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 varest
Phi(x, nstep=10, ...)
# S3 method for svarest
Phi(x, nstep=10, ...)
# S3 method for svecest
Phi(x, nstep=10, ...)
# S3 method for vec2var
Phi(x, nstep=10, ...)An array with dimension \((K \times K \times nstep + 1)\) holding the estimated coefficients of the moving average representation.
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().
An integer specifying the number of moving error coefficient matrices to be calculated.
Currently not used.
Bernhard Pfaff
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.
Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.
Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.
Psi, VAR, SVAR,
vec2var, SVEC
data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
Phi(var.2c, nstep=4)
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