2. PROOFS AND DISCUSSION

Let X_t be a p-dimensional vector of I(1) series and let the number of cointegrating relations among them be r, where r < p, so that the series share m = p − r common trends. In that case, X_t has a VECM representation of infinite order that in practice is approximated by the finite-order VECM

where π(L) = I_p − π₁ L − π₂ L² − ··· − π_q−1 L^q−1, L is the lag operator, Δ is the first difference operator, and e_t satisfies E(e_t) = 0, E(e_t e_t′) = Ω, and E(e_t e_s′) = 0, t ≠ s. Both α and β are (p × r) matrices where the columns of β are the coefficients in the cointegrating relations and the rows of α are the loadings on the error-correction terms β′X_t−1, respectively. In (1), under the condition that e_t is a vector of independent Gaussian innovations, variable x_i,t can be shown to be weakly exogenous with respect to the matrix of cointegrating vectors if the r error-correction terms do not appear in the equation for Δx_i,t so that α_ik = 0 for k = 1,…, r; see Johansen (1992, Thm. 1, p. 392) and Urbain (1992, Prop. 1, p. 193).

The VECM can be inverted to obtain the moving average representation for ΔX_t,

where C(L) = I_p + C₁ L + C₂ L² + ···. The long-run impacts of the reduced-form shocks on the levels of the series in X_t are given by C(1), which is a (p × p) matrix of rank m. Specifically, as shown in Johansen (1991),

where α_⊥ and β_⊥ are (p × m) matrices of rank m such that α′α_⊥ = 0 and β′β_⊥ = 0 and the (m × m) full rank matrix γ is given by (α_⊥′π(1)β_⊥)⁻¹.

Gonzalo and Ng (2001) define the (p × 1) vector u_t, whose first m elements are the permanent shocks and remaining r elements are the transitory shocks, as u_t = Ge_t where

and E(u_tu_t′) = GΩG′. The (p × p) matrix G is assumed to be of full rank.¹

The long-run impact matrix of the permanent-transitory shocks u_t is C(1)G⁻¹. It is easy to show that the last r column vectors of this long-run impact matrix are all zero as required for a permanent-transitory decomposition after noting that

This expression for G⁻¹ appears in Gonzalo and Granger (1995) and also in Levtchenkova, Pagan, and Robertson (1996).

Gonzalo and Ng (2001) apply the following transformation to orthogonalize the permanent-transitory shocks:

where H is the unique lower triangular matrix such that

and by construction E(v_t v_t′) = I_p. The long-run impact matrix of the orthogonal permanent-transitory shocks is C(1)G⁻¹H. The permanent-transitory decomposition is preserved because postmultiplication of C(1)G⁻¹ by the lower triangular matrix H results in the matrix where the last r column vectors are all zero.

The orthogonal shocks obtained from the Gonzalo and Ng (2001) procedure and those obtained directly from the VECM are related when some of the variables in the system are weakly exogenous, as the following proposition and corollaries show.

PROPOSITION. Let X_t be a p-dimensional vector of I(1) variables with r cointegrating vectors and m = p − r common trends. Suppose there are m weakly exogenous variables with respect to the cointegrating vectors and that they are ordered first in the vector X_t. Then the orthogonal shocks obtained directly from the VECM by imposing a recursive structure on the contemporaneous impacts of the errors constitute a permanent-transitory decomposition where the permanent shocks are identical to those from the Gonzalo and Ng (2001) permanent-transitory decomposition.

Proof. Partition the (p × r) matrix α as

where α_m and α_r are, respectively, the first m and last r (= p − m) rows of α. Using Gaussian elimination with complete pivoting, the rows of α can be uniquely rearranged such that the first r rows are independent. This is achieved by premultiplying α by the unique (p × p) matrix

where I_rr and I_mm are (r × r) and (m × m) identity matrices and 0_rm and 0_mr are (r × m) and (m × r) null matrices, respectively. Then

where α_r is an (r × r) matrix of full rank. The orthogonal complement of α is chosen as α_⊥′ = [−α_mα_r⁻¹ I_mm] F. Substituting for F, this becomes

When the weakly exogenous variables are ordered first and are equal in number to m, α_m is an (m × r) null matrix. Thus (8) becomes

It then follows from (9) that (3) may be written as

Thus, the reduced-form errors constitute a permanent-transitory decomposition. Following Sims (1980), orthogonalize the errors from the VECM by imposing a recursive structure on their contemporaneous impacts by applying the transformation

where E(v_t v_t′) = I_p and A is the unique lower triangular matrix such that AA′ = Ω. From (2), (10), and (11), the long-run impact matrix of the orthogonalized shocks is

Because A is lower triangular, the orthogonal shocks constitute a permanent-transitory decomposition because the last r column vectors of C(1)A are all zero.

Now partition the matrices H, A, and G as follows:

where H_mm, H_rr, A_mm, and A_rr are lower triangular, because H and A are lower triangular, and where G is obtained from (4) and (9). Partitioning Ω conformably with H and A, (7) shows that

for the case under consideration. The lower triangular matrix H_mm is the unique Cholesky factor of Ω_mm. But by construction

so that the lower triangular matrix A_mm is also the unique Cholesky factor of Ω_mm. Thus H_mm⁻¹ and A_mm⁻¹ are equal to the inverse of the unique Cholesky factor of Ω_mm and therefore equal to each other. Using the formula for the inverse of a partitioned matrix (Dhrymes, 2000, Prop. 2.31, p. 43), the first m orthogonal shocks are v_mt = A_mm⁻¹e_mt, where e_mt is the vector of the first m elements of e_t and, for the Gonzalo and Ng (2001) decomposition, the permanent shocks are v_mt = H_mm⁻¹e_mt. Because H_mm⁻¹ = A_mm⁻¹, it immediately follows that v_mt = H_mm⁻¹e_mt = A_mm⁻¹e_mt. █

COROLLARY 1. In addition, when there is only one cointegrating relation (r = 1), the orthogonal shocks obtained by imposing a contemporaneous recursive structure on the VECM errors are identical to those in a Gonzalo and Ng (2001) orthogonal permanent-transitory decomposition.

Proof. In this case, G is lower triangular. It follows that H⁻¹G is also lower triangular and G⁻¹H is the unique Cholesky factor of Ω. Because A is also the unique Cholesky factor of Ω, A⁻¹ = H⁻¹G so that this is an exact special case of the Gonzalo and Ng (2001) procedure. █

COROLLARY 2. Suppose there are 0 < j < m weakly exogenous variables with respect to the cointegrating vectors and they are ordered first in X_t. Then the orthogonal shocks obtained by imposing a contemporaneous recursive structure on the VECM errors do not constitute a permanent-transitory decomposition; however, the first j of these shocks are identical to the first j permanent shocks in the Gonzalo and Ng (2001) orthogonal permanent-transitory decomposition.

Proof. Partition the (m × r) matrix α_m as

where α_j and α_m−j are, respectively, the first j and last (m − j) rows of α_m. Given that there are j < m weakly exogenous variables and that they are ordered first, α_j is the (j × r) null matrix. Then from (8), the matrix G can be written as

Each row of the matrix α_m−j has at least one nonzero element; otherwise there would be more than j weakly exogenously variables. Thus, the ((m − j) × r) matrix −α_m−jα_r⁻¹ is not null, and therefore the (m × r) matrix −α_mα_r⁻¹ is not null either. Using an expression analogous to (8) for β_⊥, it is straightforward to show from (3) that the upper right-hand block of C(1) is the (m × r) matrix −γα_mα_r⁻¹, which is not null as γ is of full rank. Thus, the last r column vectors of C(1) are not null so that a permanent-transitory decomposition does not pertain in this case.

Now partition H and A conformably with G as follows:

where all the matrices along the principal diagonal of H and A are, respectively, lower triangular because H and A are both lower triangular. Also partition Ω conformably with H, A, and G. Then it follows from (7) and (15) that H_jj H_jj′ = Ω_jj so that H_jj is the unique Cholesky factor of Ω_jj. But by construction A_jj A_jj′ = Ω_jj so that A_jj is also the unique Cholesky factor of Ω_jj. It immediately follows that v_jt = H_jj⁻¹e_jt = A_jj⁻¹e_jt. █