Returns the estimated orthogonalised coefficient matrices of the moving average representation of a stable VAR(p) as an array.
# S3 method for varest
Psi(x, nstep=10, ...)
# S3 method for vec2var
Psi(x, nstep=10, ...)An array with dimension \((K \times K \times nstep + 1)\) holding the estimated orthogonalised coefficients of the moving average representation.
An object of class ‘varest’, generated by
VAR(), or an object of class ‘vec2var’,
generated by vec2var().
An integer specifying the number of othogonalised moving error coefficient matrices to be calculated.
Dots currently not used.
Bernhard Pfaff
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 \(s = 1, 2, 3, \ldots\).
Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.
Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.
Phi, VAR, SVAR,
vec2var
data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
Psi(var.2c, nstep=4)
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