Corrigendum to Inference on impulse. response functions in structural VAR models. [J. Econometrics 177 (2013), 1-13]

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1 Corrigendum to Inference on impulse response functions in structural VAR models [J. Econometrics 177 (2013), 1-13] Atsushi Inoue a Lutz Kilian b a Department of Economics, Vanderbilt University, Nashville TN b University of Michigan, Department of Economics, Ann Arbor, MI JEL classification: C32; C52; E37 Keywords: Vector autoregression; Simultaneous inference; Impulse responses; Sign restrictions; Median; Mode; Credible set Corresponding author: Lutz Kilian, University of Michigan, Department of Economics, 309 Lorch Hall, 611 Tappan Street, Ann Arbor, MI Tel: Fax: lkilian@umich.edu. 1

2 Proposition 1 in Inoue and Kilian (2013) for the posterior density of the set of structural impulse responses in the fully sign-identified vector autoregressive (VAR) model is not correct as stated. The correct statement and proof is provided below. The correction does not affect the substance of the empirical findings based on this proposition. The statement and proof of Proposition 2 in the original article and the empirical results based on that proposition are not affected. Consider an n-dimensional VAR(p) model with an intercept. Ignoring the intercept for notational convenience, let B = [B 1 B p ] denote the slope parameters of the VAR model where B i is an n n matrix for i = 1, 2,..., n. Let Σ denote the error covariance matrix. Let A be the lower-triangular Cholesky decomposition of Σ such that AA = Σ, and let vech(a) denote the n(n + 1)/2 1 vector that consists of the on-diagonal elements and the below-diagonal elements of A. Let n n rotation matrix U be an element of O(n) (i.e., U is orthogonal and satisfies U U = I n ) and be Haar-distributed over O(n). By construction, we have U = 1 with probability with probability 1 2 Define Ũ = U if U = 1 W U if U = 1 2

3 and à = A if U = 1 AW if U = 1 where W is the n n diagonal matrix whose diagonal elements whose first diagonal element is -1 and other diagonal elements are 1. Then Ũ belongs to SO(n) (Ũ Ũ = 1 and Ũ = 1) and we have ÃŨ = AU. Note that U is Haardistributed when U = 1 where U belongs to O(n). Note also that W U is Haar-distributed when U = 1. Thus, Ũ is also Haar-distributed on SO(n) even unconditionally. Moreover, conditional on U = 1 or U = 1, Ũ has a one-to-one relationship with U and the Jacobian of the transformation from U to Ũ is always one. Because Ũ belongs to SO(n), S = I n 2(I n + Ũ) 1 (1) is an n n skew-symmetric matrix (León et al., 2006, p. 414). Let s denote the n(n 1)/2 1 vector that consists of the below-diagonal elements of S. the density of s is given by f(s) = ( ) Π n Γ(i/2) 2 (n 1)(n 2)/2 i=2 (2) π i/2 I n + S n 1 (see equation 4 of León et al. (2006), p. 415). 1 1 León et al. (2006, p. 416) define s as the vector that consists of the above-diagonal elements of S. Let s leon denote this vector. Because S is skew-symmetric, there is a n(n 1)/2 n(n 1)/2 permutation matrix P such that s leon = P s. Because the 3

4 Let Φ = [Φ 1 Φ 2 Φ p] where Φ i is the ith reduced-form vector moving average coefficient matrix. There is a one-to-one mapping between the first p + 1 structural impulse responses Θ = [A, Φ 1 A, Φ 2 A,, Φ p A], where A = ÃŨ, on the one hand, and the tuple formed by the reduced-form VAR parameters and s, (B, vech(a), s), on the other. The nonlinear function Θ = h(b, vech(ã), s) is known. Using the change-of-variables method, the posterior density f of Θ can be written as f( Θ) = = [vec(b) vech(ã) s ] vec( Θ) f(b, vech(ã), s) vec( Θ) 1 vech(σ) [vec(b) vech(ã) s ] f(b, Σ, s) vech(ã) vec( Θ) 1 vech(σ) [vec(b) vech(ã) s ] f(b Σ)f(Σ)f(s), (3) vech(ã) where B, Σ = ÃÃ, and Ũ are the unique values that satisfy the nonlinear function Θ = h(b, vech(ã), s). The conditioning on the data is omitted for notational simplicity. Let D n denote the n 2 n(n + 1)/2 duplication matrix of zeros and ones such that vec(m) = D n vech(m) for any n n symmetric matrix M (see Definition 4.1 of Magnus, 1988, p. 55). D + n denotes the Moore-Penrose inverse of D n, i.e., D + n = (D nd n ) 1 D n, so that we can write vech(m) = D + n vec(m) (see Theorem 4.1 of Magnus, 1988, p. 56). Similarly, D n is the n 2 n(n 1)/2 absolute value of the determinant of P is always 1, the density remains the same as in (2) even when s is defined to be the vector obtained from the below-diagonal elements of S. 4

5 matrix such that vec(s) = D n s (see Definition 6.1 in Magnus, 1988, p. 94) and D n + = ( D n D n ) 1 D n. Let E ij = e i e j denote a square matrix with the (i, j)th element equal to one and zeros elsewhere. Let L n denote the n(n + 1)/2 n 2 elimination matrix of zeros and ones such that vec(m) = L nvech(m) for any lower triangular matrix M (see Definition 5.1 of Magnus, 1988, p. 76). K n denotes the n 2 n 2 communication matrix such that vec(m ) = K n vec(m) for any n n matrix M (see Magnus and Neudecker, 1999, pp ). Proposition 1. The posterior density of Θ is f( Θ) = 2 n(n+1) 2 Ã (np 1) I n + Ũ (n 1) f(b Σ)f(Σ)f(s). (4) Proof of Proposition 1: First, we will show that f( Θ) = = vec( Θ) 1 vech(σ) [vec(b) vech(ã) s ] f(b Σ)f(Σ)f(s) ( vech(ã) ( Ũ I n )L n (I n Ã)J U Ã np ) 1 D n + [(Ã I n) + (I n Ã)K n]l n f(b Σ)f(Σ)f(s), (5) where J U = 2[(I n S ) 1 (I n S) 1 ] D n. Because Ũ = 2(I n S) 1 I n, dũ = 2(I n S) 1 (ds)(i n S) 1, (6) 5

6 and thus vec(dũ) = 2[(I n S) 1 (I n S) 1 ]vec(ds) = 2[(I n S) 1 (I n S) 1 ] D n ds = J U ds. (7) Because d(ãũ) = (dã)ũ + Ã(dŨ) = (dã)ũ + 2Ã(I n S) 1 (ds)(i n S) 1,(8) d(φãũ) = (dφ)ãũ + Φ(dÃ)Ũ + ΦÃ(dŨ) = (dφ)ãũ + Φ(dÃ)Ũ + 2ΦÃ(I n S) 1 (ds)(i n S) 1, (9) it follows that dvec(ãũ) = (Ũ I n )dvec(ã) + 2(I n Ã)((I n S ) 1 (I n S) 1 )dvec(s), = (Ũ I n )L ndvech(ã) + 2(I n Ã)((I n S ) 1 (I n S) 1 ) D n ds, = (Ũ I n )L ndvech(ã) + (I n Ã)J Uds, (10) dvec(φãũ) = (Ũ Ã I np )dvec(φ) + (Ũ Φ)dvec(Ã) +2(I n ΦÃ)((I n S ) 1 (I n S) 1 )dvec(s) = (Ũ Ã I np )dvec(φ) + (Ũ Φ)L ndvech(ã) + (I n ΦÃ)J Uds. (11) 6

7 It follows from expressions (10) and (11) that J 1 vec( Θ) [vech(ã) s vec(φ) ] = (Ũ I n )L n (I n Ã)J U O n 2 n 2 p (Ũ Φ)L n (I n ΦÃ)J U Ũ Ã I np (12). Since the upper-left submatrix, [(Ũ I n )L n (I n Ã)J U], is n 2 n 2, the upper-right submatrix is the n 2 n 2 p submatrix of zeros, and the lower-right submatrix, Ũ Ã I np, is n 2 p n 2 p, J 1 is block lower triangular. Thus its determinant is given by the product of the determinants of its blocks: J 1 = = (Ũ I n )L n (Ũ I n )L n (I n Ã)J U Ũ Ã I np (I n Ã)J U Ã np. (13) Because of the recursive relationship defined by equations (11), (14) and (17) in Inoue and Kilian (2013), the Jacobian matrix of Φ with respect to B is block-diagonal and each diagonal block has a unit determinant. Thus J 2 vec(φ) vec(b) = 1. (14) Since the Jacobian of vec(σ) with respect to vec(ã) is [(Ã I n) + (I n Ã)K n], (15) the determinant of the Jacobian of vech(σ) with respect to vech(ã) is given 7

8 by J 3 D n + [(Ã I n) + (I n Ã)K n]l n (16) Thus, expression (5) follows from expressions (13), (14) and (16). Next, we will show that the absolute values of two of the determinants in expression (5) can be written as: ( ) n(n 1) (Ũ I n )L n (I n Ã)J 1 2 U = I n + 2 Ũ n 1 Π n 1 i=1 ãn i ii,(17) D n + [(Ã I n) + (I n Ã)K n]l n = 2 n Π n i=1ã n i+1 ii, (18) where ã ii is the (i, i)th element of Ã. Express the left-hand side of equation (17) as [(Ũ I n )L n (I n Ã)J U] = (Ũ I n )Z, (19) where Z = [L n 2(F G) D n ], (20) F = Ũ(I n S ) 1 = 1 2 (I n + Ũ) = (I n S) 1, (21) G = Ã(I n S) 1 = ÃF. (22) where F and G are well-defined because I n +Ũ is nonsingular with probability 8

9 one. Then the determinant of (Ũ I n )Z equals that of Z. Because Z Z = I n(n+1) 2 2L n (F G) D n 2 D n(f G )L n 4 D n(f F G G) D n, (23) we have Z 2 = Z Z = 2 n(n 1) D n(f G )(I n 2 L nl n )(F G) D n (24) Because L nl n is a diagonal matrix with zeros and ones on the diagonal, so is I n 2 L nl n. Thus I n 2 L nl n = i>j E ii E jj, (25) where the summation is over i, j = 1, 2,...n such that i > j. There exists an n(n 1)/2 n matrix, say Q, that contains only zeros and ones and has full row rank such that QQ = I n(n 1), Q Q = I n 2 L nl n = (E ii E jj ). (26) 2 i>j Thus, using Q, we can write D n(f G )(I n 2 L nl n )(F G) D n = D n(f G )Q Q(F G) D n 9

10 = D n(f F )(I n à )Q Q(I n Ã)(F F ) D n = ( D + n (F F ) D n ) ( D n(i n à )Q )(Q(I n Ã) D n )( D + n (F F ) D n ), (27) where Theorem 6.11(i) of Magnus (1988, p. 100) is used in deriving the last equality. Let ũ ij denote the vector of the below-diagonal elements of E ij where i > j. Then Q = i<j ũ ji vec(e ij ) = i>j ũ ij vec(e ji ), (28) and D n = i>j vec(e ij E ji )ũ ij, (29) where the latter equality follows from Theorem 6.1 of Magnus (1988, p. 95). Hence Q(I n Ã) D n = ũ ij vec(e ji ) (I n Ã)vec(E st E ts )ũ st i>j s>t = ũ ij (e i e j)(i n Ã)(e t e s e s e t )ũ st i>j s>t = ũ ij (e ie t )(e jãe s)ũ st ũ ij (e ie s )(e jãe t)ũ st i>j s>t i>j s>t = ã js ũ ij ũ st ã jt ũ ij ũ it = ã jt ũ ij ũ it s>i>j i>j i>t i>j t = ã jj ũ ij ũ ij ã jt ũ ij ũ it. (30) i>j i>j>t 10

11 Element i + f(j) of ũ ij is unity, where f(j) = (j 1)n j(j + 1), (31) 2 and all other elements are zero. Note that f(j + 1) = f(j) j + n 1 such that f(j + 1) > f(j) if j < n 1 f(j + 1) = f(j) if j = n 1 f(j + 1) < f(j) if j = n. (32) This implies that the matrix ũ ij ũ it is lower triangular for i > j > t. Moreover, ũ ij ũ ij is diagonal for i > j. Hence the matrix Q(I n Ã) D n is lower triangular and thus Q(I n Ã) D n = ( 1) n(n 1) 2 ã jj ũ ij ũ ij = ( 1) n(n 1) 2 Π n 1 j=1 ãn j jj (33) i>j It follows from (27) and (33) that Z 2 = 2 n(n 1) D n(f G )(I n L nl n )(F G) D n = 2 n(n 1) D + n (F F ) D n 2 Q(I n Ã) D n 2 = 2 n(n 1) F 2(n 1) Q(I n Ã) D n 2 = 2 n(n 1) I n + Ũ 2(n 1) Q(I n Ã) D n 2 = 2 n(n 1) I n + Ũ 2(n 1) (Π n 1 j=1 ãn j jj ) 2, (34) 11

12 where Theorem 6.13(iii) of Magnus (1988, p. 100) is used in deriving the third equality. Thus equation (17) follows from equations (19) and (34). Turning to equation (18), it follows from Theorem 3.1 and Theorem 4.2(ii) in Magnus (1988) that D + n [(Ã I n) + (I n Ã)K n]l n = D + n [(Ã I n) + K n (Ã I n)]l n = D + n (I n + K n )(Ã I n)l n = 2D + n (Ã I n)l n = 2(D nd n ) 1 (L n (Ã I n )D n ). (35) Given Theorem 4.4(iii) and Theorem 5.12 in Magnus (1988), we have that D n + [(Ã I n) + (I n Ã)K n]l n = 2(D nd n ) 1 (L n (Ã I n )D n ) = 2 n(n+1)/2 2 n(n 1)/2 Ln (Ã I n )D n ) = 2 n Π n i=1ã n i+1 ii (36) which proves equation (18). Lastly, combining equations (5), (17) and (18), we obtain the desired result. Acknowledgments We thank Tom Doan, Jan Magnus and Jonas Arias for helpful discussions. 12

13 References 1. Inoue, A., Kilian, L., Inference on impulse response functions in structural VAR models. Journal of Econometrics 177, León, C.A., Massé, J.-C., Rivest, L.P., A statistical model for random rotations. Journal of Multivariate Analysis 976, Magnus, J.R., Linear structures. Oxford University Press, Oxford, UK. 4. Magnus, J.R., Neudecker, H., Matrix differential calculus with applications in statistics and econometrics. second edition. Wiley, Chichester, UK. 13