vars (version 1.1-9)

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, ...)
## S3 method for class 'vec2var':
Psi(x, nstep=10, ...)

Arguments

x

An object of class varest, generated by VAR(), or an object of class vec2var, generated by vec2var().

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
• VECM

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.

See Also

Phi, VAR, SVAR, SVAR2, vec2var

Examples

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