NAG Library Routine Document
G13DXF calculates the zeros of a vector autoregressive (or moving average) operator.
This routine is likely to be used in conjunction with G05PJF
||K, IP, IWORK(K*IP), IFAIL
||PAR(IP*K*K), RR(K*IP), RI(K*IP), RMOD(K*IP), WORK(K*K*IP*IP)
Consider the vector autoregressive moving average (VARMA) model
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
matrices of autoregressive (AR) parameters and
denotes a sequence of
matrices of moving average (MA) parameters.
is a vector of length
containing the series means. Let
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.
G13DXF 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
- 1: K – INTEGERInput
On entry: , the dimension of the multivariate time series.
- 2: IP – INTEGERInput
On entry: the number of AR (or MA) parameter matrices, (or ).
- 3: PAR() – REAL (KIND=nag_wp) arrayInput
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 ).
- 4: RR() – REAL (KIND=nag_wp) arrayOutput
On exit: the real parts of the eigenvalues.
- 5: RI() – REAL (KIND=nag_wp) arrayOutput
On exit: the imaginary parts of the eigenvalues.
- 6: RMOD() – REAL (KIND=nag_wp) arrayOutput
On exit: the moduli of the eigenvalues.
- 7: WORK() – REAL (KIND=nag_wp) arrayWorkspace
- 8: IWORK() – INTEGER arrayWorkspace
- 9: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
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 .
9.1 Program Text
Program Text (g13dxfe.f90)
9.2 Program Data
Program Data (g13dxfe.d)
9.3 Program Results
Program Results (g13dxfe.r)