Description

Estimates an SVAR (either ‘A-model’, ‘B-model’ or ‘AB-model’) by using a scoring algorithm or by directly minimising the negative log-likelihood with optim().

Usage

Arguments

Details

Consider the following structural form of a k-dimensional vector autoregressive model:

A \bold{y}_t = A_1^*\bold{y}_{t-1} + … + A_p^*\bold{y}_{t-p} + C^*D_t + B\bold{\varepsilon}_t

The coefficient matrices (A_1^* | … | A_p^* | C^*) might now differ from the ones of a VAR (see ?VAR). One can now impose restrictions on ‘A’ and/or ‘B’, resulting in an ‘A-model’ or ‘B-model’ or if the restrictions are placed on both matrices, an ‘AB-model’. In case of a SVAR ‘A-model’, B = I_K and conversely for a SVAR ‘B-model’. Please note that for either an ‘A-model’ or ‘B-model’, K(K-1)/2 restrictions have to be imposed, such that the models' coefficients are identified. For an ‘AB-model’ the number of restrictions amounts to: K^2 + K(K-1)/2.
For an ‘A-model’ a (K \times K) matrix has to be provided for the functional argument ‘Amat’ and the functional argument ‘Bmat’ must be set to ‘NULL’ (the default). Hereby, the to be estimated elements of ‘Amat’ have to be set as ‘NA’. Conversely, for a ‘B-model’ a matrix object with dimension (K \times K) with elements set to ‘NA’ at the positions of the to be estimated parameters has to be provided and the functional argument ‘Amat’ is ‘NULL’ (the default). Finally, for an ‘AB-model’ both arguments, ‘Amat’ and ‘Bmat’, have to be set as matrix objects containing desired restrictions and ‘NA’ values. The parameters are estimated by minimising the negative of the concentrated log-likelihood function:

\ln L_c(A, B) = - \frac{KT}{2}\ln(2π) + \frac{T}{2}\ln|A|^2 - \frac{T}{2}\ln|B|^2 - \frac{T}{2}tr(A'B'^{-1}B^{-1}A\tilde{Σ}_u)

Two alternatives are implemented for this: a scoring algorithm or direct minimization with optim(). If the latter is chosen, the standard errors are returned if SVAR() is called with ‘hessian = TRUE’.

If ‘start’ is not set, then 0.1 is used as starting values for the unknown coefficients.

The reduced form residuals can be obtained from the above equation via the relation: \bold{u}_t = A^{-1}B\bold{\varepsilon}_t, with variance-covariance matrix Σ_U = A^{-1}BB'A^{-1'}.

Finally, in case of an overidentified SVAR, a likelihood ratio statistic is computed according to:

LR = T(\ln\det(\tilde{Σ}_u^r) - \ln\det(\tilde{Σ}_u)) \quad ,

with \tilde{Σ}_u^r being the restricted variance-covariance matrix and \tilde{Σ}_u being the variance covariance matrix of the reduced form residuals. The test statistic is distributed as χ^2(nr - 2K^2 - \frac{1}{2}K(K + 1)), where nr is equal to the number of restrictions.

Value

A list of class ‘svarest’ with the following elements is returned:

Author(s)

Bernhard Pfaff

References

Amisano, G. and C. Giannini (1997), Topics in Structural VAR Econometrics, 2nd edition, Springer, Berlin.

Breitung, J., R. Brüggemann and H. Lütkepohl (2004), Structural vector autoregressive modeling and impulse responses, in H. Lütkepohl and M. Krätzig (editors), Applied Time Series Econometrics, Cambridge University Press, Cambridge.

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR, SVEC, logLik, irf, fevd

Examples