vars (version 0.1.7)

Phi: Coefficient matrices of the MA represention

Description

Returns the estimated coefficient matrices of the moving average representation of a stable VAR(p) or of an SVAR as an array.

Usage

## S3 method for class 'varest':
Phi(x, nstep=10, ...)
## S3 method for class 'svarest':
Phi(x, nstep=10, ...)

Arguments

x

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

nstep

An integer specifying the number of moving error coefficient matrices to be calculated.

...

Currently not used.

Value

• An array with dimension $(K \times K \times nstep + 1)$ holding the estimated coefficients of the moving average representation.

encoding

latin1

concept

• VAR
• Vector autoregressive
• Moving Average Representation
• Impulse Response Function
• Impulse Responses

Details

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: $$\Theta_i = \Phi_i A^{-1} B \quad .$$

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

Psi, VAR, SVAR

Examples

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