# NAG Library Function Document

## 1Purpose

nag_tsa_arma_roots (g13dxc) calculates the zeros of a vector autoregressive (or moving average) operator.

## 2Specification

 #include #include
 void nag_tsa_arma_roots (Integer k, Integer ip, const double par[], double rr[], double ri[], double rmod[], NagError *fail)

## 3Description

Consider the vector autoregressive moving average (VARMA) model
 $Wt-μ=ϕ1Wt-1-μ+ϕ2Wt-2-μ+⋯+ϕpWt-p-μ+εt-θ1εt-1-θ2εt-2-⋯-θqεt-q,$ (1)
where ${W}_{t}$ denotes a vector of $k$ time series and ${\epsilon }_{t}$ is a vector of $k$ residual series having zero mean and a constant variance-covariance matrix. The components of ${\epsilon }_{t}$ are also assumed to be uncorrelated at non-simultaneous lags. ${\varphi }_{1},{\varphi }_{2},\dots ,{\varphi }_{p}$ denotes a sequence of $k$ by $k$ matrices of autoregressive (AR) parameters and ${\theta }_{1},{\theta }_{2},\dots ,{\theta }_{q}$ denotes a sequence of $k$ by $k$ matrices of moving average (MA) parameters. $\mu$ is a vector of length $k$ containing the series means. Let
 $Aϕ= ϕ1 I 0 . . . 0 ϕ2 0 I 0 . . 0 . . . . . . ϕp-1 0 . . . 0 I ϕp 0 . . . 0 0 pk×pk$
where $I$ denotes the $k$ by $k$ 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
 $Bθ= θ1 I 0 . . . 0 θ2 0 I 0 . . 0 . . . . . . θq-1 0 . . . 0 I θq 0 . . . 0 0 qk×qk .$
Then the model is said to be invertible if the eigenvalues of $B\left(\theta \right)$ lie inside the unit circle.
nag_tsa_arma_roots (g13dxc) 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 one.

## 4References

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

## 5Arguments

1:    $\mathbf{k}$IntegerInput
On entry: $k$, the dimension of the multivariate time series.
Constraint: ${\mathbf{k}}\ge 1$.
2:    $\mathbf{ip}$IntegerInput
On entry: the number of AR (or MA) parameter matrices, $p$ (or $q$).
Constraint: ${\mathbf{ip}}\ge 1$.
3:    $\mathbf{par}\left[{\mathbf{ip}}×{\mathbf{k}}×{\mathbf{k}}\right]$const doubleInput
On entry: 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, ${\mathbf{par}}\left[\left(\mathit{l}-1\right)×k×k+\left(i-1\right)×k+j-1\right]$ must be set equal to the $\left(i,j\right)$th element of ${\varphi }_{l}$, for $\mathit{l}=1,2,\dots ,p$ (or the $\left(i,j\right)$th element of ${\theta }_{\mathit{l}}$, for $\mathit{l}=1,2,\dots ,q$).
4:    $\mathbf{rr}\left[{\mathbf{k}}×{\mathbf{ip}}\right]$doubleOutput
On exit: the real parts of the eigenvalues.
5:    $\mathbf{ri}\left[{\mathbf{k}}×{\mathbf{ip}}\right]$doubleOutput
On exit: the imaginary parts of the eigenvalues.
6:    $\mathbf{rmod}\left[{\mathbf{k}}×{\mathbf{ip}}\right]$doubleOutput
On exit: the moduli of the eigenvalues.
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_EIGENVALUES
An excessive number of iterations have been required to calculate the eigenvalues.
NE_INT
On entry, ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ip}}\ge 1$.
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The accuracy of the results depends on the original matrix and the multiplicity of the roots.

## 8Parallelism and Performance

nag_tsa_arma_roots (g13dxc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_tsa_arma_roots (g13dxc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken is approximately proportional to $k{p}^{3}$ (or $k{q}^{3}$).

## 10Example

This example finds the eigenvalues of $A\left(\varphi \right)$ where $k=2$ and $p=1$ and ${\varphi }_{1}=\left[\begin{array}{rr}0.802& 0.065\\ 0.000& 0.575\end{array}\right]$.

### 10.1Program Text

Program Text (g13dxce.c)

### 10.2Program Data

Program Data (g13dxce.d)

### 10.3Program Results

Program Results (g13dxce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017