Psi: Coefficient matrices of the orthogonalised MA represention

In vars: VAR Modelling

Coefficient matrices of the orthogonalised MA represention

Description

Returns the estimated orthogonalised coefficient matrices of the moving average representation of a stable VAR(p) as an array.

Usage

## S3 method for class 'varest'
Psi(x, nstep=10, ...)
## S3 method for class 'vec2var'
Psi(x, nstep=10, ...)

Arguments

x	An object of class ‘varest’, generated by VAR(), or an object of class ‘vec2var’, generated by vec2var().
nstep	An integer specifying the number of othogonalised moving error coefficient matrices to be calculated.
...	Dots currently not used.

Details

In case that the components of the error process are instantaneously correlated with each other, that is: the off-diagonal elements of the variance-covariance matrix \Sigma_u are not null, the impulses measured by the \Phi_s matrices, would also reflect disturbances from the other variables. Therefore, in practice a Choleski decomposition has been propagated by considering \Sigma_u = PP' and the orthogonalised shocks \bold{\epsilon}_t = P^{-1}\bold{u}_t. The moving average representation is then in the form of:

\bold{y}_t = \Psi_0 \bold{\epsilon}_t + \Psi_1 \bold{\epsilon}_{t-1} + \Psi \bold{\epsilon}_{t-2} + \ldots ,

whith \Psi_0 = P and the matrices \Psi_s are computed as \Psi_s = \Phi_s P for s = 1, 2, 3, \ldots.

Value

An array with dimension (K \times K \times nstep + 1) holding the estimated orthogonalised coefficients of the moving average representation.

Note

The first returned array element is the starting value, i.e., \Psi_0. Due to the utilisation of the Choleski decomposition, the impulse are now dependent on the ordering of the vector elements in \bold{y}_t.

Author(s)

Bernhard Pfaff

References

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

Phi, VAR, SVAR, vec2var

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
Psi(var.2c, nstep=4)

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