# SVEC: Estimation of a SVEC In cheaton/vars2: VAR Modelling

## Description

Estimates an SVEC by utilising a scoring algorithm.

## Usage

 1 2 3 4 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 γ. 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 Δ \bold{y}_t = Π \bold{y}_{t-1} + Γ_1 Δ \bold{y}_{t-1} + … + Γ_p Δ \bold{y}_{t-p + 1} + \bold{u}_t \quad .

This VECM has the following MA representation:

\bold{y}_t = Ξ ∑_{i=1}^t \bold{u}_i + Ξ^*(L)\bold{u}_t + \bold{y}_0^* \quad ,

with Ξ = β_{\perp} (α_{\perp}'(I_K - ∑_{i=1}^{p-1}Γ_i)β_{\perp} )^{-1}α_{\perp}' and Ξ^*(L) signifies an infinite-order polynomial in the lag operator with coefficient matrices Ξ^*_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 Ξ 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(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)

with \tilde{Σ}_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{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(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., Σ_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().

Bernhard Pfaff

## References

Amisano, G. and C. Giannini (1997), Topics in Structural VAR Econometrics, 2nd edition, Springer, Berlin.

Breitung, J., R. BrÃ¼ggemann and H. LÃ¼tkepohl (2004), Structural vector autoregressive modeling and impulse responses, in H. LÃ¼tkepohl and M. KrÃ¤tzig (editors), Applied Time Series Econometrics, Cambridge University Press, Cambridge.

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

LÃ¼tkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

SVAR, irf, fevd
  1 2 3 4 5 6 7 8 9 10 11 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)