SVAR {vars} | R Documentation |
Estimates an SVAR (either ‘A-model’, ‘B-model’ or
‘AB-model’) by numerically minimising the negative
log-likelihood using optim()
.
SVAR(x, Amat = NULL, Bmat = NULL, start = NULL, ...)
x |
Object of class ‘varest ’; generated by
VAR() . |
Amat |
Matrix with dimension (K times K) for A- or AB-model. |
Bmat |
Matrix with dimension (K times K) for B- or AB-model. |
start |
Vector with starting values for the parameters to be optimised. |
... |
Arguments that arr passed to optim() . |
Consider the following structural form of a k-dimensional vector autoregressive model:
A y_t = A_1^*y_{t-1} + ... + A_p^*y_{t-p} + C^*D_t + Bvarepsilon_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. The
reduced form residuals can be obtained from the above equation
via the relation: u_t =
A^{-1}Bvarepsilon_t, with variance-covariance matrix
Σ_U = A^{-1}BB'A^{-1'}.
Hence, 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}Atilde{Σ}_u)
If ‘start
’ is not set, then 0.1
is used as
starting values for the unknown coefficients. If the function is
called with ‘hessian = TRUE
’, the standard errors of the
coefficients are returned as list elements ‘Ase
’ and/or
‘Bse
’, where applicable.
Finally, in case of an
overidentified SVAR, a likelihood ratio statistic is computed according to:
LR = T(lndet(tilde{Σ}_u^r) - lndet(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 chi^2(nr - 2K^2 - frac{1}{2}K(K + 1)), where nr is equal to the number of restrictions.
A list of class ‘svarest
’ with the following elements is
returned:
A |
If A- or AB-model, the matrix of estimated coeffiecients. |
Ase |
If ‘hessian = TRUE ’, the standard errors of
‘A ’, otherwise a null-matrix is returned. |
B |
If A- or AB-model, the matrix of estimated coeffiecients. |
Bse |
If ‘hessian = TRUE ’, the standard errors of
‘B ’, otherwise a null-matrix is returned. |
LRIM |
For Blanchard-Quah estimation LRIM is the estimated
long-run impact matrix; for all other SVAR models LRIM is
NULL . |
Sigma.U |
The variance-covariance matrix of the reduced form residuals times 100, i.e., Σ_U = A^{-1}BB'A^{-1'} times 100. |
LR |
Object of class ‘code{htest}’, holding the Likelihood ratio overidentification test. |
opt |
List object returned by optim() . |
start |
Vector of starting values. |
type |
SVAR-type, character, either ‘A-model’, ‘B-model’ or ‘AB-model’. |
var |
The ‘varest ’ object ‘x ’. |
call |
The call to SVAR() . |
Bernhard Pfaff
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.
data(Canada) var.2c <- VAR(Canada, p = 2, type = "const") amat <- diag(4) diag(amat) <- NA amat[2, 1] <- NA amat[4, 1] <- NA SVAR(var.2c, Amat = amat, Bmat = NULL, hessian = TRUE, method="BFGS")