SVAR {vars}R Documentation

Estimation of a SVAR

Description

Estimates an SVAR (either ‘A-model’, ‘B-model’ or ‘AB-model’) by numerically minimising the negative log-likelihood using optim().

Usage

SVAR(x, Amat = NULL, Bmat = NULL, start = NULL, ...)

Arguments

x Object of class ‘varest’; generated by VAR().
Amat Matrix with dimension (K times K) for A- or AB-model.
Bmat Matrix with dimension (K times K) for B- or AB-model.
start Vector with starting values for the parameters to be optimised.
... Arguments that arr passed to optim().

Details

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

A y_t = A_1^*y_{t-1} + ... + A_p^*y_{t-p} + C^*D_t + Bvarepsilon_t

The coefficient matrices (A_1^* | ... | 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. The reduced form residuals can be obtained from the above equation via the relation: u_t = A^{-1}Bvarepsilon_t, with variance-covariance matrix Σ_U = A^{-1}BB'A^{-1'}.

Hence, 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(2π) + frac{T}{2}ln|A|^2 - frac{T}{2}ln|B|^2 - frac{T}{2}tr(A'B'^{-1}B^{-1}Atilde{Σ}_u)

If ‘start’ is not set, then 0.1 is used as starting values for the unknown coefficients. If the function is called with ‘hessian = TRUE’, the standard errors of the coefficients are returned as list elements ‘Ase’ and/or ‘Bse’, where applicable.
Finally, in case of an overidentified SVAR, a likelihood ratio statistic is computed according to:

LR = T(lndet(tilde{Σ}_u^r) - lndet(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 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 coeffiecients.
Ase If ‘hessian = TRUE’, the standard errors of ‘A’, otherwise a null-matrix is returned.
B If A- or AB-model, the matrix of estimated coeffiecients.
Bse If ‘hessian = TRUE’, the standard errors of ‘B’, otherwise a null-matrix is returned.
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., Σ_U = A^{-1}BB'A^{-1'} times 100.
LR Object of class ‘code{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’.
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, SVAR2, SVEC

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
SVAR(var.2c, Amat = amat, Bmat = NULL, hessian = TRUE, method="BFGS")

[Package vars version 0.7-7 Index]