Identification structural shocks

According to Breitung et al. [24], if the process yt is I(0), the effects of shocks in the

variables of a given system are most seen in its Wold moving average (MA) represen- tation, yt= ut+Ψ1ut−1+Ψ2ut−2+ ..., where Ψ0= IK, Ψs=Psj=1Ψs−jΦj and

Φj is K × K matrix of coefficients of VAR(p) model, equation 3.1. The coefficients

ψij of the matrices Ψs is interpreted as the response to the impulse of structural

shocks in the system. Note that, obtain forecast error impulse responses is feasible if Σu is diagonal, otherwise, the forecast error impulse responses are not valid, because

the underlying shocks are not likely to occur in isolation if the components of ut are

instantaneously correlated.

In practice, one way to obtain orthogonal innovations is to use Choleski decom- position of the white noise covariance matrix Σu = PP0 such that P is a lower

triangular matrix with positive elements on the main diagonal.

According to Breitung et al. [24] and Lütkepohl [77] there are different procedures used to non-sample information in specifying and then obtaining unique impulse responses. The three most used are associated with the so-called A - model, B - model and AB - model (see also Amisano and Giannini [1]).

The A - model

Lets the structural errors from equation 3.34 have the following MA representation

where the matrices Θj= ΨjA−1 for j = 1, 2, ...; represent the response to structural

shocks εt assumed to be mutually uncorrelated. The relation between structural

innovations and residuals from the VAR model 3.1 are εt= Aut, which is called the

A - model. For this model, the identifying restrictions are imposed on A matrix such that, the product Aut= εt has a diagonal covariance matrix. If it is plausible

that A has a one in the main diagonal, they are necessary K(K − 1)/2 restrictions on A to ensure just identified shocks, consequently identified impulse response.

Summarizing, according to Breitung et al. [24] and Lütkepohl [77], in A - model, the set of innovations εt is modelled in the system of equations as a function of

the residuals such that Aut = εt and the linear restrictions on A can be write in

explicit form as vec(A) = RAγA + rA, where γA represent all unrestrited elements

of A, RA is a suitable matrix with zero and one elements and rA is the vector of

normalized constants.

The B - model

In SVAR model equation 3.34, because the shocks are not observed, some assump- tions are need to identify them. For instance we required that the structural shocks should be orthogonal, and that the structural shock εt are assumed to be related to

the model residuals by linear relations ut= Bεt, where B is a K × K matrix. Note

that we assume that for Aut= Bεt and the matrix A = IK.

Using the relation between model residuals and structural innovations, one can write equation 3.34 as follows:

Ay_t=Φ∗₁yt−1+ ... +Φ∗pyt−p+ Bεt (3.36)

where εt∼ N (0, IK), Φ∗ are K × K coefficient matrices, B is K × K matrix such

that Σu= BΣεB0. Because the covariance matrix of εt are normalized with one in

diagonal, the covariance matrix of residuals became P

u= BB0, where B is lower

triangular chosen by Choleski decomposition.

According to Lütkepohl and Krätzig [78], the equality ut= Bεt where the struc-

tural shock εt ∼ N (0, IK) is the so called B-model. Due to the symmetry of the

covariance matrix of P

u, we specify only K(K + 1)/2 different equations and need

additional K(K − 1)/2 restrictions to be imposed to identify all K2 elements of B. In summary, for B-model, the identification consists to exclude some linear combinations of the structural shocks by imposing restriction of the form vec(B) = RBγB+ rB, where γB contains unrestricted elements of B, RB are the restricted

3 . 5 . S T RU C T U R A L VA R A N D S V E C A N A LY S I S

The AB model

According to Lütkepohl [77], situation where is possible to consider both types of restrictions, that is, from equation 3.36, Aut = Bεt where εt ∼ N (0, IK), we are

facing a model known as AB - model which relates the reduced form errors ut to

the underlying structural shocks εt.

Identification the structural shocks can be done, imposing restrictions on A and B parameter matrices. Here for K-variables K(K − 1)/2 restrictions are necessary for orthogonalizing the shocks because there are K(K − 1)/2 different instantaneous covariances. Detail of the procedures can be found in Breitung et al. [24] and Galí [50].

For this model, from a previous relation between residuals of the model and structural shock, we get ut= A−1Bεt, and the corresponding covariance matrix is

P

u= A−1BB0A−10. Thus, we have K(K + 1)/2 equations. Meaning that we need

K2 elements for each matrix A and B, additionally 2K2−1₂K(K + 1) restrictions

to identify all 2K2 elements of A and B at least locally. Even considering that the matrix A has elements one on main diagonal, 2K2− K −1₂K(K + 1) restrictions are

needed for identification of the AB - model.

In order to avoid the large number of restrictions in most applications is used the special cases just discussed. The A - model where B = IK or the B-model with

A = I_K. According to Breitung et al. [24] and Lütkepohl [77], considering the general case AB-model, the constraints are normally normalized and they can be written in the form of linear equations.

vec(A) = RAγA+ rA and vec(B) = RBγB+ rB (3.37)

where RA and RB are possible restricted matrices with zero and ones, γA and γB

are unrestricted vectors with free parameters and finally rA and rB are vectors of

fixed parameters.

Identification SVEC model

If all or some variables in the system are integrated with r cointegrating vectors, which describe the long-run relationships between variables, the VEC model, equation 3.30 is an appropriate model and the corresponding SVEC model is given by,

A∆yt=Π∗yt−1+Γ∗1∆yt−1+ ... +Γ∗p−1∆yt−p+1+ Bεt (3.38)

where Π∗ and Γ∗_i are structural form parameters matrices and εt is a structural

error term, B is a K × K matrix of impact of shocks and matrix A models the instantaneous relations among the variables in the system.

According to Pfaff [98], in order to identify the structural form parameters, we need to considers the Beveridge-Nelson (Beveridge and Nelson [10]) moving average representation of the variables yt if they adhere to the VEC model process as in

equation 3.30. yt=Ξ t X i=1 ui | {z } I(1) + ∞ X j=0 Ξ∗_jut−j | {z } I(0) +y∗₀ (3.39)

where the matrix Ξ is defined as Ξ = β_⊥α0_⊥(Ik−Pp−1i=1Γi)β⊥

−1

α0_⊥, with α⊥ and

β⊥ indicate orthogonal components of α and β respectively. The first term I(1) is

the common trend that drives the system of yt, the second term I(0) it bounded by

the infinite sum because the series ofΞ∗_j converges to zero as j → ∞ and y∗₀ contains all initial values.

According to Breitung et al. [24] and Pfaff [98], since the long-run effects of shocks is captured on the common trend, this is the centre of modelling Structural VEC model.

Replacing ut in equation 3.39 by A−1Bεt and then, assuming that εt∼ N (0, IK)

and the matrix A = I_k, then the impact of the short-run or transitory orthogonal shocks are obtained by Ξ∗_jB and the long-run by ΞB in the same way as in the stationary VAR model.

For example in B - model using the relation ut = Bεt, is necessary at least

K(K − 1)/2 restrictions to identify B. But if rank(ΞB) = K − r, where r is the rank

of the system yt, means that r columns have zeros in the matrix of long-run, which

corresponds to r structural innovations having transitory shocks, the remainder

m = K −r structural innovations have permanent shocks. To identify the permanent

shocks exactly we need m × (m − 1)/2 additional restrictions. Similarly r(r − 1)/2 additional contemporaneous restrictions needed to identify the transitory shocks. Continuing with the assumption that A = I_K, we have just enough restrictions to identify B and ΞB. In this case we have a model exactly identified, if we impose additional restrictions on the parameters, so it would be an overidentified model, and the overidentified restriction could be tested.