Structural (Identified) VARs—733

the cointegrating vector in the system, you may edit the specification, but you should make certain that the resulting system is identified.

You should also note that while the standard VAR can be estimated efficiently by equation- by-equation OLS, this is generally not the case for the modified specification. You may wish to use one of the system-wide estimation methods (e.g. SUR) when estimating nonstandard VARs using the system object.

Estimate Structural Factorization

This procedure is used to estimate the factorization matrices for a structural (or identified) VAR. The full details of this procedure is given in “Structural (Identified) VARs” on

page 733. You must first estimate the structural factorization matrices using this proc in order to use the structural options in impulse responses and variance decompositions.

Structural (Identified) VARs

The main purpose of structural VAR (SVAR) estimation is to obtain non-recursive orthogonalization of the error terms for impulse response analysis. This alternative to the recursive Cholesky orthogonalization requires the user to impose enough restrictions to identify the orthogonal (structural) components of the error terms.

Let yt be a k -element vector of the endogenous variables and let Σ = E[ etet′] be the residual covariance matrix. Following Amisano and Giannini (1997), the class of SVAR models that EViews estimates may be written as:

estimated. The structural innovations ut are assumed to be orthonormal, i.e. its covariance matrix is an identity matrix E[ utut′] = I . The assumption of orthonormal innovations ut imposes the following identifying restrictions on A and B :

Noting that the expressions on either side of (24.13) are symmetric, this imposes

k( k + 1) ⁄ 2 restrictions on the 2k2 unknown elements in A and B . Therefore, in order to identify A and B , you need to supply at least 2k2 − k( k + 1 ) ⁄ 2 = k( 3k − 1) ⁄ 2 additional restrictions.

Specifying the Identifying Restrictions

As explained above, in order to estimate the orthogonal factorization matrices A and B , you need to provide additional identifying restrictions. We distinguish two types of identi-

734—Chapter 24. Vector Autoregression and Error Correction Models

fying restrictions: short-run and long-run. For either type, the identifying restrictions can be specified either in text form or by pattern matrices.

Short-run Restrictions by Pattern Matrices

For many problems, the identifying restrictions on the A and B matrices are simple zero exclusion restrictions. In this case, you can specify the restrictions by creating a named “pattern” matrix for A and B . Any elements of the matrix that you want to be estimated should be assigned a missing value “NA”. All non-missing values in the pattern matrix will be held fixed at the specified values.

For example, suppose you want to restrict A to be a lower triangular matrix with ones on the main diagonal and B to be a diagonal matrix. Then the pattern matrices (for a k = 3 variable VAR) would be:

You can create these matrices interactively. Simply use Object/New Object... to create two new 3 3 matrices, A and B, and then use the spreadsheet view to edit the values. Alternatively, you can issue the following commands:

matrix(3,3) pata

’ fill matrix in row major order pata.fill(by=r) 1,0,0, na,1,0, na,na,1 matrix(3,3) patb = 0

patb(1,1) = na patb(2,2) = na patb(3,3) = na

Once you have created the pattern matrices, select Proc/Estimate Structural Factorization... from the VAR window menu. In the SVAR Options dialog, click the Matrix button and the Short-Run Pattern button and type in the name of the pattern matrices in the relevant edit boxes.

Short-run Restrictions in Text Form

For more general restrictions, you can specify the identifying restrictions in text form. In text form, you will write out the relation Aet = But as a set of equations, identifying each element of the et and ut vectors with special symbols. Elements of the A and B matrices to be estimated must be specified as elements of a coefficient vector.

736—Chapter 24. Vector Autoregression and Error Correction Models

able to the first structural shock to be zero C2, 1 = 0 . Then the long-run response matrix will have the following pattern:

You can create this matrix with the following commands:

matrix(2,2) patc = na

patc(2,1) = 0

Once you have created the pattern matrix, select Proc/Estimate Structural Factorization... from the VAR window menu. In the SVAR Options dialog, click the Matrix button and the Long-Run Pattern button and type in the name of the pattern matrix in the relevant edit box.

To specify the same long-run restriction in text form, select Proc/Estimate Structural Factorization... from the VAR window and click the Text button. In the edit window, you would type the following:

@lr2(@u1)=0 ’ zero LR response of 2nd variable to 1st shock

where everything on the line after the apostrophe is a comment. This restriction begins with the special keyword “@LR#”, with the “#” representing the response variable to restrict. Inside the parentheses, you must specify the impulse keyword “@U” and the innovation number, followed by an equal sign and the value of the response (typically 0). We caution you that while you can list multiple long-run restrictions, you cannot mix short-run and long-run restrictions.

Note that it is possible to specify long-run restrictions as short-run restrictions (by obtaining the infinite MA order representation). While the estimated A and B matrices should be the same, the impulse response standard errors from the short-run representation would be incorrect (since it does not take into account the uncertainty in the estimated infinite MA order coefficients).

Some Important Notes

Currently we have the following limitations for the specification of identifying restrictions:

•The A and B matrices must be square and non-singular. In text form, there must be exactly as many equations as there are endogenous variables in the VAR. For short-run restrictions in pattern form, you must provide the pattern matrices for both A and B matrices.

738—Chapter 24. Vector Autoregression and Error Correction Models

number of endogenous variables in the VAR. In order to identify A and B , you need to provide at least k( k + 1) ⁄ 2 − 2k2 = k( 3k − 1) ⁄ 2 additional identifying restrictions. This is a necessary order condition for identification and is checked by counting the number of restrictions provided.

As discussed in Amisano and Giannini (1997), a sufficient condition for local identification can be checked by the invertibility of the “augmented” information matrix (see Amisano and Giannini, 1997). This local identification condition is evaluated numerically at the starting values. If EViews returns a singularity error message for different starting values, you should make certain that your restrictions identify the A and B matrices.

We also require the A and B matrices to be square and non-singular. The non-singularity condition is checked numerically at the starting values. If the A and B matrix is non-sin- gular at the starting values, an error message will ask you to provide a different set of starting values.

Sign Indeterminacy

For some restrictions, the signs of the A and B matrices are not identified; see Christiano, Eichenbaum, and Evans (1999) for a discussion of this issue. When the sign is indeterminate, we choose a normalization so that the diagonal elements of the factorization matrix A−1B are all positive. This normalization ensures that all structural impulses have positive signs (as does the Cholesky factorization). The default is to always apply this normalization rules whenever applicable. If you do not want to switch the signs, deselect the

Normalize Sign option from the Optimization Control tab of the SVAR Options dialog.

Estimation of A and B Matrices

Once you provide the identifying restrictions in any of the forms described above, you are ready to estimate the A and B matrices. Simply click the OK button in the SVAR Options dialog. You must first estimate these matrices in order to use the structural option in impulse responses and variance decompositions.

A and B are estimated by maximum likelihood, assuming the innovations are multivariate normal. We evaluate the likelihood in terms of unconstrained parameters by substituting out the constraints. The log likelihood is maximized by the method of scoring (with a Marquardt-type diagonal correction—See “Marquardt” on page 958), where the gradient and expected information matrix are evaluated analytically. See Amisano and Giannini (1997) for the analytic expression of these derivatives.

Optimization Control

Options for controlling the optimization process are provided in the Optimization Control tab of the SVAR Options dialog. You have the option to specify the starting values, maximum number of iterations, and the convergence criterion.