| Phi {vars} | R Documentation |
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.
## 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, ...)
x |
An object of class ‘ |
nstep |
An integer specifying the number of moving error coefficient matrices to be calculated. |
... |
Currently not used. |
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.
An array with dimension (K \times K \times nstep + 1) holding the estimated coefficients of the moving average representation.
The first returned array element is the starting value, i.e., Φ_0.
Bernhard Pfaff
Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.
Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.
data(Canada) var.2c <- VAR(Canada, p = 2, type = "const") Phi(var.2c, nstep=4)