SVAR: Estimation of a SVAR

In vars: VAR Modelling

Estimation of a SVAR

Description

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().

Usage

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), ...)

Arguments

x	Object of class ‘varest’; generated by VAR().
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 NULL for just-identified system.
max.iter	Integer, maximum number of iteration (used if estmethod = "scoring").
conv.crit	Real, convergence value of algorithm (used if estmethod = "scoring").
maxls	Real, maximum movement of the parameters between two iterations of the scoring algorithm (used if estmethod = "scoring").
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.

Details

Consider the following structural form of a k-dimensional vector autoregressive model:

A \bold{y}_t = A_1^*\bold{y}_{t-1} + \ldots + A_p^*\bold{y}_{t-p} + C^*D_t + B\bold{\varepsilon}_t

The coefficient matrices (A_1^* | \ldots | 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(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)

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 \Sigma_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{\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(nr - 2K^2 - \frac{1}{2}K(K + 1)), where nr is equal to the number of restrictions.

Value

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 ‘A’.
B	If A- or AB-model, the matrix of estimated coefficients.
Bse	The standard errors of ‘B’.
LRIM	For Blanchard-Quah estimation LRIM is the estimated long-run impact matrix; for all other SVAR models LRIM is NULL.
Sigma.U	The variance-covariance matrix of the reduced form residuals times 100, i.e., \Sigma_U = A^{-1}BB'A^{-1'} \times 100.
LR	Object of class ‘htest’, holding the Likelihood ratio overidentification test.
opt	List object returned by optim().
start	Vector of starting values.
type	SVAR-type, character, either ‘A-model’, ‘B-model’ or ‘AB-model’.
var	The ‘varest’ object ‘x’.
iter	Integer, the count of iterations.
call	The call to SVAR().

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

VAR, SVEC, logLik, irf, fevd

Examples

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") 

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