SVEC: Estimation of a SVEC

In vars: VAR Modelling

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)
## S3 method for class 'svecest'
print(x, digits = max(3, getOption("digits") - 3), ...)

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.
digits	the number of significant digits to use when printing.
...	further arguments passed to or from other methods.

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.

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 ‘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.
iter	Integer, the count of iterations.
call	The call to SVEC().

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

SVAR, irf, fevd

Examples

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)

vars documentation built on June 24, 2024, 9:08 a.m.