naginterfaces.library.tsa.uni_arma_roots(k, ip, par)[source]

uni_arma_roots calculates the zeros of a vector autoregressive (or moving average) operator. This function is likely to be used in conjunction with rand.times_mv_varma, uni_arima_resid(), multi_varma_estimate() or multi_varma_diag().

For full information please refer to the NAG Library document for g13dx


, the dimension of the multivariate time series.


The number of AR (or MA) parameter matrices, (or ).

parfloat, array-like, shape

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 ).

rrfloat, ndarray, shape

The real parts of the eigenvalues.

rifloat, ndarray, shape

The imaginary parts of the eigenvalues.

rmodfloat, ndarray, shape

The moduli of the eigenvalues.

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

An excessive number of iterations have been required to calculate 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 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

where denotes the 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.

uni_arma_roots 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 .


Wei, W W S, 1990, Time Series Analysis: Univariate and Multivariate Methods, Addison–Wesley