Identification in Dynamic Models Using Sign Restrictions

Transcription

1 Identification in Dynamic Models Using Sign Restrictions Bulat Gafarov Pennsylvania State University, National Research University Higher School of Economics September 30, 2013 Abstract Sign restrictions on impulse response functions are used in the literature to identify structural vector autoregressions and structural factor models. I extend the rank condition used for exclusion restrictions and provide a necessary and sufficient conditions for point identification for sign restrictions in this class of models. The necessary condition for point identification implies that as the number of sign restrictions grows a subset with sufficient number of sign restrictions becomes binding in the limit. However, one does not need to possess information about this subset to achieve point identification. So when exclusion restrictions are not justified by theory, sign restrictions can provide an alternative way to get point-identified impulse response functions. Also further, I present a closed form representation of the set of all impulse response functions satisfying a set of sign restrictions. I demonstrate that restrictions on responses to all shocks can dramatically shrink this set when compared to restrictions only on a small number of shocks. Keywords: SVAR, SFM, point identification, sign restrictions. JEL Classification Numbers: C1, C32, C38, E47 The first draft : September 4, 2012 Address: 605 Kern Building, University Park, PA 16802, USA, telephone: , bzg134@psu.edu I am grateful to Joris Pinkse for posing the question and for his help and encouragement. I also would like to thank Saroj Bhattarai, Anna Mikusheva, Sung Jae Jun, James Stock, Jonathan Wright, Konstantin Styrin, Nail Kashaev, Bruno Salcedo, Mark Razhev, Ismael Mourifie and participants of the PERC 2012 conference at the PSU for discussion of the paper and helpful comments that shaped substantially the focus of my research. 1

2 1 Introduction I consider the problem of identification of impulse response functions in two prominent time series dynamic models, structural vector autoregressions (SVAR) and structural factor models (SFM), using sign restrictions. These models appear naturally in several areas of economics, but especially as linearized versions of dynamic stochastic general equilibrium models (DSGE) in macroeconomics and as models for asset prices in finance (Stock and Watson (2005); Forni et al. (2009); Kilian (2013)). In the SVAR and SFM frameworks, the object of interest for empirical macroeconomists is usually impulse response functions (IRF) of macroeconomic variables to structural shocks. For example, one can evaluate the impact of a shock in monetary policy on output using these models. Without any additional restrictions on the IRF it is impossible to decompose the estimated reduced form shocks into the set of the shocks implied by economic theory. The standard way to achieve this aim, the recursive approach, described in Forni et al. (2009); Kilian (2013), is to impose a finite number of zero restrictions on IRF motivated by underlying structural models. Zero restrictions do identify IRF in SVAR and SFM, but they are sometimes hard to justify from the viewpoint of economic theory. Sign restrictions, as proposed by Faust (1998); Canova and Nicoló (2002); Uhlig (2005), are easier to justify using calibrated DSGE models (Paustian (2007); Canova and Paustian (2011)). Generically, a finite number of sign restrictions provides only set identification of the IRF and requires special treatment (Fry and Pagan (2011); Moon et al. (2013)). The most common way of dealing with the problem of partial identification in SVAR and SFM is to impose prior distributions on the parameters of the model (Uhlig (2005)) and follow the Bayesian approach. This way to solve the problem has some drawbacks. The first problem, as Moon et al. (2013) show, is that in the case of partial identification, Bayesian credible intervals do not have the correct coverage rate. The second and more important drawback is that the estimates of IRF in this framework depend on the choice of priors and these choices affect the results, even in large samples. Kilian (2013) mentions in his review that conditions under which the estimates of SVAR models do not depend on priors is an open research topic. Giacomini and Kitagawa (2013) address this problem by using robust Bayes analysis. Another way to solve this problem would be to propose conditions for point identification under sign restrictions, because in that case any non-dogmatic priors over the parameter space would lead to the same result in large samples. According to practitioners (Canova (2011)), sign restrictions can produce a unique set of structural shocks. Paustian (2007) executes a Monte-Carlo experiment with some popular DSGE models to show that the set of parameter values in SVAR models consistent with a sufficiently large number of sign restrictions motivated by these models is tight and can, effectively, be considered as a point. This result, however, lacks formal conditions for point identification in dynamic models with sign restrictions. I provide analytic bounds on IRF implied by a set of sign restrictions and show that restrictions 2

3 on all shocks shrink the set of parameter values dramatically. Using these results I establish necessary and sufficient conditions for point identification using only sign restrictions and show that restrictions on responses to all shocks can dramatically shrink this set when compared to restrictions only on some shocks. It turns out that a unique solution to the identification problem with sign restrictions will exist only if the limiting set of the sign restrictions has a subset with a number of restrictions that are binding, i.e. the inequality restrictions become equalities. It does not imply, however, that the point identified case with sign restrictions is equivalent to the zero restriction approach because the choice of the equality restrictions in the first case can be data driven rather than assumed a priori. Sign restrictions can also provide point identification even if the rank conditions for global identification with zero restrictions of Rubio-Ramirez et al. (2010) are not satisfied, since one can make the impulse response functions in the model, that are locally identified by zero restrictions, globally identified by adding sign restrictions. I show how to use the results to analyze the effects of a monetary policy shock. My results complement results of Moon et al. (2013) by demonstrating that sign restrictions can uniquely identify SVARs and SFMs and results of Rubio-Ramirez et al. (2010) on conditions for point identification in SVARs based on exclusion restrictions by considering conditions for point identification in case of sign restrictions. I demonstrate analytically the importance of sign restrictions on all shocks found by practitioners Peersman (2005). The structure of the rest of the paper is as follows. Section 2 describes the environment. Section 3 contains my main result about necessary and sufficient condition for point identification of the impulse response functions when restrictions imposed only on one shock. Section 4 contains conditions for point identification when restrictions imposed on several shocks. Section 5 describes how the condition for point identification by sign restrictions are related to the conditions for point identification with zero restrictions. In Section 6 I give two examples to show underlying ideas. Section 7 discusses implications for estimation and inference. The last section concludes. 2 Model framework 2.1 Structural Vector Autoregression Consider a q dimensional stationary vector process X t with zero mean that follows A 0 X t = A 1 X t A p X t p +u t = A(L)X t +u t, (2.1) where A(L) is a matrix lag polynomial, the vector of structural shocks u t is a zero mean vector process with E(u jt u js ) = 0 for any j and t s and normalization E(u t u t ) = I q, and A 0 is invertible. The 3

4 representation (2.1) is called structural as opposed to the reduced form representation X t = A 1 0 A 1X t A 1 0 A px t p +A 1 0 u t = Ã1X t ÃpX t p +ε t = Ã(L)X t +ε t, (2.2) where once again Ã(L) is a matrix lag polynomial, and the vector of reduced form shocks ε t = A 1 0 u t is also a zero mean process with E(ε jt ε js ) = 0 for any j and t s, but without restrictions on the contemporaneous covariance. The parameters in the VAR model (2.2) are always identified (Hsiao (2001)) and can be consistently estimated while (2.1) is not identified because the number of implied moment restrictions is less than the number of coefficients by q(q 1)/2. 1 According to the necessary condition of Rothenberg (1971) there exists more than one structural representation (2.1) consistent with the same reduced form representation (2.2). The reduced form representation (2.2) can be inverted into a vector moving average process of infinite order (VMA( )): X t = ε t +C 1 ε t 1 +C 2 ε t = C(L)ε t, (2.3) where the matrix lag polynomial C(L) is called a matrix of impulse response functions (IRF), i.e. elements (i,j) of a sequence of matrices {C l } l=1 correspond to impulse response function of the time series x it to the reduced form shock ε jt. For the purpose of counterfactual analysis, researchers usually evaluate the impact of structural shocks u t on X t : X t = C(L)A 1 0 u t = B 0 u t +B 1 u t 1 +B 2 u t = B(L)u t. (2.4) I will call this representation structural. However, the parameters in the SVAR representation 2.4 are not identified, so one needs additional restrictions to identify the matrix A 1 0 to pin down this representation. By Proposition A.1 from Uhlig (2005) the matrix A 1 0 can be represented as A 1 0 = T ε H, where H is some orthogonal matrix and T ε is the unique Cholesky matrix for Σ ε = T ε T ε. The matrix A 1 0 will be identified if some set of restrictions on the parameters of the model will pin down the matrix H. For example, one has to impose restrictions on the matrix of structural IRF B(L) = C(L)T ε H = C(L)H to determine the matrix H uniquely. 2.2 Structural Factor model The SFM framework of Forni et al. (2009) allows one to study SVAR models in the high-dimensional case thanks to convenient restrictions on the rank of the covariance matrix of structural shocks. In this 1 See Proposition 9.2 in Lütkepohl (2005) 4

5 context an identification problem similar to the problem in the SVAR model appears. Consider a collection of time series X = {x it,i N,t Z}. Here i is the index of the time series and the index t denotes the time period. Every series is assumed to be weakly stationary and to have zero mean. It is assumed that the process x it can be represented as x it = χ it +η it, where χ it is a common component and η it is an idiosyncratic component. It is also assumed that all dependence between any two series x it and x jt for i j is due to the common components and so the idiosyncratic components are independent across time series and independent of the common components. The common components are assumed to have the following latent factor form: k χ it = a is f st = a if t, s=1 where a i = (a i1,...,a ik ) is a vector of factor loadings, corresponding to series number i, while f t = (f 1t,...,f kt ) is a stationary vector stochastic process that affects every time series x 1t,...,x nt through the common component. The elements of f t are called static factors. The vector process f t is a VAR process that can be represented as a VMA process of infinite order such that f t = N(L)u t, (2.5) where the vector of common shocks, or dynamic factors, 2 u t, is a q dimensional vector process with E(u jt u js ) = 0 for any j and t s covariance matrix I q and N(L) is a k q matrix of lag polynomials of infinite order. The dimension q is fixed. The representation (2.5) is called fundamental if there exists a q k one-sided filter G(L) such that G(L)N(L) = I q. It means that the common shocks can be recovered by observing only past values of f t (possibly infinitely many of them), G(L)f t = G(L)N(L)u t = u t. As Forni et al. (2009) argue, fundamental representations of shocks are generic, so I follow their suggestion and restrict my attention to fundamental representations only. Let b il j The vector B i (L) = a i N(L) consists of impulse response functions of the time series i on shocks u t. be the j th component of this vector corresponding to the l th power of the lag operator, which is equal to the impact of the unit shock in the j-th common shock on the i th time series with lag l. So in this notation B i (L) = b i0 1. b i0 q + b i1 1. b i1 q L+...+ b il 1. b il q L l This term for the common shocks is used in Forni et al. (2009). 5

6 Note that there can be many different fundamental representations of the model. Consider any fundamental representation with shocks u t and the matrix lag polynomial N(L). Then for any orthogonal matrix H of size q q the pair Hu t and N(L)H will be another fundamental representation of the structural factor model. Furthermore, Forni et al. (2009) state that for any two fundamental representations of the static factors F t = N(L)u t = M(L)v t with two different fundamental white noise processes {u t } and {v t }, there exists a q q orthogonal matrix H such that: v t = Hu t, N(L) = M(L)H. This fact implies non-identification of the vector polynomials B i (L) = a in(l). Indeed, any sequence of IRF of the form C i (L) = a i M(L) = Bi (L)H will be consistent with some fundamental representation unless I impose some additional restrictions on it. In the rest of the paper B i (L) will be a fundamental representation that satisfies some specific set of restrictions motivated by some economic model, while C i (L) corresponds to any other fundamental representation of IRF. To be consistent with the SVAR case I call this representation structural while any other fundamental representation, that is consistent with parameters {a i } i=1 and reduced form VAR representation of f t, I call nonstructural. I can fix some q zero restrictions on the IRF that give exact point identification of some non-structural C i (L). Such restrictions exist according to Forni et al. (2009). The resulting IRF, C i (L), has no structural economic interpretation as the restrictions were chosen arbitrarily. By Proposition 2 of Forni et al. (2009), any other fundamental representation of the common component IRF can be obtained through a suitable orthogonal transformation H. I can consider this transformation as an additional matrix of parameters to identify. Note, that identification of the infinite dimensional object B i (L) becomes equivalent to identification of the finite dimensional object H which is easier and makes the problem equivalent to the SVAR case considered earlier. Macroeconomists are often only interested in identification of the IRF corresponding to one shock (Uhlig (2005), Kilian (2013)), a monetary policy shock for example. It means that they are only interested in identification of one column of H. Since the choice of a column is arbitrary, one can apply the same procedure to identify other columns. So I consider in this paper only the problem of identification of the first column in H. Recall that in my notation the scalar b il j is the component of the vector lag polynomial B i (L) that is equal to the impact of a positive unit shock in the common shock j on the time series i with lag l. Let h p j be the j th component of the first column hp of the matrix H. Let J be the set of all triples of indices (i,l,p) that correspond to a given set of sign restrictions on the IRF of series i to a common shock p with lag l. Then any sign restriction (i,l,p) J will have the form of an inequality, 6

7 b il p 0 or bil p 0 The second type of inequality is equivalent to b il p 0. So without loss of generality I normalize inequalities for the first shock to be of the first type by multiplying IRF vectors with opposite first sign restriction by 1. These inequalities are equivalent to the following set of inequalities in terms of the pilot IRF c il j for every (i,l,p) J q j=1 q j=1 c il j h1 j 0, (2.6) c il j h p j 0. (2.7) Note, that for every (i,l,p) J by normalization b il 1 0, i.e. one can impose (non-zero) sign restrictions on other p 1 if one imposes sign restriction on the first component. A zero restriction b il p = 0 is equivalent to conjunction combining b il p 0 and b il p 0, so zero restrictions are just a particular combination of sign restrictions and can be considered in this framework. The matrix H is orthogonal 1 if p = k h p h k =. (2.8) 0 if p k Let Θ be the set of all solutions h 1 to the system (2.6) (2.8). By restriction (2.8) Θ is a set on a q dimensional sphere. If the set Θ is a singleton then the IRF for the first shock is point identified or globally identified. If the set Θ has isolated points, then he IRF for the first shock is locally identified. Consider the set C of vectors c il = (c il 1,...,c il q ), for which there exists at least one restriction on response to some shock p. To simplify notation assume without loss of generality that these vectors have unit length, since such a normalization does not affect any sign restriction. Note that in case of an infinite number of sign restrictions, I can consider the limiting points of the set C as additional sign restrictions. Proposition 1. For every vector c = (c 1,...,c q ) from the closure of C and any solution h of the system (2.6) (2.8) it must be true that q c j h 1 j 0 (2.9) j=1 7

8 If also further for some p 1 there is a sequence of vectors {c m } m=1 C such that c = lim m cm and for every m the following sign restrictions hold q j=1 c m j hp j ( )0, (2.10) then it must be true that q c j h p j j=1 ( )0. (2.11) Proof. See Appendix. 3 Conditions for Point Identification When Restrictions Imposed Only on One shock Suppose that restrictions are imposed only on the first shock, i.e. p = 1. Then the set is obtained by intersection of the half-spaces (2.6) and the surface of the sphere (2.8). So it can be an empty set, a point, or a connected compact manifold on a multidimensional sphere. In this section I state and prove necessary and sufficient conditions for the point identification of the vector h 1. To understand the idea behind my result I suggest the following example where the solution to (2.6) (2.8) can be shown graphically. Consider the setup with three restrictions, J = 3, and two shocks, q = 2. Consider particular parameter values such that the structural impact of both common shocks of the time series with superscript 1 with zero lag are non-negative, b 1 1 0,b 1 2 0; for the time series with superscript 2 it is equal to zero for the first factor, b 2 1 = 0, and non-negative for the second factor, b 2 2 0; and the impact of both shocks on the time series with superscript 3 is negative b 3 1 0, b I can represent these IRF in one diagram in the form of vectors from the origin. In a non-structural representation these vectors are transformed by an orthogonal matrix H. In the two dimensional case this transformation is a rotation around the origin on an angle φ = arccos ( h1) 1. This operation is represented in Figure 1. Figure 2 depicts the same vectors of IRF but in a different fundamental representation. Suppose that I impose sign restrictions only on the impact of the first common shock. Then each vector determines a half-space through the inequalities (2.9). Note that the vector with superscript 3 is directed to the opposite side because b In the picture I assume that b The red arc shows all possible angles φ corresponding to all orthogonal transformations H consistent with the inequalities. Figure 2 shows the partially identified case, where many different angles are consistent with the same reduced form representation. To understand the conditions for point identification suppose that the specific numerical values are 8

9 such that b 3 = b 2. Also suppose that b Then the set of possible angles would shrink to one point (a solid red dot in Figure (3)). Indeed, such a pair of restrictions would be equivalent to the zero restriction b 3 1 = 0, since b2 1 = q j=1 c2 j h1 j = q j=1( c 3 j ) h 1 j 0. So under certain conditions point identification becomes possible with the use of sign restrictions only. In this case, in fact, the structural impact of the series with superscript 3 on the first factor would be equal to zero, b 3 1 = 0, i.e. point identification is possible only if there is a pair of sign restrictions in the closure C, that is equivalent to a zero restriction. I generalize this condition from the simple example to the multidimensional case in Assumption H. b 2 c 1 b 1 2 b 2 2 c 2 b 3 1 b 1 1 b 1 φ c 3 b 3 2 Figure 1: Impulse Response Functions in the case J = 3, q = 2. Solid vectors correspond to the structural representation, dashed vectors correspond to a non-structural representation. 9

10 cone( C) h 1 2 c 3 c 1 c 2 φ h 1 1 Figure 2: The set-identified case with J = 3, q = 2. The red arc corresponds to the set Θ, the solution to the system (2.6) (2.8) in terms of a non-structural representation. It determines the set of all possible angles φ and the set of all structural representations consistent with the reduced form. cone( C) h 1 2 c 1 c 2 φ c 3 h 1 1 Figure 3: The point-identified case with J = 3, q = 2. The red dot corresponds to the set Θ, the solution to the system (2.6)-(2.8) in terms of a non-structural representation. It determines the unique angle φ and the structural representation consistent with the reduced form. Assumption H. The conic hull of C defined as cone( C) = { m i=1 λ ic i λ i 0,c i C,i N,m N } contains a half-space. For example, if the number of shocks q is 2 then the half-space implied by Assumption H is a half- 10

11 plane (see Figure (3)). Note that this assumption is independent of the choice of the fundamental representation and thus it has to be true for any fundamental representation. Remark 1. Suppose that C contain q vectors that lie on the same q 1 dimensional hyperplane, any q 1 of them are linearly independent, and the conic hull of these vectors covers the entire hyperplane. If, in addition, C contains at least one vector which does not belong to the hyperplane, then Assumption H holds. The following Lemma shows the main idea behind the proof of Theorem 1 in the paper, which will be stated later. Lemma 1. a cone( C) if and only if cone(θ) = { h c C,c h 0} {h a h 0 }. Proof. See Appendix. Now I can state the theorem. Theorem 1 (Necessary and sufficient conditions for point identification with restrictions on one shock). Suppose that Assumption H holds, then the set Θ has at most one element. If the set Θ is a singleton, then Assumption H holds. Proof. See Appendix. 3.1 Examples of Point Identification with Sign Restrictions Short and long lived shocks Consider a bivariate VAR model and a corresponding structural VMA( ) representation. x t y t = B(L) u 1t u 2t Suppose that one wants to identify impacts of the shocks u 1t and u 2t on both time series. The set of restrictions is such that for all l > L b xl 1 0 and b yl 1 0. Suppose that numerical values are such that for all l > L b xl 1,b xl 2,b yl 1 0 and byl 2 0 the impact of u 1t on both time series vanishes faster than the impact of u 2t, i.e. lim b xl 1 /b xl 2 = 0 and lim b yl 1 /byl 2 = 0. The limits of sequences (bxl 1,b xl 2 ) / b xl and l l (b yl 1,byl 2 ) / b yl are (0,1) and (0, 1), respectively, which are both orthogonal to (1,0) and directed in opposite sides. Moreover, if not all vectors in (b xl 1,b xl 2 ) / b xl and (b yl 1,byl 2 ) / b yl are linearly dependent, then I can choose one of them, which is linearly independent from (0,1), so Assumption H is satisfied in this case by (1). Note that unlike long-run identification a la Blanchard and Quah (1989) this case does not require the series to be non-stationary or cointegrated. 11

12 3.1.2 Multiple series with heterogeneous responses Consider a structural factor model with two common shocks, u 1t and u 2t, in structural form: x it = B i (L) u 1t u 2t +ξ it Here, as in Section 2, i indexes the time series and take natural numbers, ξ it is idiosyncratic shock. Once again one wants to identify impact of the common shocks on the observed time series. A set of restrictions that allows to identify these impacts is the following. To achieve this aim it is sufficient to consider only contemporaneous responses b i0 1 and bi0 2. Suppose that there are two subsequences of time series with indices { i + k} k=1 and { i k} such that for any natural k the following set of sign k=1 restrictions holds: b i+ k 0 1,b i+ k 0 2,b i k and b i k Also assume that zero is a limit point of the { } { } { } sequences b i+ k 0 1 and b i k 0 1, k=1 k=1 (0,1) is a limit vector of the sequence (b i+ k 0 1,b i+ k 0 2 ) / b i+ k 0 { }, k=1 and (0, 1) is a limit vector (b i k 0 1,b i k 0 2 ) / b i k 0. Here, as in the previous example, Assumption k=1 H holds, so the IRF are identified. 4 Sign Restrictions on Several Shocks 4.1 Analytic bound implied by sign restrictions on several components of IRF In the previous section I considered the setup in which sign restrictions are imposed only on IRF for the first shock. In the set-identified case additional restrictions on other IRF can make the identified set of the IRF for the shock of interest smaller. Indeed, the columns of every orthogonal matrix are orthogonal to each other and one can shrink the set of all impulse response functions consistent with the data by imposing additional restrictions on the IRF for other shocks. The results presented in this section provide analytic bounds on Θ for the cases when this improvement happens. As before, I impose the normalization that the first component of any IRF vector is positive and every IRF vector has unit length. Theorem 2. Suppose that there are q linearly independent IRF vectors ( c 1,...,c q) from C such that there are positive sign restrictions on their p-th components with p 1 for the first m vectors and negative for the rest of them. Then the set Θ does not have any points in the interior of cone ( c 1,...,c m, c m+1,..., c q) and cone ( c 1,..., c m,c m+1,...,c q). Proof. See Appendix. The following Theorem and Corollary show that if restrictions imposed on signs of all shocks, then just two restricted IRF vectors is enough to get non-trivial identified set or even point identification. 12

13 Theorem 3. Suppose that there are two IRF vectors c 1 and c 2 with sign restrictions on every component such that for p 1 the following inequality is true c 1 h p c 2 h p 0. Then the set Θ belongs to the set defined by the system h h = 1 ( 1 (c 1 h) 2)( 1 (c 2 h) 2) c 1 c 2 ( c 1 h )( c 2 h ) 0 (4.1) Proof. See Appendix.. Corollary 1. Suppose that c 1 = c 2, then Θ {c 1 }. Proof. See Appendix. Theorems 4 5 show that the coneθ can be non-convex as opposed to the results form Section 3. Theorem 4. Consider the set A of vectors c from C with restrictions on every component and the same signs restrictions on the corresponding components. Then the set Θ does not have any points inside the interior of the coneā. Proof. See Appendix. The Theorem has Corollary that demonstrates that under curtain conditions the set Θ is disjoint, has finite number of points and corresponds to the so called locally identified case discussed in Rubio-Ramirez et al. (2010). Corollary 2. Suppose that there exist q different orthogonal vectors ( c 1,...,c q) in C with restrictions on every component and the same signs restrictions on the corresponding components. Then Θ { c 1,...,c q}. Proof. See Appendix. Theorem 5. Suppose that there are two IRF vectors c 1 and c 2 with restrictions on every component and the same signs restrictions on the corresponding components. Then the set Θ does not contain any elements inside the interior of the set defined by the system: h h = 1 ( 1 (c 1 h) 2)( 1 (c 2 h) 2) c 1 c 2 ( c 1 h )( c 2 h ) 0 (4.2) Proof. See Appendix. 13

14 Corollary 3. Suppose that there exist two orthogonal vectors c 1 and c 2 in C with the same sign restrictions on every component. Then every element of the set Θ must be orthogonal to either c 1 or c 2. Proof. See Appendix. The last result in this Section shows sufficient condition under which the dimension of the set Θ is less than q. Theorem 6. Suppose that there exist m linearly independent vectors ( c 1,...,c m) in C, 1 m q 1, with the same sign restriction on some component p 1 and a vector c in C with different sign restrictions on the p-th component and c cone ( c 1,...,c m). Then the p-th component of the vector c is equal to zero in the structural representation. If the number of components of c with different signs is k, m = q k and c belongs to the interior of cone ( c 1,...,c q k), then the corresponding components of ( c 1,...,c q k) are equal to zero in the structural representation and Θ span ( c 1,...,c q k). Proof. See Appendix. Remark 2. Note that I do not need to impose restrictions on every component of the IRF vectors to apply Theorem 6 unlike Theorems 2 5 with Corollaries1 3. If the signs of some components of some IRF vector are unknown, one can consider all possible combinations of signs of the unknown components indexed with i and apply results from this section to get conditional sets Θ i. The set Θ then would be a union the sets Θ i. For some sign patterns the resulting set Θ i can be empty. Remark 3. Theorem 6 provides a way to get zero restrictions implied by a set of sign restrictions. Once a number of zero restrictions is available one can employ the necessary and sufficient rank condition for point identification under zero restrictions proposed by Rubio-Ramirez et al. (2010). If the set of implied zero restrictions will satisfy the rank condition then the original sign restrictions would imply point identification of h Two dimensional case In the case q = 2 one can get a closed form representation of the set Θ. It is instructive to consider the case with two IRF vectors and two shocks similar to the setup of Figure 2. For now restrictions b 1 1 0, b1 2 0, b2 1 0, b Note that in the two dimensional case the second column of the orthogonal matrix H can take only two values ( h 1 2, h1 1) or ( h 1 2,h 1 1). In the first case the following 14

15 system corresponds to the sign restrictions b 2 1 = c2 1 h1 1 +c2 2 h1 2 0 b 2 2 = c2 1 h1 2 +c2 2 h1 1 0 (4.3) b 1 1 = c1 1 h1 1 +c1 2 h1 2 0 b 1 2 = c1 1 h1 2 c1 2 h = h 1 h 1 (4.4) One can think about the second and the fourth inequalities as of scalar products of additional vectors 3 c 2 = ( c 2 2 1), c2 and c 1 = ( c 1 2 1),c1 and the column h 1. That allows one to think about the setup with restrictions on two shocks using the setup developed in Section 3 for the case of restrictions on one shock only. cone( C) h 1 2 c 1 c z c 2 φ h 1 1 c 1 Figure 4: The set-identified case with J = 4, q = 2 and restrictions imposed on signs of both components of IRF vectors. The red arc corresponds to the set Θ, the solution to the system (4.3) (4.4) in terms of a non-structural representation. It determines the set of all possible angles φ and the set of all structural representations consistent with the reduced form. Compare Figure 4 with Figure 2. Additional restrictions on signs of IRF for the second shocks made the set Θ considerably smaller. There is a nuance, however. If one would choose ( h 1 2,h1) 1 instead of ( h 1 2, h1) 1 then the set Θ would be different. In particular, for Figure 4 the set Θ under choice of ( h 2,h 1 ) would be an empty set. In fact, if the restrictions are consistent with each other then either only one choice of orthogonal vectors is consistent with the restrictions or both options give the same set 3 Note that I use the index for the right orthogonal vector and for left orthogonal vector. 15

16 Θ to be consistent with Theorems 2 and Three dimensional case In the case of three shocks one can get explicit geometric characterization of the set Θ. In the structural representation after normalization b 1 0 every IRF vector lies in one the four octants (b 2 0,b 3 0), (b 2 0,b 3 0), (b 2 0,b 3 0), (b 2 0,b 3 0) depending on imposed restrictions. Every orthogonal matrix H can be described in terms of three orthonormal vectors that define the structural coordinate system in terms of non-structural. To find all possible matrices H = ( h 1 h 2 h 3) consistent with the sign restrictions on all components of IRF vectors one need to consider all coordinate systems defined by a basis ( h 1 h 2 h 3) in which every IRF vector lies in the octants prescribed by the system of sign restrictions. In general, for vectors h 1 Θ there will be many pairs of orthogonal vectors. For example, consider the case with three IRF vectors b ++,b + and b. The first index corresponds to the sign restriction on the second components and the second index correspond to the sign restriction on the third component. In the non-structural representation the corresponding IRF vectors c ++,c + and c, in general, will not have the same signs as in the structural representation. Figure 5 depicts normalized IRF vectors c ++,c + and c on the unit sphere and one particular choice of the vector h 1 consistent with the sign restrictions. The set Θ constructed based on Theorem 2 and Proposition 2, which is the Corollary of the Theorem 5. Note that the vectorsc ++,c + and c lie in the corresponding octants showed on the picture with red dashed lines. h 3 h 2 c ++ 0 c + h 1 c c ++ c Figure 5: The set-identified case with J = 9, q = 3 and restrictions on all components of the IRF. The red solid line corresponds to the set Θ of all vectors h 1 consistent with the sign restrictions. In this case the set Θ consists of two parts. 16

17 In the three dimensional case one can get a parametric curve representation of the set 4.1 which can be helpful in applications: Proposition 2. If q = 3 then the set (4.1) has form h 1 = xc 1 +yc 2 +zc 3 x [0,1] y = (1+cos2 Γ)x+ x 2 sin 4 Γ+4cos 2 Γ 2cosΓ z = ± 1 x 2 y 2 2xycosΓ (4.5) where c 3 = c2 c 1 c 1 c 2 and Γ = arcsin( c 1 c 2 ). Proof. See Appendix. 4.4 Implications for point identification In the previous subsections I provide several results demonstrating that additional sign restrictions on other components of IRF vectors can dramatically shrink the set Θ and thus the set of all structural impulse response functions for the the first shock consistent with the restrictions. Corollaries 1 3 provide additional sufficient conditions for (local) point identification which would not be possible if one would impose restrictions only on the IRF components for the first shock. These conditions, however, are not necessary as the following Theorem and Remark 4 show. Theorem 7 (Necessary conditions for point identification). Suppose that the set Θ = { h 1}. Then for every shock p 1 there are two restrictions b il p 0 and b mk p conditions holds: 0. Also at least one of the following Exists a binding sign restriction on the first component, i.e. a restricted IRF vector c il such that q j=1 cil j h1 j = 0 For every shock p 1 exists a binding sign restriction on p-th component, i.e. a restricted IRF vector c il such that q j=1 cil j hp j = 0. Proof. See Appendix. Remark 4. Theorem 1 and Corollaries Corollary 1 Corollary 3 provide sufficient conditions for the set Θ to be a singleton. These conditions are consistent with the necessary conditions from Theorem 7. However, there are other setups in which the set Θ is a singleton. In particular, if one can propose a set 17

18 of restrictions under which the set Θ have isolated points, additional sign restrictions can eliminate all points but one and these additional sign restrictions do not have to be binding on the points from Θ. Consider the example of a system which is locally identified 4 but not globally identified considered in Rubio-Ramirez et al. (2010). Although the original setup for this example is a system of simultaneous equations, exactly the same problem appears for a three dimensional VAR system with the IRF matrix B 0 in structural form 2.4 satisfying zero restrictions: B 0 = (4.6) Rubio-Ramirez et al. (2010) show that these restrictions generically lead to local identification but not to global identification, or simply identification. They use a numerical example to demonstrate this point. Consider matrices B 0 = and P = 2/3 2/3 1/3 1/3 2/3 2/3 2/3 1/3 2/3 The matrix B 0 satisfies the restrictions(4.6). Suppose that there is an orthogonal matrix H that converts the corresponding reduced form parameters C 0 to structural form representation B 0 = C 0 H. But since B 0 = B 0 P = B 0, H = HP also satisfies the restrictions (4.6) so the set Θ has at least two points and the other IRF are not identified globally. However Rubio-Ramirez et al. (2010) shows that generically the restrictions (4.6) identify the IRF locally. One can consider this setup using the framework, developed ( ) in Sections Section 3 and Section 4. One can treat the rows of the matrix C 0 = c 2 c 3 c 1 as the reduced form IRF vectors. The zero restrictions (4.6) are equivalent to the following set of sign restrictions after normalization c i h 1 0 c 1 h 1 0 and c 1 h 1 0 (4.7) c 2 h 2 0 and c 2 h 2 0 (4.8) c 3 h 3 0 and c 3 h 3 0. (4.9) Suppose, in addition, that c 2 h 3 0 and c 3 h 2 0. Combined with c 2 h 2 0 and c 3 h 3 0 by Theorem 5 they restrict the set Θ to lie outside of the interior of the set (4.1) or (4.5) constructed around c 2 and c 3. In this case c 2 h 2 0 and c 3 h 3 0 Theorem 3 implies that Θ lies inside the set (4.5) constructed around c 2 and c 3. So the set Θ lies on the border of the set (4.5). The inequalities(4.7) 4 i.e. there is no other point in the neighborhood of the solution in the parameter space that is consistent with the restrictions 18

19 imply that the set Θ lies on the plane c 1 h 1 = 0. Figure (6) shows that these restrictions imply that the set Θ consists of two isolated points, i.e. the IRF for the first shock is locally identified. If one can add additional sign restriction c 4 h 1 0 which would separate the two points in the set Θ, then one would get point identification. Figure (6) demonstrates one particular example of such c 4. Note, while finite number of sign restrictions generically do not provide even local identification, as Rubio-Ramirez et al. (2010) claim, if one can locally identify a IRF using zero restrictions, a small number of sign restrictions can help to eliminate multiplicity even when the sufficient condition for global identification of Rubio-Ramirez et al. (2010) is not satisfied. c 1 c 2 0 c 4 c 3 Figure 6: In this case the set Θ has two isolated points shown as a bold dot and a cross. The additional sign restriction corresponds to the green hyperplane which leaves only one solution. 5 Relationship to Exclusion Restrictions Theorem 7 shows that one can obtain point identification only if a subset of the sign restrictions become binding, i.e. the limiting inequalities become equalities as number of restrictions grows to infinity. If information about these restrictions would be available in advance, the set of sign restrictions could be easily replaced by these equality restrictions and theory for zero restrictions from Rubio-Ramirez et al. (2010) could be applied. So in cases when the point identification using sign restrictions is possible, in fact, in the limit there are implicit exclusion restrictions, which are sufficient to achieve point identification of the IRF. This interpretation gives a clue on how to choose sign restrictions efficiently. The best sign restrictions are those close to zero restrictions. The main reason why one would use sign restrictions instead of zero restrictions to get point identi- 19

20 fication is the following. In some cases, it is a priori unclear which IRF are equal to zero, while instead there can be a set of candidates with non-negativity restrictions. If among these candidates there will be combinations of time series and lags for which the impact is in fact zero or very close to zero, then one can get point identification without imposing zero restrictions from the very beginning, i.e. choice of the equality restrictions can be data driven. If information about exclusion restrictions is available a priori then one should utilize it since every zero restriction represented a pair of opposite binding inequalities, as (4) demonstrates. Theorem 6 demonstrates particular usefulness of such restrictions.these restrictions can reduce the dimensionality of the sign identification problem as it was shown in Moon et al. (2013). Note also that a binding sign restriction is not equivalent to a zero restriction, since every exclusion restriction is equivalent to two binding sign restrictions. For this reason, one need to have more potentially binding sign restrictions to get point identification than in the case of zero restrictions. For example, Figure 3 shows that in the two dimensional case one need to have two sign restrictions instead of only one zero restriction required by the recursive identification scheme. It is not surprising, since the corresponding zero restriction would imply exactly the same pair of opposite sign restrictions. In higher dimensional cases the number of sign restrictions necessary for point identification is smaller than the number of sign restrictions implied by zero restrictions necessary for point identification. In this sense assumptions about signs are less restrictive than zero restrictions. Remark 4 shows a case where exclusion restrictions are not enough to get point identification but with additional sign restriction one gets unique solution. It shows that not only binding sign restrictions are important for point identification and non-binding restrictions can be employed to make a locally identified IRF globally identified. A particular example of exclusion restrictions is restrictions on exogenous instrumental variables. Such variables are correlated only with one particular shock and uncorrelated with other shocks. The assumption about exogenous instruments were recently used for identification in dynamic models by Stock and Watson (2012) and Mertens and Ravn (2013). Stock and Watson (2012) show, however, that statistical tests employing overidentifying restrictions reject exogeneity of their external instruments and assumption zero restrictions is misspecified. Instead of assuming exogenous instruments one can include potentially valid instruments into the VAR or SFM model and impose only one-sided sign restrictions on IRF for these variables. If instruments are valid, then the sign restrictions will be binding and can provide point identification by results provided in Section 4. If some of them will turn out to be non exogenous, still one can get tight set estimates for IRF. 20

21 6 Application 6.1 Implication for Estimation of the an Impact of a Monetary Shock The framework of Example 2 can be used to study impact of monetary policy shock. One can consider an extension of the hybrid scheme of Moon et al. (2013) in the SFM framework. Consider the SFM model with three common shocks(u rt,u πt,u yt ) and four sets of observed time series,{r it },{m it },{π it } and {y it }. The shock u rt is the shock in monetary policy, u πt is the shock in inflation expectations, and u rt is the productivity shock. The set {r it } consists of panel of different interest rates from the same economy. It includes Fed funds rate, T-bills returns with different maturities and other interest rates rates. The set {π it }, like the one considered in Reis and Watson (2010), consists of differences in logarithms of individual prices. The set {m it } consists of real money balances corresponding to different combinations of monetary aggregates and deflators. The set {y it } consists of different measures of economic activity which includes industrial production indices, unemployment rate and others. The model has the following MA form: r it = B ri (L) u rt u πt +ξr it, π it = B πi (L) u rt u πt +ξr it, u yt u yt m it = B mi (L) u rt u πt +ξm it, y it = B yi (L) u rt u πt +ξy it, u yt u yt where the matrix lag polynomials ( B ri (L),B πi (L),B mi (L),B yi (L) ) are products of factor loadings and IRF for the underlying static factors, and ξit r,ξπ it,ξm it,ξy it are idiosyncratic corresponding to the groups of time series {r it },{m it },{π it } and {y it }. Consider identification restrictions consistent with those of Moon et al. (2013): b (r)i0 r b (π)i0 r b (m)i0 r b (y)i0 r b (r)i0 π b (π)i0 π b (m)i0 π b (y)i0 π b (r)i0 y b (π)i0 y b (m)i0 y b (y)i0 y = , (6.1) where, for example, b (r)i0 π is the simultaneous impact of the shock u πt on the interest rate series with number i, and in the RHS 0 stands for an exclusion restriction, +/ for sign restrictions, and for no restriction. The exclusion restrictions can uniquely identify the impact of the productivity shock and the dimension of the sign identification problem is q = 2. 21

22 { } Assume that (1,0) 1 is a limiting vector of the set b π (r)i0 (b(r)i0 r,b π (r)i0 ), and (0, 1) is a limit { } 1 vector of the set,b π (m)i0 ). The first assumption implies that there exist an interest b (m)i0 r (b(m)i0 r rate that do not react to the inflation expectation shock when compared to their reaction to the monetary policy shock. Probably yields for long-term bonds can be relatively insensitive to short-term shocks in inflationary expectations since they depend on expectations of average inflation over long period of time. The second assumption implies that some monetary aggregate react to short term interest with at least one period lag. Note that one does not have to know these series in advance and that the data can select them. The larger the number of series in the sets {r it } and {m it }, the higher the chance that these assumptions will be satisfied in data. If the assumptions are satisfied, then by Corollary 2 in the set Θ there will be only two isolated points. Sign restrictions on B πi (0) will rule out one of these points and will point-identify the IRF for monetary and expectations shocks. This identification scheme can take advantage of big data sets employed in Bernanke et al. (2005) and Forni et al. (2009) while being agnostic like the scheme of Uhlig (2005) and Moon et al. (2013). My approach is also different from Amir Ahmadi and Uhlig (2009) because it utilizes a large set of sign restrictions on disaggregated data to achieve point identification. 7 Implications for Estimation and Inference Examples from Section 3 and Section 6 are the cases where point identification is achievable using only sign restrictions. They suggest that sign restrictions are more likely to provide identification in cases where the number of reasonable sign restrictions is large and when many of them are nearly binding restrictions. This setup allows one to use estimation and inference methods for the point-identified case (Canova (2011); Forni et al. (2009)). In other cases if one chooses to use sign restrictions, one has to rely on methods for set-identified parameters proposed in Moon et al. (2013). Uniform inference methods that can give valid confidence intervals under both point and partial-identified cases for infinite number of restrictions in the dynamic models are not developed yet. Moon et al. (2013) provide an estimation procedure for the SVAR model. The estimation procedure for SFM can be made in a similar fashion in two steps. In the first step one can estimate any non-structural fundamental representation of the model. To estimate consistently the non-structural representation of SFM, one can use methods for estimation of exactly identified parameters developed in Forni et al. (2009). In the second step one imposes sign restrictions on the non-structural estimates to convert it into the structural one. To do so, one can use the algorithms proposed in Rubio-Ramirez et al. (2010). In the finite sample the set of sign inequalities can be inconsistent even if it is consistent in the population. So one will have to introduce some loss function that would penalize for violation of the constraints like in 22

23 Moon et al. (2013) to estimate the rotation matrix H. The only difference here from the set-estimator of Moon et al. (2013) is that if Assumption H or any sufficient condition from Section 4 is satisfied then any point from their set-estimator will provide a consistent estimate of the IRF. Thus for bootstrap-type inference I can just resample point estimates, corresponding to a particular selection rule for a point from Θ. The corresponding confidence intervals on IRF by construction will be tighter that ones obtained by bootstraping set estimates. The results from Sections 3 and 4 provide alternative way to do the second step. One can collect the sign restrictions into groups that satisfy the assumptions of the theorems from Section 4 to find analytic bounds on the set Θ. Such analytic bounds can be faster to compute than the bounds based on a loss function minimization employed by Moon et al. (2013). However one need to develop an numerical algorithm to do implement this method. The analytic bounds on the set Θ also can be used potentially to improve the inference about the set estimates for IRF. 8 Conclusion In this paper I provide the necessary and sufficient conditions for point identification of impulse response functions in the SVAR and SFM using sign restrictions. The necessary condition implies that implicitly a set exclusion restrictions holds in the limit. One might use this approach in cases when a justifiable set exclusion restrictions cannot be proposed based on theory while a large set of sign restrictions can be justified. In general, the sufficient conditions for point identification with sign restrictions are less restrictive than the necessary conditions for zero restrictions. For example, sign restrictions can be employed to make a IRF that are locally-identified with zero restrictions globally-identified. I provide examples where sign restrictions can guarantee point identification. One of them can be used in evaluation of dynamic effects of monetary policy that can take advantage of disaggregated data while being agnostic about identification assumptions. The closed form representation of the identified set of IRF under set identification demonstrate that if restrictions imposed on all shocks the set of IRF for a given shock can be significantly tighter when compared with restrictions on IRF for this shock only. References Amir Ahmadi, P. and H. Uhlig (2009): Measuring the dynamic effects of monetary policy shocks: A Bayesian FAVAR approach with sign restriction, Tech. rep., mimeo, Humboldt University of Berlin. Bernanke, B. S., J. Boivin, and P. Eliasz (2005): Measuring the effects of monetary policy: a factor-augmented vector autoregressive (FAVAR) approach, The Quarterly Journal of Economics, 120,