2. THE REDUCED FORM MODEL

Let {X_t} be a p-dimensional real valued vector time series generated by the unrestricted vector autoregression of finite order k:²

where A(L) is a p × p matrix polynomial of order k in the lag operator (L):

, with I_p a p × p identity matrix. Equation (1) can be extended to include drift terms, dummy variables, or exogenous variables. As these extensions merely complicate the notation without affecting the main results of this paper, only the simplest model is considered. The roots of det[A(z)] = 0 are all either equal to one or outside the unit circle, thereby excluding the possibility of explosive processes or seasonal unit roots. Moreover, {X_t} is assumed to be cointegrated of order (1,1) with cointegrating rank equal to r (Johansen, 1991). It then follows that {X_t} has a vector error correction model (VECM) representation (Engle and Granger, 1987):

where Δ = 1 − L is the first difference operator,

, and α and β are p × r matrices of full column rank such that −αβ′ = A(1). Assuming normality, this model can be estimated by maximum likelihood (Johansen, 1988, 1991).

From Granger's representation theorem (see, e.g., Johansen, 1991, Sect. 4) it follows that ΔX_t can also be expressed in terms of past disturbances:

with total impact matrix (C(1)) equal to

where Ψ is the mean lag matrix of the VAR representation and α_⊥ and β_⊥ are orthogonal complements of α and β, respectively: α_⊥′α = β_⊥′ β = 0.

3. THE STRUCTURAL VECM WITHOUT LONG-RUN RESTRICTIONS

To define the structural model, the reduced form disturbances {ε_t} are expressed in terms of the structural innovations {e_t}, which can be given an economic interpretation:

where B₀ is a nonsingular p × p matrix. From (5) and (1) it follows that

As Ω has p(p + 1)/2 distinct elements, this relationship imposes p(p + 1)/2 independent nonlinear restrictions on B₀, leaving p(p − 1)/2 elements free. Consequently p(p − 1)/2 additional independent restrictions have to be imposed on B₀ to exactly identify the structural model. These restrictions have to come from economic theory as they are not testable. It is also possible to impose more than p(p − 1)/2 restrictions, in which case the overidentifying restrictions are testable (Amisano and Giannini, 1997). Assuming these restrictions are all linear, they can be expressed in the following general implicit form:

where R denotes a g × p² matrix, imposing the g independent restrictions on B₀, vec is the column stacking operator, and d denotes a g-dimensional vector of deterministic constants (usually zeros). The implicit form (7) can be transformed into explicit form by calculating the null space of R:

where λ represents the free parameters of B₀ and R_⊥′ can be calculated as

Throughout this paper, H_xi matrices denote 0–1 matrices that select appropriate columns from the matrix x. The term H_R1 selects columns of R such that RH_R1 is nonsingular, whereas H_R2 selects the remaining columns.

The structural model is estimated in a two-step procedure. In the first step the reduced form model (2) is estimated, and in the second step λ is estimated conditional on the estimated reduced form parameters. The second step is performed by maximizing the following log-likelihood with respect to λ (in the just-identified case this is a concentrated likelihood function):

where c is a constant and

is the estimated covariance matrix of the residuals of the first step.

This procedure is consistent and also asymptotically efficient as long as no cross restrictions on the reduced form parameters on the one hand and B₀ on the other are imposed. This requirement ensures the asymptotic block diagonality of the covariance matrix of the structural model, with the estimators of the mean parameters in the one block and those of

in the other. The joint asymptotic distribution of

is given by

where

denotes convergence in distribution,

where [otimes ] is the Kronecker product and Σ_λ can be computed as the inverse of the information matrix of λ in (10). These covariances can be consistently estimated in a straightforward manner. It is important to realize that the β matrix may be assumed fixed—even if it is estimated—provided that it is properly identified. This result is due to the fact that the β matrix converges at rate T to its true value in that case (Lütkepohl and Reimers, 1992). Moreover, both covariances will be nonsingular if the B₀ matrix is properly identified.

To derive the impulse response functions, the VAR(k) representation given in (1) is rewritten in VAR(1) form:

where

and

Repeated substitution of Z_t−1 in (12) and subsequent substitution of Z_t and ν_t by X_t and e_t according to (13), (14), and (5), respectively, leads to

The asymptotic distribution of the impulse response functions (B₀,B₁,…,B_h) can be derived along the lines of Baillie (1987) and Lütkepohl (1989, 1990) and is given in Lütkepohl and Reimers (1992). Let b(h) denote vec(B₀,B₁,…,B_h). Then

where the (i + 1,j + 1)th (p² × p²) block of Σ_b(h) is given by

with

and

where W denotes the matrix that relates the VAR parameters to the relevant freely estimated stochastic VECM parameters (β may be assumed known):

A necessary precondition for (17) to hold is that the partial derivatives D_1,i and D_2,i in (18), evaluated at the true parameter values, do not contain zero rows. Benkwitz, Lütkepohl, and Neumann (2000) emphasize the importance of this requirement by showing that it is far from trivial in the case of a stationary univariate AR(1) process. They conclude that a method that does not work well for the easiest case cannot be expected to work well in a more complicated environment. However, the problem arises only if all elements in a row of [D_1,i,D_2,i] are equal to zero. Complex models involve more elements in the rows and are thus less likely to result in zero elements in the diagonal of Σ_b(h).

4. LONG-RUN RESTRICTIONS ON THE IMPULSE RESPONSES

From an economic point of view, it is often desirable to combine contemporaneous with long-run restrictions to identify B₀. These combinations are easily incorporated in the current setting. To do so the restriction matrix R in (7) and its outcome d are split into two parts:

where R_l and R_c refer to the long-run and contemporaneous restrictions of dimensions g_l × p² and g_c × p², respectively.

The long-run restrictions on the impulse responses of the structural VECM are directly related to the total impact matrix, C(1), which gives the ultimate effect of a shock in the reduced form disturbances on the level of the endogenous variables (see (4)). If the jth structural innovation is supposed to have a known (usually zero) impact on variable i in the long run, this can be imposed by restricting the i,j element of C(1)B₀ to this value.

in (21) equals

As the rank of C(1) equals (p − r), B₀ can be identified solely with long-run restrictions only if r < 2 (or r = p; see note 3).

The α_⊥ and β_⊥ matrices are not uniquely defined. To derive the asymptotic distribution of C(1) an explicit expression for α_⊥ is needed. In accordance with the procedure for R_⊥′ (see equation (9)), the following expression is used:

where H_α1′ selects columns of α′ such that α′H_α1′ is nonsingular and H_α2′ selects the remaining columns.

The asymptotic distribution of the impulse response function is considerably complicated by the introduction of long-run restrictions, as these restrictions on B₀ are functions of the first step parameters. Consequently, as shown by Mittnik and Zadrozny (1993), the partial derivative of B_j with respect to γ contains an extra element reflecting its impact on B₀. Moreover, the convenient block-diagonality of the covariance matrix of parameter estimates, with the ones of the mean on the one hand and those of

on the other, is destroyed. However, the following theorem shows that the two-step procedure can still be used to derive the asymptotic distribution of the impulse responses. As the stochastic nature of the restrictions is ignored in the second step of the estimation procedure, the block-diagonality still holds. The interaction between

is nevertheless taken into account, not via their covariance but via an extra term in the partial derivative of B₀ with respect to R.

THEOREM 1. The asymptotic distribution of the impulse response functions, subject to the identifying restrictions given in (7), (21), and (22) and estimated by the two-step procedure as outlined in Section 3, is given by

The (i + 1,j + 1)th (p² × p²) block of Σ_b(h) is given by

where the partial derivatives D_1,j and D_2,j and the two-step covariance matrices Σ_γ and Σ_λ are equivalent to the corresponding matrices in (18) and

where

where K_pp is the commutation matrix that transforms vec(B′) into vec(B) (Magnus and Neudecker, 1986),

with

where ι_p represents a column vector with ones of length p, and⁴

Proof. The proof is based on a first-order Taylor approximation of

around vec(B_j):

where

denote, respectively, the B_j matrix and λ vector derived under the assumption that all restrictions are nonstochastic. Notice that Σ_λ in (25) represents the variance of

estimated in the two-step procedure, which indeed embodies this assumption.

Provided that no row of [D_1,j + D_3,j,D_2,j] equals zero and that D_1,j + D_3,j and D_2,j are continuous functions, the result follows immediately from the well-known result for continuously differentiable functions of consistently estimated normally distributed parameters (e.g., Serfling, 1980, Theorem 3.3.A).

We start our proof by showing the correctness of the partial derivatives of (26). These results are derived by straightforward application of some basic rules for matrix differential calculus (Magnus and Neudecker, 1986):

Result (27) follows directly by applying (31) on (16). The partial derivative of B₀ with respect to R (equation (28)) has two elements, a direct effect and also an indirect effect via λ.⁵

Disregarding the impact of R on λ, the derivative reads:

For the indirect effect, we make use of a first-order Taylor approximation of Ω (see equation (6)). As B₀ is only a function of λ and R, one gets

As Ω is determined in the first step of the estimation procedure, it is not affected by changes in R. Consequently,

in (36) may be set to zero for the purpose of deriving the partial derivative of λ with respect to vec(R):

where x⁺ denotes the Moore–Penrose generalized inverse of x, which for a full-column matrix x is given by (x′x)⁻¹x′. The partial derivative of vec(Ω) with respect to vec(B₀) is given by (I_p² + K_pp)(B₀ [otimes ] I_p) (see equations (6) and (31)) using the fact that (B [otimes ] A)K_pp = K_pp(A [otimes ] B) for arbitrary p × p matrices A and B. Result (28) follows by filling in the partial derivatives of vec(B₀) with respect to R (equation (35)) and λ (= R_⊥′; see equation (8)), using the fact that ½(Ip² + K_pp) is symmetric and idempotent (Magnus and Neudecker, 1986), premultiplying (37) by the latter derivative, and summing the direct and indirect effects.

The partial derivative of R with respect to C(1) (equation (29)) follows from multiplying the partial derivative of R with respect to R_l (= I_p² [otimes ] [I_gl,0_gl×gc]′) and the partial derivative of R_l with respect to C(1) (which follows directly from (22)). As the contemporaneous restrictions are deterministic, these do not influence the result. The part related to (I_p [otimes ] C(1)) can only be checked manually.

The derivation of the partial derivative of C(1) with respect to γ will be split into the derivatives with respect to Γ_i on the one hand and with respect to α on the other.⁶

which completes the proof of the derivatives. █

For the asymptotic distribution to hold, all derivatives have to be continuous, and none of the rows of [D_1,i + D_3,i,D_2,i] is allowed to be zero. The continuity does not pose any problem here. With respect to the nonzero condition on the derivatives, there probably exist parameter values for which this condition is not fulfilled. Given the complexity of the models considered, however, and the fact that all parameters in a row of [D_1,i + D_3,i,D_2,i] have to be zero for the distribution to become degenerate, the practical problem of this assumption is not likely to be severe.