conditional on te zero restrictions and, second, we know ow to to numerically compute tis density. Tus, tey justify te following importance sampler algoritm to independently draw from te N GN(ν, Φ, Ψ, Ω) distribution over te structural parameterization conditional on te sign and zero restrictions. Algoritm 3. Te following algoritm independently draws from te N GN(ν, Φ, Ψ, Ω) distribution over te structural parameterization conditional on te sign and zero restrictions. 1. Use Algoritm 2 to independently draw (A 0, A + ). 2. If (A 0, A + ) satisfies te sign restrictions, ten set its importance weigt to NGN (ν,φ,ψ,ω) (A 0, A + ) NIW (ν,φ,ψ,ω) (B, Σ)v (g f ) Z (A 0, A + ) det(a 0) (2n+m+1) v (g f ) Z (A 0, A + ), were (B, Σ, Q) = f (A 0, A + ) and Z denotes te set of all structural parameters tat satisfy te zero restrictions. Oterwise, set its importance weigt to zero. 3. Return to Step 1 until te required number of draws as been obtained. As was te case wit Algoritms 1 and 2, Algoritm 3 follows te steps igligted in Section 3.4: it draws from a distribution over te ortogonal reduced-form parameterization conditional on te sign and zero restrictions and ten transforms te draws into te structural parameterization. It follows from te discussion in Section 5.2 tat te independent draws of (A 0, A + ) produced by Algoritm 3 will be from te N GN(ν, Φ, Ψ, Ω) distribution over te structural parameterization conditional on te sign and zero restrictions. Algoritm 3 inerits te key features of Algoritm 1. First, being able to independently draw (A 0, A + ) from te normal-generalized-normal family of distributions over te structural parameterization conditional on te sign and zero restrictions means tat we can use Algoritm 3 to independently 20