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

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 Σ_u are not null, the impulses measured by the Φ_s matrices, would also reflect disturbances from the other variables. Therefore, in practice a Choleski decomposition has been propagated by considering Σ_u = PP' and the orthogonalised shocks old{ε}_t = P^{-1}old{u}_t. The moving average representation is then in the form of:

old{y}_t = Ψ_0 old{ε}_t + Ψ_1 old{ε}_{t-1} + Ψ old{ε}_{t-2} + … ,

whith Ψ_0 = P and the matrices Ψ_s are computed as Ψ_s = Φ_s P for s = 1, 2, 3, ….

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., Ψ_0. Due to the utilisation of the Choleski decomposition, the impulse are now dependent on the ordering of the vector elements in old{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)

Results