Choosing between identification schemes in noisy-news models

Joshua C. C. Chan; Eric Eisenstat; Gary Koop

doi:10.1515/snde-2020-0016

Article

Choosing between identification schemes in noisy-news models

Joshua C. C. Chan , Eric Eisenstat and Gary Koop

Published/Copyright: October 26, 2020

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Studies in Nonlinear Dynamics & Econometrics Volume 26 Issue 1

Abstract

This paper is about identifying structural shocks in noisy-news models using structural vector autoregressive moving average (SVARMA) models. We develop a new identification scheme and efficient Bayesian methods for estimating the resulting SVARMA. We discuss how our identification scheme differs from the one which is used in existing theoretical and empirical models. Our main contributions lie in the development of methods for choosing between identification schemes. We estimate specifications with up to 20 variables using US macroeconomic data. We find that our identification scheme is preferred by the data, particularly as the size of the system is increased and that noise shocks generally play a negligible role. However, small models may overstate the importance of noise shocks.

Keywords: Bayesian estimation; noise shocks; savage dickey density ratio; structural identification; vector autoregressive moving average models

Corresponding author: Eric Eisenstat, The University of Queensland School of Economics, St Lucia, QLD, Australia, E-mail: e.eisenstat@uq.edu.au

Funding source: Australian Research Council Discover Project

Award Identifier / Grant number: DP180102373

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: Joshua C. C. Chan and Eric Eisenstat gratefully acknowledge support from the Australian Research Council Discover Project grant #DP180102373.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

A Proofs of propositions

To prove Proposition 1, we first state a lemma characterizing uniqueness when observationally equivalent representations are restricted to be equivalent up to constant orthogonal rotations.

Lemma 1.

Let K ( L ) ε t , K ˜ ( L ) ε ˜ t and Γ be defined as in Proposition 1 . Assume K ˜ ( L ) = K ( L ) Γ . If K ( L ) and K ˜ ( L ) both satisfy Assumptions 1–5, then Γ 12 = Γ 21 = 0 and Γ 11 is diagonal.

Proof.

By Assumptions 3 and 5, K ˜ ( L ) satisfies

K ˜ 0 , 12 = K 0 , 11 Γ 12 + K 0 , 12 Γ 22 = 0 ,

K ˜ 1 j ( L ) = K 11 ( L ) Γ 1 j + K 12 ( L ) Γ 2 j = 0 , j ≥ 3 .

Since K 0 , 12 = 0 , but K 0 , 11 ≠ 0 and K l , 12 ≠ 0 for some l ≥ 1 (as implied by the rank conditions in Assumption 5), we immediately obtain Γ 12 = ⋯ = Γ 1 n = 0 and Γ 23 = ⋯ = Γ 2 n = 0 . Orthogonality of Γ then implies Γ 21 = ⋯ = Γ n 1 = 0 and Γ 32 = ⋯ = Γ n 2 = 0 as well, with | Γ 11 | = 1 and | Γ 22 | = 1 .

By Assumption 4, we obtain for some constant c ˜ > 0

(19) K ˜ 0 , i 3 − c ˜ K ˜ 0 , i 2 = ∑ j = 1 n K 0 , i j ( Γ j 3 − c ˜ Γ j 2 ) = 0 , i = 2 , … , n .

Taking into account the restrictions on Γ established above and the fact that

K 0 , i 2 = 1 c K 0 , i 3 ,

the restrictions in Eq. (19) can be represented by

( K 0 , 23 ⋯ K 0 , 2 n ⋮ ⋱ ⋮ K 0 , n 3 ⋯ K 0 , n n ) ( Γ 33 ± c ˜ / c Γ 43 ⋮ Γ n 3 ) = 0 .

Since the first matrix in this expression is n − 1 × n − 2, and Assumption 4 implies it is of rank n − 2, we obtain Γ i 3 = 0 for all i ≥ 4, | Γ 33 | = c ˜ / c , and by orthogonality of Γ , Γ 3 j = 0 for all j ≥ 4, | Γ 33 | = 1 .□

Proof of Proposition 1

Following the derivations in Section 5, define

C 1 = ( 0 0 1 c 1 + c 2 1 1 + c 2 0 − 1 1 + c 2 c 1 + c 2 0 I n − 3 ) , C 2 = ( 0 0 1 c ˜ 1 + c ˜ 2 1 1 + c ˜ 2 0 − 1 ˜ 1 + c ˜ 2 c ˜ 1 + c ˜ 2 0 I n − 3 ) ,

and let

Φ ( L ) = K ( L ) C 1 ( L − τ I n − 1 ) , Φ ˜ ( L ) = K ˜ ( L ) C 2 ( L − τ I n − 1 ) ,

where L − 1 denotes the forward operator and L − τ = ( L − 1 ) τ .

Observe that if ω = 0 in Assumption 2, then Φ ( L ) and Φ ˜ ( L ) are fundamental representations; otherwise they are basic nonfundamental representations (Lippi and Reichlin 1994, Definition 3) corresponding to the fundamental one associated with ω = 0. Accordingly, Lippi and Reichlin (1994, Theorem 3) yields Φ ˜ ( L ) = Φ ( L ) C 0 , where C ₀ is a constant orthogonal matrix.

Moreover, if K (L) satisfies Assumptions 3–5, then Φ ( L ) satisfies

(20) Φ 14 ( L ) = ⋯ = Φ 1 n ( L ) = 0 ,

(21) Φ 11 ( L ) L τ − c Φ 12 ( L ) = 0 , Φ 11 ( 0 ) ≠ 0 ,

(22) Φ ˆ 11 ( ζ ) = Φ ˆ 12 ( ζ ) = 0 , Φ ˆ 13 ( ζ ) ≠ 0 , for some ζ ≠ 0 ,

where Φ ˆ 1 i = χ ( L ) Φ 1 i for i = 1, 2, 3. The same holds for Φ ˜ ( L ) if K ˜ ( L ) satisfies Assumptions 3–5 as well, with χ ˜ ( L ) being the least common multiple of the the denominators of K ˜ 11 ( L ) and K ˜ 12 ( L ) .

The restrictions in Eq. (20) imply

c 1 + c 2 K ˆ 12 ( L ) L − τ C 0 , 1 i + 1 1 + c 2 K ˆ 12 ( L ) C 0 , 2 i + K ˆ 11 ( L ) C 0 , 3 i = 0 , i = 4 , … , n .

But by Assumption 5, K ˆ 11 ( ζ ) C 0 , 3 i = 0 , for some ζ ≠ 0 and K ˆ 11 ( ζ ) ≠ 0 . Hence, C 0 , 3 i = 0 for all i = 4, … , n. At the same time K ˆ 12 ( 0 ) = 0 by Assumption 3, which together with C 0 , 3 i = 0 yields K ˆ τ , 12 C 0 , 1 i = 0 , and consequently, C 0 , 1 i = 0 . Finally, it follows from the remaining term, 1 1 + c 2 K ˆ 12 ( L ) C 0 , 2 i = 0 , that C 0 , 2 i = 0 .

Hence, it holds that C 0 , 1 i = C 0 , 2 i = C 0 , 3 i = 0 for all i ≥ 4, and by orthogonality of C ₀, this implies C 0 , i 1 = C 0 , i 2 = C 0 , i 3 = 0 for all i ≥ 4 as well. Another important implication is that the functions K ˆ 12 ( L ) L − τ , K ˆ 12 ( L ) , and K ˆ 11 ( L ) are linearly independent.

The restrictions in Eq. (21) further imply

(23) 1 1 + c 2 K ˆ 12 ( L ) ( C 0,11 − c ˜ c C 0,22 ) + 1 1 + c 2 K ˆ 12 ( L ) L τ C 0 , 21 + K ˆ 11 ( L ) L τ C 0 , 31 − c ˜ c 1 + c 2 K ˆ 12 ( L ) L − τ C 0 , 21 − c ˜ K ˆ 11 ( L ) C 0 , 32 = 0 .

If the three functions K ˆ 12 ( L ) L − τ , K ˆ 12 ( L ) , and K ˆ 11 ( L ) are linearly independent, then so are the four functions K ˆ 12 ( L ) L − τ , K ˆ 12 ( L ) , K ˆ 12 ( L ) L τ , and K ˆ 11 ( L ) L τ . Hence, if C _0,32 = 0 in Eq. (23), it follows immediately that C 0 , 21 = C 0 , 31 = C 0 , 12 = 0 and | C 0 , 11 | = | C 0 , 22 | = 1 .

Likewise, if all five functions K ˆ 11 ( L ) , K ˆ 12 ( L ) L − τ , K ˆ 12 ( L ) , K ˆ 12 ( L ) L τ , and K ˆ 11 ( L ) L τ in Eq. (23) are linearly independent, then we again trivially obtain C 0 , 21 = C 0 , 31 = C 0 , 12 = C 0 , 32 = 0 and | C 0 , 11 | = | C 0 , 22 | = 1 . Otherwise, there exist constants ξ 1 , ξ 2 , ξ 3 , and ξ 4 such that

K ˆ 11 ( L ) = ξ 1 K ˆ 12 ( L ) + ξ 2 K ˆ 12 ( L ) L τ + ξ 3 K ˆ 11 ( L ) L τ + ξ 4 K ˆ 12 ( L ) L − τ .

We aim to show that in this case, C 0 , 32 ≠ 0 can only hold for coefficients of K(L) in a set with Lebesgue measure zero, so that C 0 , 21 = C 0 , 31 = C 0 , 12 = C 0 , 32 = 0 and | C 0 , 11 | = | C 0 , 22 | = 1 holds almost everywhere.

To this end, observe that ξ 3 = 1 / ζ ≠ 0 and ξ 4 = K 0 , 11 / K τ , 12 ≠ 0 . Substituting into Eq. (23) yields

(24) 1 1 + c 2 K ˆ 12 ( L ) ( C 0 , 11 − c ˜ c C 0 , 22 − 1 + c 2 ξ 1 c ˜ C 0 , 32 ) + 1 1 + c 2 K ˆ 12 ( L ) L τ ( C 0 , 21 − 1 + c 2 ξ 2 c ˜ C 0 , 32 ) + K ˆ 11 ( L ) L τ ( C 0 , 31 − ξ 3 c ˜ C 0 , 32 ) − c ˜ c 1 + c 2 K ˆ 12 ( L ) L − τ ( C 0 , 21 + 1 + c 2 c ξ 4 C 0 , 32 ) = 0 .

Since, K ˆ 12 ( L ) L − τ , K ˆ 12 ( L ) , K ˆ 12 ( L ) L τ , and K ˆ 11 ( L ) L τ are linearly independent, however, this results in:

C 0 , 11 = ( C 0 , 22 C 0 , 32 + 1 + c 2 ξ 1 ) c ˜ C 0 , 32 , C 0 , 21 = 1 + c 2 ξ 2 c ˜ C 0 , 32 , C 0 , 31 = ξ 3 c ˜ C 0,32 , C 0 , 12 = − 1 + c 2 c ξ 4 C 0 , 32 ,

whereas orthogonality of C ₀ further requires:

1 = C 0 , 11 2 + C 0 , 21 2 + C 0 , 31 2 , 1 = C 0 , 12 2 + C 0 , 22 2 + C 0 , 32 2 , 0 = C 0 , 11 C 0 , 12 + C 0 , 21 C 0 , 22 + C 0 , 31 C 0 , 32 .

This system of seven equations in seven unknowns (C _0,11, C _0,21, C _0,31, C _0,12, C _0,22, C _0,32, and c ˜ ) admits a unique solution, up to sign normalization, of the form:

C 0 , 11 = ( 1 + c 2 c ) ξ 1 + ξ 5 ( ( 1 + c 2 c ) ξ 1 + ξ 5 ) 2 + ( 1 + c 2 ) ξ 2 2 + ξ 3 2 , C 0 , 12 = − ( 1 + c 2 c ) ξ 4 1 + ( 1 + c 2 c 2 ) ξ 4 2 + ξ 5 2 , C 0 , 21 = 1 + c 2 ξ 2 ( ( 1 + c 2 c ) ξ 1 + ξ 5 ) 2 + ( 1 + c 2 ) ξ 2 2 + ξ 3 2 , C 0 , 22 = ξ 5 1 + ( 1 + c 2 c 2 ) ξ 4 2 + ξ 5 2 , C 0 , 31 = ξ 3 ( ( 1 + c 2 c ) ξ 1 + ξ 5 ) 2 + ( 1 + c 2 ) ξ 2 2 + ξ 3 2 , C 0 , 32 = 1 1 + ( 1 + c 2 c 2 ) ξ 4 2 + ξ 5 2 ,

where

ξ 5 = ξ 3 − ( 1 + c 2 c 2 ) ξ 1 ξ 4 ( 1 + c 2 c ) ξ 4 − 1 + c 2 ξ 2 .

In addition, there exist unique (up to sign normalization) constants C _0,13, C _0,23, and C _0,33 satisfying

0 = C 0 , 13 C 0 , 11 + C 0 , 23 C 0 , 21 + C 0 , 33 C 0,31 , 0 = C 0 , 13 C 0 , 12 + C 0 , 23 C 0 , 22 + C 0 , 33 C 0 , 32 , 1 = C 0 , 13 2 + C 0 , 23 2 + C 0 , 33 2 .

Clearly, C 0 , 13 ≠ 0 , C 0 , 23 ≠ 0 and C 0 , 33 ≠ 0 almost everywhere.

The restrictions in Eq. (22), however, require that there exists a constant ζ ˜ ≠ 0 such that Φ ˜ ˆ 11 ( ζ ˜ ) = Φ ˜ ˆ 12 ( ζ ˜ ) = 0 but Φ ˜ ˆ 13 ( ζ ˜ ) ≠ 0 , where Φ ˜ ˆ 1 i ( L ) = χ ˜ ( L ) Φ ˜ ˆ 1 i ( L ) for i = 1, 2, 3. This, in turn, implies

0 = c 1 + c 2 K ˇ 12 ( ζ ˜ ) ζ ˜ − τ C 0 , 11 + 1 1 + c 2 K ˇ 12 ( ζ ˜ ) C 0 , 21 + K ˇ 11 ( ζ ˜ ) C 0 , 31 , 0 = c 1 + c 2 K ˇ 12 ( ζ ˜ ) ζ ˜ − τ C 0 , 12 + 1 1 + c 2 K ˇ 12 ( ζ ˜ ) C 0 , 22 + K ˇ 11 ( ζ ˜ ) C 0 , 32 , 0 ≠ c 1 + c 2 K ˇ 12 ( ζ ˜ ) ζ ˜ − τ C 0 , 13 + 1 1 + c 2 K ˇ 12 ( ζ ˜ ) C 0 , 23 + K ˇ 11 ( ζ ˜ ) C 0 , 33 ,

where K ˇ 1 i ( L ) = χ ˜ ( L ) K 1 i ( L ) = χ ˜ ( L ) / χ ( L ) K ˆ 1 i ( L ) for i = 1, 2. Since Φ ˜ ˆ 11 ( L ) , Φ ˜ ˆ 12 ( L ) , and Φ ˜ ˆ 13 ( L ) are polynomials, χ ˇ ( L ) = χ ˜ ( L ) / χ ( L ) is a polynomial with χ ˇ ( z ) < ∞ for all z ∈ ℂ ; since K ˇ 12 ( ζ ˜ ) and K ˇ 11 ( ζ ˜ ) cannot vanish simultaneously, χ ˇ ( ζ ˜ ) ≠ 0 either. It follows that

c 1 + c 2 χ ˇ ( ζ ˜ ) K ˆ 12 ( ζ ˜ ) ζ ˜ − τ = C 0 , 13 ξ ‾ , 1 1 + c 2 χ ˇ ( ζ ˜ ) K ˆ 12 ( ζ ˜ ) = C 0 , 23 ξ ‾ , χ ˇ ( ζ ˜ ) K ˆ 11 ( ζ ˜ ) = C 0 , 33 ξ ‾ ,

for some ξ ‾ ≠ 0 . But these equations can only be satisfied for

K ˆ 12 ( ( c C 0 , 23 C 0 , 13 ) 1 τ ) = 1 + c 2 C 0 , 23 C 0 , 33 K ˆ 11 ( ( c C 0 , 23 C 0 , 13 ) 1 τ ) .

As this condition does not hold for almost all K ˆ 11 ( L ) , K ˆ 12 ( L ) , and c, we conclude that C 0 , 32 ≠ 0 almost everywhere, and in this case, C _0,21 = C _0,31 = C _0,12 = 0 as well. Orthogonality of C ₀ then yields C _0,13 = 0, and together with the previously established restrictions, we obtain

C 0 , 12 = ⋯ = C 0 , 1 n = 0 , C 0 , 21 = ⋯ = C 0 , n 1 = 0 ,

such that C ₀ commutes with ( L − τ I n − 1 ) . Therefore,

K ˜ ( L ) = Φ ˜ ( L ) ( L τ I n − 1 ) C 2 ′ = Φ ( L ) C 0 ( L τ I n − 1 ) C 2 ′ = Φ ( L ) ( L τ I n − 1 ) C 0 C 2 ′ = Φ ( L ) ( L τ I n − 1 ) C 1 ′ C 1 C 0 C 2 ′ = K ( L ) Γ ,

almost everywhere, with Γ = C 1 C 0 C 2 ′ . Lemma 1 confirms the remainder of the proposition.□

Proof of Proposition 2.

Define

C 1 = ( 1 0 0 0 c 1 + c 2 1 1 + c 2 0 − 1 1 + c 2 c 1 + c 2 I n − 3 ) , C 2 = ( 1 0 0 0 c ˜ 1 + c ˜ 2 1 1 + c ˜ 2 0 − 1 1 + c ˜ 2 c ˜ 1 + c ˜ 2 I n − 3 ) ,

and let

Φ ( L ) = K ( L ) C 1 ( 1 L − τ I n − 2 ) , Φ ˜ ( L ) = K ˜ ( L ) C 2 ( 1 L − τ I n − 2 ) .

Focusing for the moment on

(25) Φ ( L ) = ( K 11 ( L ) c L − τ K 12 ( L ) 1 + c 2 K 12 ( L ) 1 + c 2 0 K 21 ( L ) K 21 ( L ) c L − τ K 12 ( L ) 1 + c 2 K 11 ( L ) K 22 ( L ) + c K 23 ( L ) 1 + c 2 K 24 ( L ) ⋮ ⋮ ⋮ ⋮ K n 1 ( L ) K n 1 ( L ) c L − τ K 12 ( L ) 1 + c 2 K 11 ( L ) K n 2 ( L ) + c K n 3 ( L ) 1 + c 2 K n 4 ( L ) ) ,

observe that although it is singular (with normal rank n − 1), it can be decomposed as Φ ( L ) = Ψ ( L ) ϒ ( L ) , where Ψ ( L ) is n × (n − 1) and ϒ ( L ) is (n − 1) × n. In particular,

(26) Ψ ( L ) = ( 1 χ ( L ) K 12 ( L ) 1 + c 2 0 K 21 ( L ) K ˆ 11 ( L ) K 22 ( L ) + c K 23 ( L ) 1 + c 2 K 24 ( L ) ⋮ ⋮ ⋮ K n 1 ( L ) K ˆ 11 ( L ) K n 2 ( L ) + c K n 3 ( L ) 1 + c 2 K n 4 ( L ) ) ,

(27) ϒ ( L ) = ( K ˆ 11 ( L ) c L − τ K ˆ 12 ( L ) 1 + c 2 0 0 0 I n − 2 ) ,

Note that Ψ ( L ) is a full column rank rational transfer matrix, and ϒ ( L ) is a full row rank polynomial matrix (since K ˆ 11 ( L ) and L − τ K ˆ 12 ( L ) are scalar polynomials by Assumption 7.1). In addition, since K ˆ 11 ( L ) and K ˆ 12 ( L ) have no common roots, ϒ ( L ) clearly has no finite zeros, and it can be further verified that it has no infinite zeros either.

Specifically, ϒ ( L ) has a zero at infinity if and only if there exists a rational vector υ ( L ) such that for z ∈ ℂ ,

0 < lim z → ∞ | | υ ( z ) | | < ∞

and

lim z → ∞ υ ( z ) ′ ϒ ( z ) = 0 .

Let υ ( z ) = ( υ 1 ( z ) , υ 2 ( z ) ) ′ . It is immediately evident that lim z → ∞ υ 2 ( z ) = 0 so that we must have lim z → ∞ υ 1 ( z ) ≠ 0 . But since both lim z → ∞ K ˆ 11 ( z ) and lim z → ∞ z − τ K ˆ 12 ( z ) diverge, the conditions above cannot be satisfied.

Consequently, ϒ ( L ) has no zeros (finite or infinite) and no finite poles. It does, however, have a pole at infinity of multiplicity equal to max { deg K ˆ 11 ( L ) , deg L − τ K ˆ 12 ( L ) } . Moreover, we can construct the rational transfer matrix

Ξ ( L ) = Ψ ( L ) ( ψ ( L ) I n − 2 ) ,

where

ψ ( L ) ψ ( L − 1 ) = K ˆ 11 ( L ) K ˆ 11 ( L − 1 ) + c 2 1 + c 2 K ˆ 12 ( L ) K ˆ 12 ( L − 1 )

and

Ξ ( L ) Ξ ( L − 1 ) ′ = Φ ( L ) Φ ( L − 1 ) ′ = K ( L ) K ( L − 1 ) ′ .

Observe that Ξ ( L ) is an n × (n − 1) rational transfer matrix of normal rank n − 1.

Although ψ ( L ) in this formulation has max { deg K ˆ 11 ( L ) , deg L − τ K ˆ 12 ( L ) } roots, we can always choose ψ ( L ) to have only roots with modulus greater than or equal to unity (since ψ ( L ) ψ ( L − 1 ) = ψ ( L ) ϖ ( L ) ϖ ( L − 1 ) ψ ( L − 1 ) for an arbitrary ϖ ( L ) ϖ ( L − 1 ) = 1 ). On the other hand, Ψ ( L ) , being a tall transfer matrix, is zero-free for almost all K(L) (Anderson and Deistler 2008). In the non-generic case, all the zeros (finite or infinite) along with the finite poles of Ψ ( L ) are equivalent to the zeros (finite or infinite) along with the finite poles of Φ ( L ) .

The locations of the zeros of Φ ( L ) and therefore Ψ ( L ) , if they exist, are determined by ω. If ω = (0, … , 0), then Ψ ( L ) has no zeros of modulus less than unity, and neither does Ξ ( L ) . Accordingly, an identical procedure applied to Φ ˜ ( L ) yields the rational transfer matrix Ξ ˜ ( L ) with no zeros of modulus less than unity, given ω = (0, … , 0). In this case, Baggio and Ferrante (2016, Theorem 2) confirms that Ξ ˜ ( L ) = Ξ ( L ) C 0 , where C 0 is a constant orthogonal matrix.^[10]

If ω ≠ ( 0 , … , 0 ) , then Ξ ( L ) will have some zeros inside the unit circle. However, we can apply the arguments in Lippi and Reichlin (1994, Proof of Theorem 3) to obtain a rational transfer matrix N ( L ) with no zeros of modulus less than unity and

N ( z ) = Ξ ( z ) D ( z ‾ − 1 ) * ,

where D ( z ) is a Blaschke matrix containing factors

R ( α h , z ) = ( z − α h 1 − α ‾ h z I n − 2 )

for each zero α _h of Ξ ( z ) satisfying | α h | < 1 . Manipulating Ξ ˜ ( L ) in an identical way yields N ˜ ( L ) with no zeros of modulus less than unity.

Baggio and Ferrante (2016, Theorem 2) may now be applied to obtain N ˜ ( L ) = N ( L ) C 00 for a constant orthogonal matrix C ₀₀, which also implies that N (L) and N ˜ ( L ) have identical zeros. Therefore, Ξ ( L ) and Ξ ˜ ( L ) corresponding to the same ω must also have identical zeros. Applying again the arguments in Lippi and Reichlin (1994, Proof of Theorem 3), we obtain that Ξ ˜ ( L ) = Ξ ( L ) C 0 for some constant orthogonal matrix C ₀.

From Ξ ˜ ( 0 ) = Ξ ( 0 ) C 0 and given that Ξ ˜ 1 j ( 0 ) = Ξ 1 j ( 0 ) = 0 for all j ≥ 2, it follows that C 0 , 1 j = C 0 , j 1 = 0 for all j ≥ 2 and | C 0 , 11 | = 1 . Accordingly, since for some z ≠ 0 , Ξ ˜ 12 ( z ) ≠ 0 , Ξ 12 ( z ) ≠ 0 and Ξ ˜ 1 j ( z ) = Ξ 1 j ( z ) = 0 for all j ≥ 3, we obtain C _0,2j = C _0,j2 = 0 and | C 0 , 22 | = 1 .

Therefore,

(28) ψ ˜ ( L ) χ ˜ ( L ) = ± ψ ( L ) χ ( L ) ,

(29) K ˜ 12 ( L ) 1 + c ˜ 2 = ± K 12 ( L ) 1 + c 2 ,

(30) ψ ˜ ( L ) K ˜ i 1 ( L ) K ˜ ˆ 11 ( L ) = ± ψ ( L ) K i 1 ( L ) K ˆ 11 ( L ) , i = 2 , … , n

(31) K i 2 ( L ) + c K i 3 ( L ) 1 + c 2 = ± K ˜ i 2 ( L ) + c ˜ K ˜ i 3 ( L ) 1 + c ˜ 2 , i = 1 , ... , n .

Applying Assumption 7.2,

1 + c ˜ 2 c ˜ K ˜ i 3 ( L ) + K ˜ i 1 ( L ) K ˜ 11 ( L ) K ˜ 12 ( L ) 1 + c ˜ 2 = ± ( 1 + c 2 c K i 3 ( L ) + K i 1 ( L ) K 11 ( L ) K 12 ( L ) 1 + c 2 ) ,

and using the above equalities,

(32) K ˜ i 1 ( L ) K ˜ 11 ( L ) = ψ ˜ ( L ) K ˜ i 1 ( L ) K ˜ ˆ 11 ( L ) ψ ˜ ( L ) χ ˜ ( L ) = ψ ( L ) K i 1 ( L ) K ˆ 11 ( L ) ψ ( L ) χ ( L ) = K i 1 ( L ) K 11 ( L ) , K ˜ i 1 ( L ) K ˜ 11 ( L ) K ˜ 12 ( L ) 1 + c ˜ 2 = ± K i 1 ( L ) K 11 ( L ) K 12 ( L ) 1 + c 2 ,

where the ± in the last line is the sign of C _0,22, which is the same as the ± in Eq. (31).

It follows that

1 + c ˜ 2 c ˜ K ˜ i 3 ( L ) = ± 1 + c 2 c K i 3 ( L ) .

Then, applying g to both sides of this equality and observing that

K ˜ τ * , 12 1 + c ˜ 2 = ± K τ * , 12 1 + c 2

yields c ˜ = c . Consequently, straightforward algebra leads to

(33) K ˜ i 1 ( L ) = ± K i 1 ( L ) ,

(34) K ˜ i 2 ( L ) = ± K i 2 ( L ) ,

(35) K ˜ i 3 ( L ) = ± K i 3 ( L ) ,

(36) K ˜ i 4 ( L ) = K i 4 ( L ) C 0,33 ,

for i = 1, … , n and where C 0 , 33 ′ C 0 , 33 = C 0 , 33 C 0 , 33 ′ = I n − 3 . Note that Eq. (33) follows from K ˜ 11 ( L ) K ˜ 11 ( L − 1 ) = K 11 ( L ) K 11 ( L − 1 ) and the assumption that K 11 ( L ) has no roots of modulus less than unity. The equality for i = 2, … , n then follows directly from Eq. (32). Finally, setting

Γ = ( ± 1 ± 1 ± 1 C 0 , 33 )

yields K ˜ ( L ) = K ( L ) Γ as desired. □

B Econometric methods

B.1 Imposing structural identification restrictions

We begin by detailing 5 of the algorithm outlined in Section 5, which entails implementing restrictions R2–R4. In fact, this involves applying three types of orthogonal rotations, Γ 1 , Γ 2 , and Γ 3 , such that their product Γ = Γ 1 Γ 2 Γ 3 yields the comprehensive set of orthogonal rotations that transform A ˜ ( L ) into the structural representation of interest A (L), where the two VMA polynomials are related by A j = A ˜ j Γ for j = 0, … , q. In our implementation, we set the horizon in 5.2 to be 20 quarters after impact.

Accordingly, Γ 1 is determined by setting the first column Γ 1 , 1 = A ˜ 0 , ( 1 ) ′ / | | A ˜ 0 , ( 1 ) | | 2 , where A ˜ 0 , ( 1 ) denotes the first row of A ˜ 0 , and the remaining columns Γ 1 , i for i = 2, … , n equal to the n − 1 vectors that are orthogonal to A 0 , ( 1 ) ′ (normalized such that | | Γ 0 , i | | = 1 ).

Next, let K ˜ ( L ) = B ( L ) − 1 A ˜ ( L ) Γ 1 be the impulse responses obtained after applying the first set of orthogonal rotations Γ 1 , and define K ˜ j , 1 , 2 : n for j ≥ 0 as the 1 × n − 1 row vector constructed from the first row and columns 2 to n of K ˜ j . Compute the eigenvalue decomposition of ∑ j = 1 20 K ˜ j , 1 , 2 : n ′ K ˜ j , 1 , 2 : n , with eigenvalues sorted in descending order, and store the eigenvectors in E ₂. The orthogonal matrix that identified news shocks according to restriction R3 is then given by

Γ 2 = ( 1 0 0 E 2 ) .

We normalize the sign of the news shock by requiring that the maximum impulse response (over the horizon 0:20) of TPF to news is positive.

Let A ˇ ( L ) = A ˜ ( L ) Γ 1 Γ 2 be the VMA representation obtained after applying the first two sets of orthogonal rotations. At this stage, non-news and news shocks are identified according to restrictions R2 and R3, but the noise shock is not identified in the sense that following multiplication by Γ 1 Γ 2 , the third column of A ˇ 0 will generally not be proportional to the second. To enforce the proportionality restriction, we construct a third orthogonal matrix

Γ 3 = ( I 2 0 0 E 3 ) ,

where the first column E _3,1 of the n − 2 × n − 2 orthogonal matrix E ₃ must satisfy ( A ˇ 0 , 3 , … , A ˇ 0 , n ) E 3 , 1 = c A ˇ 0 , 2 .

By construction, the n×n−1 matrix ( A ˇ 0 , 2 , … , A ˇ 0 , n ) has rank n − 2 and, therefore, there exists an n − 1 × 1 vector ζ A = ( ζ A , 1 , ζ A , 2 ′ ) ′ , ζ A = 1 such that^[11]

( A ˇ 0 , 2 , … , A ˇ 0 , n ) ζ A = 0 .

Accordingly, set

(37) c = | ζ A , 1 | 1 − ζ A , 1 2

(38) E 3 , 1 = − sgn ( ζ A , 1 ) ζ A , 2 1 − ζ A , 1 2 ,

and the remaining columns E 3 , 2 , … , E 3 , n − 2 of E ₃ to be the n − 3 vectors orthogonal to E _3,1 (normalized such that | | E 3 , i | | = 1 ). Subsequently, multiplying A j = A ˇ j Γ 3 for all j = 0, … , q yields the desired representation A (L) where A 0 , 3 = c A 0 , 2 while preserving the restrictions R2 and R3.

As mentioned in Section 5, the algorithm involves finding an n × r _* matrix Δ such that Θ j Θ 0 Δ = 0 for all j = q − τ + 1 , … , q . When r * > 1 , this matrix is not unique as there exist infinitely many orthonormal matrices C Δ such that Δ ˜ = Δ C Δ is also a valid alternative. To obtain a unique Δ in the r * > 1 case, one may choose Δ to maximize the fraction of forecast error variance of TFP that is jointly explained by non-news and noise shocks at 20 quarters after impact.

Note that this objective is in line with the theoretical Assumption 3. Starting with an arbitrary Δ obtained in Step 3, the maximization may be performed numerically (over C Δ ) by iterating Step 4 and Step 5, which themselves involve computationally simple operations. Of course, the complexity of the numerical optimization approach will vary with r * (for example, for r * = 2 , C Δ has only one free parameter, and hence, the optimization is univariate). In all our empirical work with this algorithm, however, we never encountered a need to perform numerical optimization to find a unique Δ as every draw of Θ j from all our models satisfied rank Θ j = n − 1 .

B.2 Bayesian algorithms

B.2.1 Priors

A key advantage of the Bayesian approach in time-series modeling has proven to be the ability to incorporate information through prior probability distributions. Several different priors are popular in the Bayesian VAR literature, including the Minnesota prior and various hierarchical shrinkage priors (e.g., various LASSO priors, spike-and-slab priors). In this paper, we use the SSVS prior introduced to the Bayesian VAR literature by George, Sun, and Ni (2008), and used in many VAR papers (e.g., Koop 2013; Korobilis 2013). The basic idea can be explained in terms of a generic VAR or MA coefficient, say ϑ. SSVS specifies a hierarchical prior (i.e., a prior expressed in terms of parameters which in turn have a prior of their own) which is a mixture of two Normal distributions:

(39) ( ϑ | ν ) ∼ ( 1 − ν ) N ( 0 , v 0 2 ) + ν N ( 0 , v 1 2 ) ,

where ν ∈ { 0 , 1 } is an unknown parameter. If ν = 0 the prior for ϑ is given by the first Normal distribution, and if it is ν = 1 its prior is given by the second. The prior is hierarchical since ν is treated as an unknown parameter that is estimated in a data-based fashion. The first prior variance, v 0 2 , is chosen to be ‘small’ (so that the coefficient is constrained to be virtually equal to zero) and the second prior variance, v 1 2 , to be ‘large’ (implying a relatively non-informative prior for the corresponding coefficient). Thus, SSVS allows for the data to decide which coefficients should be set to zero, so as to ensure parsimony in the SVARMA. The only subjective prior information that is required is the choice of v 0 2 and v 1 2 , but standard methods exist for their choice. Details of how this is done and the MCMC algorithm which results from use of the SVARMA model with SSVS prior are given in the following subsection.

B.2.2 Bayesian VARMAs

The ultimate goal of a Bayesian approach to estimating the effects of news and noise shocks is to obtain draws from the posterior distribution of representation 6. Indeed, the Bayesian framework offers a great deal of flexibility in designing sampling algorithms for this purpose. For example, Plagborg-Møller (2016) develops an MCMC algorithm that samples directly from a truncated approximation to Eq. (6). Such an approach has the advantage of allowing restrictions on impulse responses to be imposed directly in the sampling. However, it is computationally intensive and the requisite sampling algorithm deteriorates for systems larger than three variables.

Since in our applications we wish to estimate models potentially involving tens of variables, it is more appropriate to work with a finite order VARMA representation, such as the VARMA (p, q) specified in Eq. (15). As discussed in Section 5, however, even in the reduced form Eq. (16) the parameters of B (L) and Θ ( L ) will generally not be identified without further restrictions, and the some holds for B ( L ) and A ( L ) in the structural form. The simple reason for that is as follows: even though K ( L ) = B ( L ) − 1 A ( L ) is uniquely determined by identifying restrictions such as R2–R4 above, such restrictions do not guarantee uniqueness of B (L) and A (L) since there may exist some D (L) such that B ( L ) † = D ( L ) B ( L ) is of order p, A ( L ) † = D ( L ) A ( L ) is of order q, and both lead to the same Wold representation K ( L ) = ( B ( L ) † ) − 1 A ( L ) † .

Identification issues in VARMAs are further complicated by the fact that the fundamental and numerous non-fundamental representations are observationally equivalent, as discussed in Subsection 4.1.1. Consequently, when a unique VARMA representation is required for estimation purposes, it is typically specified as a fundamental process in the canonical echelon form. This is achieved by rewriting Eq. (16) in recursive form Eq. (17) and imposing two types of restrictions:

Exclusion restrictions on B ˜ 0 , … , B ˜ p * , M 1 , … , M p * according to the row degrees p ₁, … , p _n that define the lag structure of each equation in the system (with p * = max ( p 1 , … , p n ) );
Non-linear restrictions on M 1 , … , M p * to ensure all roots of M(L) lie outside the unit circle.

However, estimating a VARMA in echelon form is challenging. First, imposing type 2 restrictions on the roots of M(L) becomes exceedingly difficult as the size of the system increases. Moreover, imposing type 1 exclusion restrictions requires knowledge of the row degrees p ₁, … , p _n, which themselves need to be estimated in practice.^[12]

Fortunately, point identification is not necessary in the Bayesian framework. To explain, the posterior distribution will be well-defined even when the likelihood does not uniquely identify the parameters in the model, as long as proper prior distributions are specified for the parameters. In the VARMA case, this means that as long as proper priors are specified for B ₁, … , B _p, Θ ˜ 1 , … , Θ ˜ q , and Σ , we can readily obtain draws from

p ( B 1 , … , B p , Θ ˜ 1 , … , Θ ˜ q , Σ | y ) ,

even though this posterior may not be characterized by a unique mode, or may simply resemble the joint prior distribution (in the extreme case where the likelihood provides no information on the model parameters).

The key insight in a Bayesian approach to analyzing VARMA models is that parameters B ₁, … , B _p, Θ 1 , … , Θ q , and Σ themselves are not of primary interest, but rather quantities such as forecasts and impulses responses, which are uniquely identified even when the AR and MA coefficients are not. Therefore, it is possible to obtain draws from the posterior of unidentified parameters, then transform them to draws from the posterior of quantities which are, in fact, identified.

In general, Bayesian routinely build sampling algorithms on unidentified parameter spaces to obtain computational efficiency (examples include Gustafson 2005; Ghosh and Dunson 2009; Imai and van Dyk 2005; Koop, León-González, and Strachan 2010, 2012, among many others). Indeed, early work such as Meng and van Dyk (1999) and Liu and Wu (1999) suggest that artificially expanding the parameter space may reduce auto-correlation in MCMC sampling algorithms, in terms of the identified quantities of interest, thus further improving computation. Nevertheless, identification is an important concept in the Bayesian framework to the extent that it provides parsimony in over-parameterized systems. From a practical viewpoint, both parsimony and identification are features of the model that are implemented entirely through the appropriate specification of prior distributions.

Building on these ideas, Chan and Eisenstat (2017) and Chan, Eisenstat, and Koop (2016) develop MCMC algorithms on the expanded VARMA representation:

(40) B ˜ 0 y t = B ˜ 1 y t − 1 + ⋯ + B ˜ p * y t − p * + Π 0 f t + Π 1 f t − 1 + ⋯ + Π p * f t − p * + η t ,

where f t ∼ N ( 0 , Ω ) , η t ∼ N ( 0 , Λ ) , Ω and Λ are diagonal, and Π 0 is lower triangular with ones on the diagonal. Expanded form parameters are related to the VARMA parameters in Eq. (17) by the mapping:

(41) ∑ l = j p * Θ l Σ Θ l - j ′ = ∑ l = j p * Π l Ω Π l - j ′ + 1 j = 0 Λ , for all j = 0 , … , p * ,

whereas B ˜ j in the expanded form is identical to the corresponding B ˜ j in the semi-structural form for j = 0 , … , p * . Consequently, draws from Eq. (17) can be obtained by sampling directly from the expanded form Eq. (40) and then computing M 1 , … , M p * , Σ from each draw of Π 0 , … , Π p * , Ω , and Λ using the mapping in Eq. (41). The exact procedure based on generalized Schur decompositions/generalized eigenvalues is provided in Section 2.3 of Chan and Eisenstat (2017) and Appendix D of Chan, Eisenstat, and Koop (2016). To economize on space, we do not reproduce it here, but only emphasize that it is a computationally simple procedure, even for large VARMA systems.

The advantage of the expanded form is that it can be regarded as a linear state space model, and therefore, admits straightforward and efficient MCMC sampling algorithms. Moreover, there is no need to impose non-linear restrictions directly in the MCMC since restrictions on the roots of M (L) can be easily implemented in the post-processing of draws (i.e., when constructing M 1 , … , M p * , Σ from Π 0 , … , Π p * , Ω , and Λ ).

At the same time, the expanded form provides an extremely flexible approach to estimating VARMAs. For example, Chan, Eisenstat, and Koop (2016) demonstrate how to construct a prior on the expanded form parameters—using SSVS methods (see George, Sun, and Ni 2008; Kuo and Mallick 1997)—such that the implied draws from the recursive form Eq. (17) satisfy the echelon form restrictions at every iteration. Hence, the expanded form can be used to estimate unique VARMA systems, although this may still lead to computationally intensive algorithms in larger VARMAs. On the other hand, it is also possible to obtain more computational efficiency by employing priors that approximate the echelon form in the sense that they lead to exact identifying restrictions holding with some probability (less than one) in the posterior. The Bayesian approach based on the expanded form, therefore, affords a great deal of flexibility in designing algorithms that target an optimal balance between computational efficiency and parsimony.

In this paper, we use such an approximate identification approach. In particular, starting from the expanded form Eq. (40) with p = q = p ^*, we impose parsimony by first setting (with probability one)

Π p * − τ + 1 , n i = ⋯ = Π p * , n i = 0 , i = 1 , … , n ,

which corresponds to restrictions Eq. (18) on M p * − τ + 1 , … , M p * . This, in turn, leads to the restrictions

Θ p * − τ + 1 , n i = ⋯ = Θ p * , n i = 0 , i = 1 , … , n ,

such that Θ p * − τ + 1 , … , Θ p * are singular matrices. Next, we specify SSVS priors on the individual free elements of B ˜ 0 , … , B ˜ p * and Π 0 , … , Π p * of the form:

( B ˜ j , i k | γ j , i k B ) ∼ γ j , i k B N ( 0 , 1 ) + ( 1 − γ j , i k B ) N ( 0 , 0.01 ) ,

( Π j , i k | γ j , i k Π ) ∼ γ j , i k Π N ( 0 , 1 ) + ( 1 − γ j , i k Π ) N ( 0 , 0.01 ) ,

subject to the restriction that det B ( L ) has no roots inside the unit circle, and with

Pr ( γ j , i k B = 1 ) = Pr ( γ j , i k Π = 1 ) = { 0.5 if n < 6 , 0.4 if n = 6 , 0.2 if n = 10 , 1.5 n if n > 10 .

Note that we specify Pr ( γ j , i k B = 1 ) and Pr ( γ j , i k Π = 1 ) as decreasing functions of n in line with the theory developed in Mol, Giannone, and Reichlin (2008) and empirical evidence reported in Korobilis (2013).

Through extensive experimentation with the resulting algorithm, we find these settings to produce satisfactory. Moreover, moderate changes to these priors (including alternative SSVS settings and rank restrictions) do not materially impact the inference on impulse responses.

To complete the prior specification, we set

Ω i i ∼ IG ( 5 , 1 ) ,

Λ i i ∼ IG ( 10 − 12 , 0.1 ) 1 l ( Λ i i ≤ 50 ) ,

where IG ( a , b ) denotes the inverse gamma distribution with shape parameter a and rate parameter b. Note that these settings imply weakly informative priors on Ω i i and extremely heavy-tailed but proper priors on Λ i i . In the paper, we report results holding fixed all of the above prior settings, but varying the dimension of the system n.

To facilitate the use of generic priors such as these, we standardize the scale of all series in y _t before commencing MCMC. Specifically, for each original series y i , t , we transform to

y ˜ i , t = y i , t 1 T ∑ t = 2 T Δ y i , t .

After obtaining MCMC draws, we adjust them such as to remove the effect of the standardization. Hence, all impulse responses are reported on the original, unscaled variables. The approach is equivalent to working directly with y _t, but adjusting the priors by the sample standard deviations, as is often done in Bayesian time-series applications (e.g., VARs with Minnesota priors).

Simulation from the posterior of the expanded form VARMA is implemented with Gibbs sampling by cycling through the following four broad steps:

Sample ( γ i , B ˜ ( i ) , Π ( i ) | f , Λ i i , y i ) for each i = 1, … , n, where B ˜ ( i ) denotes the ith row of B ˜ = ( I n − B ˜ 0 , B ˜ 1 , … , B ˜ p * ) , Π ( i ) the ith row of Π = ( Π 0 , … , Π p * ) , and γ i is the set of all SSVS indicators pertaining to B ˜ ( i ) , Π ( i ) .
Sample ( Λ i i | B ˜ ( i ) , Π ( i ) , γ i , f , y i ) for each i = 1 , … , n .
Sample ( Ω i i | f i ) for each i = 1, … , n.
Sample ( f | B ˜ , Π , Ω , Λ , γ , y ) .

Details and extensive discussion of each sampling step above are provided in Appendix B of Chan, Eisenstat, and Koop (2016).

Each MCMC draw of the expanded form parameters is transformed to a draw of B 1 , … , B p , Θ ˜ 1 , … , Θ ˜ q and Σ , which constitutes Step 1 of the algorithm outlined in Section 5. Finally, Step 2–5 applied to each draw of Θ ˜ 1 , … , Θ ˜ q yield a draw of A 0 , … , A q and K ( L ) = B ( L ) − 1 A ( L ) provides a draw of the IRFs.

C The data

We use John Fernald’s purified TFP series available from the San Francisco Fed’s website. Inflation has been computed as the log-difference of the GDP deflator (GDPCTPI) taken from the St. Louis Fed’s website. Hours worked by all persons in the nonfarm business sector (HOANBS) is from the U.S. Department of Labor: Bureau of Labor Statistics.

The seasonally adjusted series for real GDP (GDPC96) is from the U.S. Department of Commerce: Bureau of Economic Analysis. The seasonally adjusted series for consumption of non-durables and services, real chain-weighted investment, and their deflators (which we use in order to compute the chain-weighted relative price of investment) have been computed based on the data found in Tables 1.1.6, 1.1.6B, 1.1.6C, and 1.1.6D of the National Income and Product Accounts.

The seasonally adjusted civilian unemployment rate (UNRATE) is from the U.S. Department of Labor: Bureau of Labor Statistics. The vacancy rate is computed from the help wanted index. The industrial production index (INDPRO), real M2 (M2REALx), USD/GBP exchange rate (EXUSUKx ), and the University of Michigan consumer sentiment index (UMCSENTx) are obtained directly from the FRED-QD database, dated 19 August 2018.

All the preceding variables are available at the quarterly frequency. The remaining variables are available at a monthly frequency and have been converted to the quarterly frequency by taking averages within the quarter. The Federal funds rate (FEDFUNDS) and all the government bond yield (TB3MS, GS1, GS3, GS5, GS10) are taken from the St. Louis Fed’s website. They are quoted at a non-annualized rate in order to make their scale exactly comparable to that of inflation.Seasonally unadjusted nominal dividends and stock prices (the S&P 500 index) are both from Robert Shiller’s website. They have then been deflated by the GDP deflator. Civilian non-institutional population (CNP16OV), which is used in transforming some of the variables, is from the U.S. Department of Labor: Bureau of Labor Statistics.

References

Anderson, B. D. O., and M. Deistler. 2008. “Properties of Zero-free Transfer Function Matrices.” SICE Journal of Control, Measurement, and System Integration 1: 284–92.10.9746/jcmsi.1.284Search in Google Scholar

Baggio, G., and A. Ferrante. 2016. “On the Factorization of Rational Discrete-Time Spectral Densities.” IEEE Transactions on Automatic Control 61: 969–81, https://doi.org/10.1109/tac.2015.2446851.Search in Google Scholar

Barsky, R., A. Justiniano, and L. Melosi. 2014. “The Natural Rate of Interest and its Usefulness for Monetary Policy.” The American Economic Review 104: 37–43, https://doi.org/10.1257/aer.104.5.37.Search in Google Scholar

Barsky, R., and E. Sims. 2011. “News Shocks and Business Cycles.” Journal of Monetary Economics 58: 273–89, https://doi.org/10.1016/j.jmoneco.2011.03.001.Search in Google Scholar

Barsky, R., and E. Sims. 2012. “Information, Animal Spirits, and the Meaning of Innovations in Consumer Confidence.” The American Economic Review 102: 1343–77, https://doi.org/10.1257/aer.102.4.1343.Search in Google Scholar

Beaudry, P., and F. Portier. 2006. “Stock Prices, News, and Economic Fluctuations.” American Economic Review 964: 1293–307.10.3386/w10548Search in Google Scholar

Benati, L., J. C. C. Chan, E. Eisenstat, and G. Koop. 2020. “Identifying Noise Shocks.” Journal of Economic Dynamics and Control (forthcoming).10.1016/j.jedc.2019.103780Search in Google Scholar

Blanchard, O., J.-P. L’Huillier, and G. Lorenzoni. 2013. “News, Noise, and Fluctuations: An Empirical Exploration.” The American Economic Review 103: 3045–70, https://doi.org/10.1257/aer.103.7.3045.Search in Google Scholar

Chahrour, R., and K. Jurado. 2018a. “News or Noise? the Missing Link.” The American Economic Review 108: 1702–36, https://doi.org/10.1257/aer.20170792.Search in Google Scholar

Chahrour, R., and K. Jurado. 2018b. “Recoverability.” Working Paper.10.2139/ssrn.3063382Search in Google Scholar

Chan, J., and E. Eisenstat. 2017. “Efficient Estimation of Bayesian Varmas with Time-Varying Coefficients.” Journal of Applied Econometrics 32: 1277–97, https://doi.org/10.1002/jae.2576.Search in Google Scholar

Chan, K.-S., and L.-H. Ho. 2004. On the Unique Representation of Non-gaussian Multivariate Linear Processes. Department of Statistics and Actuarial Science, University of Iowa. Technical Report 341.Search in Google Scholar

Chan, J., E. Eisenstat, and G. Koop. 2016. “Large Bayesian Varmas.” Journal of Econometrics 192: 374–90, https://doi.org/10.1016/j.jeconom.2016.02.005.Search in Google Scholar

Chari, V., P. Kehoe, and E. McGrattan. 2008. “Are Structural Vars with Long-Run Restrictions Useful in Developing Business Cycle Theory?” Journal of Monetary Economics 55: 1337–52, https://doi.org/10.1016/j.jmoneco.2008.09.010.Search in Google Scholar

Cooley, T., and M. Dwyer. 1998. “Business Cycle Analysis without Much Theory: A Look at Structural Vars.” Journal of Econometrics 83: 57–88, https://doi.org/10.1016/s0304-4076(97)00065-1.Search in Google Scholar

Fernandez-Villaverde, J., J. Rubio-Ramirez, T. Sargent, and M. Watson. 2006. A’s, B’s, C’s (and D’s) for Understanding Vars. Mimeo.10.3386/t0308Search in Google Scholar

Forni, M., and L. Gambetti. 2014. “Testing for Sufficient Information in Structural Vars.” Journal of Monetary Economics 66: 124–36, https://doi.org/10.1016/j.jmoneco.2014.04.005.Search in Google Scholar

Forni, M., L. Gambetti, M. Lippi, and L. Sala. 2017. “Noisy News in Business Cycles.” American Economic Journal: Macroeconomics 9: 122–52, https://doi.org/10.1257/mac.20150359.Search in Google Scholar

George, E., D. Sun, and S. Ni. 2008. “Bayesian Stochastic Search for Var Model Restrictions.” Journal of Econometrics 142: 553–80, https://doi.org/10.1016/j.jeconom.2007.08.017.Search in Google Scholar

Ghosh, J., and D. B. Dunson. 2009. “Default Prior Distributions and Efficient Posterior Computation in Bayesian Factor Analysis.” Journal of Computational & Graphical Statistics 18: 306–20, https://doi.org/10.1198/jcgs.2009.07145.Search in Google Scholar PubMed PubMed Central

Gustafson, P. 2005. “On Model Expansion, Model Contraction, Identifiability and Prior Information: Two Illustrative Scenarios Involving Mismeasured Variables.” Statistical Science 20: 111–40, https://doi.org/10.1214/088342305000000098.Search in Google Scholar

Hamilton, J. 1994. Time Series Analysis. Princeton, NJ: Princeton University Press.10.1515/9780691218632Search in Google Scholar

Hansen, L. P., W. T. Roberds, and T. J. Sargent. 1991. “Time Series Implications of Present Value Budget Balance and of Martingale Models of Consumption and Taxes.” In Rational Expectations Econometrics, edited by L. P. Hansen, and T. J. Sargent Boulder: Westview Press.Search in Google Scholar

Hyndman, R. J. 1996. “Computing and Graphing Highest Density Regions.” The American Statistician 50: 120–6, https://doi.org/10.1080/00031305.1996.10474359.Search in Google Scholar

Imai, K., and D. A. van Dyk. 2005. “A Bayesian Analysis of the Multinomial Probit Model Using Marginal Data Augmentation.” Journal of Econometrics 124: 311–34, https://doi.org/10.1016/j.jeconom.2004.02.002.Search in Google Scholar

Koop, G. 2013. “Forecasting with Medium and Large Bayesian Vars.” Journal of Applied Econometrics 28: 177–203, https://doi.org/10.1002/jae.1270.Search in Google Scholar

Koop, G., R. León-González, and R. W. Strachan. 2010. “Efficient Posterior Simulation for Cointegrated Models with Priors on the Cointegration Space.” Econometric Reviews 29: 224–42.10.1080/07474930903382208Search in Google Scholar

Koop, G., R. León-González, and R. W. Strachan. 2012. “Bayesian Model Averaging in the Instrumental Variable Regression Model.” Journal of Econometrics 171: 237–50, https://doi.org/10.1016/j.jeconom.2012.06.005.Search in Google Scholar

Koop, G., and S. M. Potter. 1999. “Bayes Factors and Nonlinearity: Evidence from Economic Time Series.” Journal of Econometrics 88: 251–81, https://doi.org/10.1016/s0304-4076(98)00031-1.Search in Google Scholar

Korobilis, D. 2013. “Var Forecasting Using Bayesian Variable Selection.” Journal of Applied Econometrics 28: 204–30, https://doi.org/10.1002/jae.1271.Search in Google Scholar

Kuo, L., and B. Mallick. 1997. “Bayesian Semiparametric Inference for the Accelerated Failure-Time Model.” Canadian Journal of Statistics 25: 457–72, https://doi.org/10.2307/3315341.Search in Google Scholar

Kurmann, A., and C. Otrok. 2013. “News Shocks and the Slope of the Term Structure of Interest Rates.” The American Economic Review 103: 2612–32, https://doi.org/10.1257/aer.103.6.2612.Search in Google Scholar

Lippi, M., and L. Reichlin. 1994. “Var Analysis, Nonfundamental Representations, Blaschke Matrices.” Journal of Econometrics 63: 307–25, https://doi.org/10.1016/0304-4076(93)01570-c.10.1016/0304-4076(93)01570-CSearch in Google Scholar

Liu, J. S., and Y. N. Wu. 1999. “Parameter Expansion for Data Augmentation.” Journal of the American Statistical Association 94: 1264–74, https://doi.org/10.1080/01621459.1999.10473879.Search in Google Scholar

Luetkepohl, H. 2005. New Introduction to Multiple Time Series Analysis. Springer-Verlag.10.1007/978-3-540-27752-1Search in Google Scholar

Meng, X.-L., and D. van Dyk. 1999. “Seeking Efficient Data Augmentation Schemes via Conditional and Marginal Augmentation.” Biometrika 86: 301–20, https://doi.org/10.1093/biomet/86.2.301.Search in Google Scholar

Mol, C. D., D. Giannone, and L. Reichlin. 2008. “Forecasting Using a Large Number of Predictors: Is Bayesian Shrinkage a Valid Alternative to Principal Components?.” Journal of Econometrics 146: 318–28.10.1016/j.jeconom.2008.08.011Search in Google Scholar

Plagborg-Møller, M. 2016. Bayesian Inference on Structural Impulse Response Functions. Also available at http://scholar.harvard.edu/files/plagborg/files/irf_bayes.pdf.Search in Google Scholar

Poskitt, D., and W. Yao. 2017. “Vector Autoregressions and Macroeconomic Modeling: An Error Taxonomy.” Journal of Business & Economic Statistics 35: 407–19, https://doi.org/10.1080/07350015.2015.1077139.Search in Google Scholar

Verdinelli, I., and L. Wasserman. 1995. “Computing Bayes Factors Using a Generalization of the Savage-Dickey Density Ratio.” Journal of the American Statistical Association 90: 614–18, https://doi.org/10.1080/01621459.1995.10476554.Search in Google Scholar

Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/snde-2020-0016).

Received: 2020-02-06

Accepted: 2020-10-07

Published Online: 2020-10-26

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/snde-2020-0016

Keywords for this article

Bayesian estimation; noise shocks; savage dickey density ratio; structural identification; vector autoregressive moving average models