Psi: Coefficient matrices of the orthogonalised MA represention
Description
Returns the estimated orthogonalised coefficient matrices of the
moving average representation of a stable VAR(p) as an array.
Usage
## S3 method for class 'varest':
Psi(x, nstep=10, ...)
Arguments
x
An object of class varest, generated by
VAR().
nstep
An integer specifying the number of othogonalised moving error
coefficient matrices to be calculated.
...
Dots currently not used.
Value
An array with dimension $(K \times K \times nstep + 1)$ holding the
estimated orthogonalised coefficients of the moving average representation.
encoding
latin1
concept
VAR
Vector autoregressive
Moving Average Representation
Orthogonalised Impulse Responses
Impulse Response Function
Impulse Responses
Details
In case that the components of the error process are instantaneously
correlated with each other, that is: the off-diagonal elements of the
variance-covariance matrix $\Sigma_u$ are not null, the impulses
measured by the $\Phi_s$ matrices, would also reflect disturbances
from the other variables. Therefore, in practice a Choleski
decomposition has been propagated by considering $\Sigma_u = PP'$ and the
orthogonalised shocks $\bold{\epsilon}_t = P^{-1}\bold{u}_t$. The
moving average representation is then in the form of:
$$\bold{y}_t = \Psi_0 \bold{\epsilon}_t + \Psi_1
\bold{\epsilon}_{t-1} + \Psi \bold{\epsilon}_{t-2} + \ldots ,$$
whith $\Psi_0 = P$ and the matrices $\Psi_s$ are computed
as $\Psi_s = \Phi_s P$ for $s = 1, 2, 3, \ldots$.
References
Hamilton, J. (1994), Time Series Analysis, Princeton
University Press, Princeton.
L�tkepohl, H. (2006), New Introduction to Multiple Time Series
Analysis, Springer, New York.