vars (version 1.1-9)

Phi: Coefficient matrices of the MA represention

Description

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.

Usage

## S3 method for class 'varest':
Phi(x, nstep=10, ...)
## S3 method for class 'svarest':
Phi(x, nstep=10, ...)
## S3 method for class 'svecest':
Phi(x, nstep=10, ...)
## S3 method for class 'vec2var':
Phi(x, nstep=10, ...)

Arguments

x

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

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
• VECM

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

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, SVAR2, vec2var, SVEC

Examples

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