Phi {vars}

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 = \Phi_0 \bold{u}_t + \Phi_1 \bold{u}_{t-1} + \Phi \bold{u}_{t-2} + \ldots ,

whith \Phi_0 = I_k and the matrices \Phi_s can be computed recursively according to:

\Phi_s = \sum_{j=1}^s \Phi_{s-j} A_j \quad s = 1, 2, \ldots ,

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 \Phi_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:

\Theta_i = \Phi_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 \Phi_i matrices likewise with integrated variables or for the level version of a VECM. However, a convergence to zero of \Phi_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., \Phi_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)