naginterfaces.library.tsa.uni_​arma_​roots

naginterfaces.library.tsa.uni_arma_roots(k, ip, par)

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

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g13/g13dxf.html

Parameters

kint

$k$, the dimension of the multivariate time series.

ipint

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

parfloat, array-like, shape $(ip\times k\times k)$

The AR (or MA) parameter matrices read in row by row in the order ${\varphi }_1,{\varphi }_2,\dots ,{\varphi }_p$ (or ${\theta }_1,{\theta }_2,\dots ,{\theta }_q$). That is, $par[(\text{l}-1)\times k\times k+(i-1)\times k+j-1]$ must be set equal to the $(i,j)$th element of ${\varphi }_l$, for $\text{l}=1,2,\dots ,p$ (or the $(i,j)$th element of ${\theta }_{\text{l}}$, for $\text{l}=1,2,\dots ,q$).

Returns

rrfloat, ndarray, shape $(k\times ip)$

The real parts of the eigenvalues.

rifloat, ndarray, shape $(k\times ip)$

The imaginary parts of the eigenvalues.

rmodfloat, ndarray, shape $(k\times ip)$

The moduli of the eigenvalues.

Raises

NagValueError

(errno $1$)

On entry, $ip=⟨value⟩$.

Constraint: $ip\ge 1$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 1$.

(errno $2$)

An excessive number of iterations have been required to calculate the eigenvalues.

Notes

Consider the vector autoregressive moving average (VARMA) model

$$W_t-\mu ={\varphi }_1(W_{t-1}-\mu )+{\varphi }_2(W_{t-2}-\mu )+\cdots +{\varphi }_p(W_{t-p}-\mu )+{ϵ}_t-{\theta }_1{ϵ}_{t-1}-{\theta }_2{ϵ}_{t-2}-\cdots -{\theta }_q{ϵ}_{t-q}\text{,}$$

where $W_t$ denotes a vector of $k$ time series and ${ϵ}_t$ is a vector of $k$ residual series having zero mean and a constant variance-covariance matrix. The components of ${ϵ}_t$ are also assumed to be uncorrelated at non-simultaneous lags. ${\varphi }_1,{\varphi }_2,\dots ,{\varphi }_p$ denotes a sequence of $k\times k$ matrices of autoregressive (AR) parameters and ${\theta }_1,{\theta }_2,\dots ,{\theta }_q$ denotes a sequence of $k\times k$ matrices of moving average (MA) parameters. $\mu$ is a vector of length $k$ containing the series means. Let

$$\begin{array}{c}A\left(\varphi \right)={\left[\begin{array}{ccccccc}{\varphi }_1& I& 0& .& .& .& 0\\ {\varphi }_2& 0& I& 0& .& .& 0\\ .& & & .& & & \\ .& & & & .& & \\ .& & & & & .& \\ {\varphi }_{p-1}& 0& .& .& .& 0& I\\ {\varphi }_p& 0& .& .& .& 0& 0\end{array}\right]}_{pk\times pk}\end{array}$$

where $I$ denotes the $k\times k$ identity matrix.

The model (1) is said to be stationary if the eigenvalues of $A\left(\varphi \right)$ lie inside the unit circle. Similarly let

$$\begin{array}{c}B\left(\theta \right)={\left[\begin{array}{ccccccc}{\theta }_1& I& 0& .& .& .& 0\\ {\theta }_2& 0& I& 0& .& .& 0\\ .& & & .& & & \\ .& & & & .& & \\ .& & & & & .& \\ {\theta }_{q-1}& 0& .& .& .& 0& I\\ {\theta }_q& 0& .& .& .& 0& 0\end{array}\right]}_{qk\times qk}\text{.}\end{array}$$

Then the model is said to be invertible if the eigenvalues of $B\left(\theta \right)$ lie inside the unit circle.

uni_arma_roots returns the $pk$ eigenvalues of $A\left(\varphi \right)$ (or the $qk$ eigenvalues of $B\left(\theta \right)$) 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 $1$.

References

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