vars (version 1.1-9)

SVEC: Estimation of a SVEC

Description

Estimates an SVEC by utilising a scoring algorithm.

Usage

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)

Arguments

x

Object of class ca.jo; generated by ca.jo() contained in urca.

LR

Matrix of the restricted long run impact matrix.

SR

Matrix of the restricted contemporaneous impact matrix.

r

Integer, the cointegration rank of x.

start

Vector of starting values for $\gamma$.

max.iter

Integer, maximum number of iteration.

conv.crit

Real, convergence value of algorithm..

maxls

Real, maximum movement of the parameters between two iterations of the scoring algorithm.

lrtest

Logical, over-identification LR test, the result is set to NULL for just-identified system.

boot

Logical, if TRUE, standard errors of the parameters are computed by bootstrapping. Default is FALSE.

runs

Integer, number of bootstrap replications.

Value

• A list of class svecest with the following elements is returned:
• SR
  The estimated contemporaneous impact matrix.
• SRse
  The standard errors of the contemporaneous impact matrix, if boot = TRUE.
• LR
  The estimated long run impact matrix.
• LRse
  The standard errors of the long run impact matrix, if boot = TRUE.
• Sigma.U
  The variance-covariance matrix of the reduced form residuals times 100, i.e., $\Sigma_U = A^{-1}BB'A^{-1'} \times 100$.
• Restrictions
  Vector, containing the ranks of the restricted long run and contemporaneous impact matrices.
• LRover
  Object of class code{htest}, holding the Likelihood ratio overidentification test.
• start
  Vector of used starting values.
• type
  Character, type of the SVEC-model.
• var
  The ca.jo object x.
• LRorig
  The supplied long run impact matrix.
• SRorig
  The supplied contemporaneous impact matrix.
• r
  Integer, the supplied cointegration rank.
• call
  The call to SVEC().

encoding

latin1

concept

• SVEC
• SVECM
• Structural VECM
• Structural Vector Error Correction Model
• B-model

Details

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.

References

Amisano, G. and C. Giannini (1997), Topics in Structural VAR Econometrics, 2nd edition, Springer, Berlin. Breitung, J., R. Br�ggemann and H. L�tkepohl (2004), Structural vector autoregressive modeling and impulse responses, in H. L�tkepohl and M. Kr�tzig (editors), Applied Time Series Econometrics, Cambridge University Press, Cambridge. Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton. L�tkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

SVAR, SVAR2, irf, fevd

Examples

data(Canada)
vec.can <- ca.jo(Canada, K = 2, spec = "transitory")
summary(vec.can)
LR <- matrix(NA, nrow = 4, ncol = 4)
SR <- diag(NA, 4)
SR[2, 1] <- NA
SR[3, 1] <- NA
SR[4, 1] <- NA
SVEC(vec.can, r = 2, LR = LR, SR = SR, lrtest = TRUE)