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) or of an SVAR as an array.

Usage

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

Arguments

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

Details

If the process y_t is stationary (i.e. I(0), it has a Wold moving average representation in the form of:

y_t = Phi_0 u_t + Phi_1 u_{t-1} + Phi u_{t-2} + ... ,

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, ... ,

whereby A_j are set to zero for j > p. The matrix elements represent the impulse responses of the components of y_t with respect to the shocks 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 .

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

Examples

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

[Package vars version 0.1.9 Index]