Estimates an SVEC by utilising a scoring algorithm.
SVEC(x, LR = NULL, SR = NULL, r = 1, start = NULL, max.iter = 100,
conv.crit = 1e-07, maxls = 1.0, lrtest = TRUE, boot = FALSE, runs = 100)
# S3 method for svecest
print(x, digits = max(3, getOption("digits") - 3), ...)Object of class ‘ca.jo’; generated by
ca.jo() contained in urca.
Matrix of the restricted long run impact matrix.
Matrix of the restricted contemporaneous impact matrix.
Integer, the cointegration rank of x.
Vector of starting values for \(\gamma\).
Integer, maximum number of iteration.
Real, convergence value of algorithm..
Real, maximum movement of the parameters between two iterations of the scoring algorithm.
Logical, over-identification LR test, the result is set
to NULL for just-identified system.
Logical, if TRUE, standard errors of the parameters
are computed by bootstrapping. Default is FALSE.
Integer, number of bootstrap replications.
the number of significant digits to use when printing.
further arguments passed to or from other methods.
A list of class ‘svecest’ with the following elements is
returned:
The estimated contemporaneous impact matrix.
The standard errors of the contemporaneous impact matrix,
if boot = TRUE.
The estimated long run impact matrix.
The standard errors of the long run impact matrix,
if boot = TRUE.
The variance-covariance matrix of the reduced form residuals times 100, i.e., \(\Sigma_U = A^{-1}BB'A^{-1'} \times 100\).
Vector, containing the ranks of the restricted long run and contemporaneous impact matrices.
Object of class ‘htest’, holding the Likelihood
ratio overidentification test.
Vector of used starting values.
Character, type of the SVEC-model.
The ‘ca.jo’ object ‘x’.
The supplied long run impact matrix.
The supplied contemporaneous impact matrix.
Integer, the supplied cointegration rank.
Integer, the count of iterations.
The call to SVEC().
Consider the following reduced form of a k-dimensional vector error correction model:
$$ A \Delta \bold{y}_t = \Pi \bold{y}_{t-1} + \Gamma_1 \Delta \bold{y}_{t-1} + \ldots + \Gamma_p \Delta \bold{y}_{t-p + 1} + \bold{u}_t \quad .$$
This VECM has the following MA representation:
$$ \bold{y}_t = \Xi \sum_{i=1}^t \bold{u}_i + \Xi^*(L)\bold{u}_t + \bold{y}_0^* \quad ,$$
with \(\Xi = \beta_{\perp} (\alpha_{\perp}'(I_K - \sum_{i=1}^{p-1}\Gamma_i)\beta_{\perp} )^{-1}\alpha_{\perp}'\) and \(\Xi^*(L)\) signifies an infinite-order polynomial in the lag operator with coefficient matrices \(\Xi^*_j\) that tends to zero with increasing size of \(j\).
Contemporaneous restrictions on the impact matrix \(B\) must be
supplied as zero entries in SR and free parameters as NA
entries. Restrictions on the long run impact matrix \(\Xi B\) have
to be supplied likewise. The unknown parameters are estimated by
maximising the concentrated log-likelihood subject to the imposed
restrictions by utilising a scoring algorithm on:
$$ \ln L_c(A, B) = - \frac{KT}{2}\ln(2\pi) + \frac{T}{2}\ln|A|^2 - \frac{T}{2}\ln|B|^2 - \frac{T}{2}tr(A'B'^{-1}B^{-1}A\tilde{\Sigma}_u) $$
with \(\tilde{\Sigma}_u\) signifies the reduced form variance-covariance matrix and \(A\) is set equal to the identity matrix \(I_K\).
If ‘start’ is not set, then normal random numbers are used as
starting values for the unknown coefficients. In case of an
overidentified SVEC, a likelihood ratio statistic is computed according to:
$$ LR = T(\ln\det(\tilde{\Sigma}_u^r) - \ln\det(\tilde{\Sigma}_u)) \quad , $$
with \(\tilde{\Sigma}_u^r\) being the restricted variance-covariance matrix and \(\tilde{\Sigma}_u\) being the variance covariance matrix of the reduced form residuals. The test statistic is distributed as \(\chi^2(K*(K+1)/2 - nr)\), where \(nr\) is equal to the number of restrictions.
Amisano, G. and C. Giannini (1997), Topics in Structural VAR Econometrics, 2nd edition, Springer, Berlin.
Breitung, J., R. Br<U+279E5B61>nn and H. L<U+34AE5C2F>hl (2004), Structural vector autoregressive modeling and impulse responses, in H. L<U+34AE5C2F>hl and M. Kr<e4>tzig (editors), Applied Time Series Econometrics, Cambridge University Press, Cambridge.
Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.
L<U+34AE5C2F>hl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.
# NOT RUN {
data(Canada)
vecm <- ca.jo(Canada[, c("prod", "e", "U", "rw")], type = "trace",
ecdet = "trend", K = 3, spec = "transitory")
SR <- matrix(NA, nrow = 4, ncol = 4)
SR[4, 2] <- 0
SR
LR <- matrix(NA, nrow = 4, ncol = 4)
LR[1, 2:4] <- 0
LR[2:4, 4] <- 0
LR
SVEC(vecm, LR = LR, SR = SR, r = 1, lrtest = FALSE, boot = FALSE)
# }
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