g13dx calculates the zeros of a vector autoregressive (or moving average) operator. This method is likely to be used in conjunction with g05pj g13as g13dd (G13DSF not in this release).
- Type: System..::..Int32On entry: , the dimension of the multivariate time series.Constraint: .
- Type: System..::..Int32On entry: the number of AR (or MA) parameter matrices, (or ).Constraint: .
- Type: array<System..::..Double>()An array of size On entry: the AR (or MA) parameter matrices read in row by row in the order (or ). That is, must be set equal to the th element of , for (or the th element of , for ).
- Type: array<System..::..Double>()An array of size On exit: the real parts of the eigenvalues.
- Type: array<System..::..Double>()An array of size On exit: the imaginary parts of the eigenvalues.
- Type: array<System..::..Double>()An array of size On exit: the moduli of the eigenvalues.
Consider the vector autoregressive moving average (VARMA) model
where denotes a vector of time series and is a vector of residual series having zero mean and a constant variance-covariance matrix. The components of are also assumed to be uncorrelated at non-simultaneous lags. denotes a sequence of by matrices of autoregressive (AR) parameters and denotes a sequence of by matrices of moving average (MA) parameters. is a vector of length containing the series means. Let
where denotes the by identity matrix.
The model (1) is said to be stationary if the eigenvalues of lie inside the unit circle. Similarly let
Then the model is said to be invertible if the eigenvalues of lie inside the unit circle.
g13dx returns the eigenvalues of (or the eigenvalues of ) along with their moduli, in descending order of magnitude. Thus to check for stationarity or invertibility you should check whether the modulus of the largest eigenvalue is less than one.
Wei W W S (1990) Time Series Analysis: Univariate and Multivariate Methods Addison–Wesley
Errors or warnings detected by the method:
On entry, , or .
- An excessive number of iterations are needed to evaluate the eigenvalues of (or ). This is an unlikely exit. All output parameters are undefined.
The accuracy of the results depends on the original matrix and the multiplicity of the roots.
The time taken is approximately proportional to (or ).
This example finds the eigenvalues of where and and .