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

Transcription

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 Proposition 1 in Inoue and Kilian (2013) for the posterior density of the set of structural impulse responses in the fully sign-identified 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 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 model. 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. We assume that the n n real rotation matrix U is orthogonal, U U I n, and that its determinant is U 1. Then S I n 2(I n + U) 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. Then U and s have a one-to-one relationship. When U is uniformly distributed over the space of n n real matrices such that U U I n and Tel: lkilian@umich.edu. 1

2 U 1, the density of s is given by f(s) ( Π n i2 ) Γ(i/2) 2 (n 1)(n 2)/2 (2) π i/2 I n + S n 1 (see equation 4 of León et al. (2006), p. 415). 1 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 AU, 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(a), s) is known. Using the change-of-variables method, the posterior density f of Θ can be written as f( [vec(b) vech(a) s ] vec( f(b, vech(a), s) vec( f(b, Σ, s) vec( f(b Σ)f(Σ)f(s), (3) where B, Σ AA, and U are the unique values that satisfy the nonlinear function Θ h(b, vech(a), 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). 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 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 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. 2

3 vec(m ) K n vec(m) for any n n matrix M (see Magnus and Neudecker (1999), pp ). Dn is the n 2 n(n 1)/2 matrix such that vec(s) D n s (see Definition 6.1 in Magnus (1988), p. 94). Proposition 1. The posterior density of Θ is f( vec( f(b Σ)f(Σ)f(s) ( (U I n )L n (I n A)J U A np ) 1 D + n [(A I n ) + (I n A)K n ]L n f(b Σ)f(Σ)f(s) (4) where J U 2[(I n S ) 1 (I n S) 1 ] D n. Derivation of the Jacobian matrix in Proposition 1: Because U 2(I n S) 1 I n, du 2(I n S) 1 (ds)(i n S) 1, (5) and thus Because vec(du) 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. (6) d(au) (da)u + A(dU) (da)u + 2A(I n S) 1 (ds)(i n S) 1,(7) d(φau) (dφ)au + Φ(dA)U + ΦA(dU) it follows that (dφ)au + Φ(dA)U + 2ΦA(I n S) 1 (ds)(i n S) 1, (8) dvec(au) (U I n )dvec(a) + 2(I n A)((I n S ) 1 (I n S) 1 )dvec(s), (U I n )L ndvech(a) + 2(I n A)((I n S ) 1 (I n S) 1 ) D n ds, (U I n )L ndvech(a) + 2(I n A)J U ds, (9) dvec(φau) (U A I np )dvec(φ) + (U Φ)dvec(A) +2(I n ΦA)((I n S ) 1 (I n S) 1 )dvec(s) (U A I np )dvec(φ) + (U Φ)L ndvech(a) + (I n ΦA)J U ds.(10) 3

4 It follows from expressions (9) and (10) that vec( J 1 [ (U I n )L ] n (I n A)J U O n 2 n 2 p [vech(a) s vec(φ) ] (U Φ)L n (I n ΦA)J U U A. I np (11) Since the upper-left submatrix, [(U I n )L n (I n A)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, U A I np, is n 2 p n 2 p, (11) is block lower triangular. Thus its determinant is given by the product of determinants: J 1 (U I n )L n (I n A)J U U A I np (U I n )L n (I n A)J U A np. (12) Because of the recursive relationship defined by equations (11), (14) and (17) in Inoue and Kilian (2013), the Jacobian matrix of θ with respect to B is block-diagonal and each diagonal block has unit determinant. Thus vec(φ) J 2 1. (13) vec(b) Since the Jacobian of vec(σ) with respect to vec(a) is [(A I n ) + (I n A)K n ], (14) the determinant of the Jacobian of vech(σ) with respect to vech(a) is given by J 3 D n + [(A I n ) + (I n A)K n ]L n (15) Thus, Proposition 1 follows from expressions (12), (13) and (15). Acknowledgments We thank Tom Doan, Jan Magnus and Jonas Arias for helpful discussions. 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,

5 3. Magnus, J.R., Neudecker, H., Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley, Chichester, UK. 4. Magnus, J.R., Linear Structures. Oxford University Press, Oxford, UK. 5