1. MOTIVATION AND RESULTS

The long-run relationship between variables is one of the most important topics in the econometric literature. Investigations of the long-run relationship have focused on the moving-average (MA) impact matrix, which is sometimes called the C matrix, as one of the important factors. For a cointegrated vector error-correction (VEC) model, it is well known that the MA impact matrix is expressed using parameters in the model and that we can get the estimator of the C matrix by the maximum likelihood (ML) method (Johansen, 1988, 1991). Statistical inference about the MA impact matrix is investigated by Johansen (1995) and Paruolo (1997).

As a related concept, the long-run impulse response function, which is defined as the limit of the short-run impulse response function, has also been discussed in the literature. In particular, Phillips (1998) derived the limiting distribution of the ML estimator of the long-run impulse response function. It is implicit in many papers that the long-run impulse response function is equivalent to the MA impact matrix, although their parametric expressions appear to be very different at first sight. This note shows that these two expressions are in fact parametrically equivalent. Some of our calculations may be found in Faliva and Zoia (2002), Johansen (2002), and Omtzigt and Paruolo (2005), among others, and readers may also refer to these papers.

To obtain the parametric expressions, let us consider the following n-variate VEC model:

where α and β are n × r full column rank matrices and {ε_t} ∼ i.i.d.N(0,Ω). Assume that α_⊥′Γβ_⊥ are nonsingular where α_⊥ and β_⊥ are n × (n − r) full column rank matrices such that β_⊥′ β = α_⊥′α = 0, and

, so that y_t is cointegrated of order r. According to Johansen (1995) this process is expressed as

where C = β_⊥(α_⊥′Γβ_⊥)⁻¹α_⊥′ is the MA impact matrix, C₁(L) = (C(L) − C(1))/(1 − L) with C(L) being a lag polynomial when [utri ]y_t is represented as the vector MA process, [utri ]y_t = C(L)(μ + ε_t), and y₀* is a stochastic component such that β′y₀* = 0. Johansen (1995) and Paruolo (1997) showed that for p > 1 the limiting distribution of

, the maximum likelihood (ML) estimator of C, is given by

where we used vec(·) to stack the rows of a matrix into a column vector, ξ′ = [(C′Γ′ − I_n)α,C′,…,C′], α = α(α′α)⁻¹ and Σ is the variance matrix of [(β′y_t−1)′,[utri ]y_t−1′,…,[utri ]y_t−p+1′]′.

On the other hand, Phillips (1998) showed that the impulse response function of the model (1) converges to

when the lead time goes to infinity, where Q′ = [β,H,0,…,0] is n × (n(p − 1) + r) with H = [β_⊥,β] and where E₁₂ and E₂₂ are submatrices of the companion matrix E given by

Here β_⊥ and β are supposed to be orthonormal so that H′H = I_n. Phillips also showed that the limiting distribution of

, the ML estimator of Θ, is normal, and its expression is given by Theorem 3.1(ii) of Arai and Yamamoto (2000):

where P = Q′(I_n(p−1)+r − E₂₂′)⁻¹J′ with J = diag{I_r,H,…,H} being an (n(p − 1) + r) square matrix.

The equivalence between C and Θ is easily proved. Because ∂y_t+h /∂ε_t = C + C_1,h from (2) where C_1,h is the hth coefficient matrix of C₁(L), and because C_1,h → 0 as h → ∞ because C₁(L) is a convergence sequence as shown by Johansen (1995), we have ∂y_t+h /∂ε_t → C = Θ. Moreover, it is proved that the parametric functions C and Θ are the same, that is,

and

Remark 1. Because (6) and (7) hold parametrically, the same numerical equalities can be established using the ML estimators. Then the analysis based on (3) is exactly the same as that based on (5).

Remark 2. The limiting distribution given by (3) is established for p > 1, and many special cases may arise when p = 1 as given by Theorem 7.1 of Paruolo (1997), which also explains how to make inference for these special cases.

Remark 3. From the expression C = β_⊥(α_⊥′Γβ_⊥)⁻¹α_⊥′, we can easily see that C is invariant to normalizations of α_⊥ and β_⊥. Also the choice of normalization of β does not matter, as long as α is normalized accordingly. Then, as adopted by Phillips (1998), we can choose H = [β_⊥,β] as an orthonormalized matrix.