| SVAR {vars} | R Documentation |
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().
SVAR(x, estmethod = c("scoring", "direct"), Amat = NULL, Bmat = NULL,
start = NULL, max.iter = 100, conv.crit = 0.1e-6, maxls = 1.0,
lrtest = TRUE, ...)
## S3 method for class 'svarest'
print(x, digits = max(3, getOption("digits") - 3), ...)
x |
Object of class ‘ |
estmethod |
Character, either scoring for estimating the SVAR-model with the scoring algorithm (default), or directly minimizing the negative log-likelihood. |
start |
Vector with starting values for the parameters to be optimised. |
lrtest |
Logical, over-identification LR test, the result is set
to |
max.iter |
Integer, maximum number of iteration (used if
|
conv.crit |
Real, convergence value of algorithm (used if
|
maxls |
Real, maximum movement of the parameters between two
iterations of the scoring algorithm (used if
|
Amat |
Matrix with dimension (K \times K) for A- or AB-model. |
Bmat |
Matrix with dimension (K \times K) for B- or AB-model. |
digits |
the number of significant digits to use when printing. |
... |
further arguments passed to or from other methods. |
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.
A list of class ‘svarest’ with the following elements is
returned:
A |
If A- or AB-model, the matrix of estimated coefficients. |
Ase |
The standard errors of ‘ |
B |
If A- or AB-model, the matrix of estimated coefficients. |
Bse |
The standard errors of ‘ |
LRIM |
For Blanchard-Quah estimation |
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 ‘ |
opt |
List object returned by |
start |
Vector of starting values. |
type |
SVAR-type, character, either ‘A-model’, ‘B-model’ or ‘AB-model’. |
var |
The ‘ |
iter |
Integer, the count of iterations. |
call |
The |
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 ## Estimation method scoring SVAR(x = var.2c, estmethod = "scoring", Amat = amat, Bmat = NULL, max.iter = 100, maxls = 1000, conv.crit = 1.0e-8) ## Estimation method direct SVAR(x = var.2c, estmethod = "direct", Amat = amat, Bmat = NULL, hessian = TRUE, method="BFGS")