R: Coefficient matrices of the MA represention

Phi {vars} | R Documentation

## Coefficient matrices of the MA represention

### Description

Returns the estimated coefficient matrices of the moving average representation of a stable VAR(p), of an SVAR as an array or a converted VECM to VAR.

### Usage

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

### Arguments

`x` | An object of class ‘`varest`’, generated by `VAR()`, or an object of class ‘`svarest`’, generated by `SVAR()`, or an object of class ‘`svecest`’, generated by `SVEC()`, or an object of class ‘`vec2var`’, generated by `vec2var()`.
`nstep` | An integer specifying the number of moving error coefficient matrices to be calculated.
`...` | Currently not used.

### Details

If the process \bold{y}_t is stationary (i.e. I(0), it has a Wold moving average representation in the form of:

\bold{y}_t = Φ_0 \bold{u}_t + Φ_1 \bold{u}_{t-1} + Φ \bold{u}_{t-2} + … ,

whith Φ_0 = I_k and the matrices Φ_s can be computed recursively according to:

Φ_s = ∑_{j=1}^s Φ_{s-j} A_j \quad s = 1, 2, … ,

whereby A_j are set to zero for j > p. The matrix elements represent the impulse responses of the components of \bold{y}_t with respect to the shocks \bold{u}_t. More precisely, the (i, j)th element of the matrix Φ_s mirrors the expected response of y_{i, t+s} to a unit change of the variable y_{jt}.
In case of a SVAR, the impulse response matrices are given by:

Θ_i = Φ_i A^{-1} B \quad .

Albeit the fact, that the Wold decomposition does not exist for nonstationary processes, it is however still possible to compute the Φ_i matrices likewise with integrated variables or for the level version of a VECM. However, a convergence to zero of Φ_i as i tends to infinity is not ensured; hence some shocks may have a permanent effect.

### Value

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

### Note

The first returned array element is the starting value, i.e., Φ_0.

### 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

`Psi`, `VAR`, `SVAR`, `vec2var`, `SVEC`

### Examples

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

[Package vars version 1.5-3 Index]