R: Coefficient matrices of the orthogonalised MA represention

Psi {vars} | R Documentation

## 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 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 \bold{ε}_t = P^{-1}\bold{u}_t. The moving average representation is then in the form of:

\bold{y}_t = Ψ_0 \bold{ε}_t + Ψ_1 \bold{ε}_{t-1} + Ψ \bold{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 \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)

[Package vars version 1.5-3 Index]