nag_tsa_arma_roots (g13dxc) (PDF version)
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g13 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_tsa_arma_roots (g13dxc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagg13.h>
void  nag_tsa_arma_roots (Integer k, Integer ip, const double par[], double rr[], double ri[], double rmod[], NagError *fail)

3  Description

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 Wt denotes a vector of k time series and εt is a vector of k residual series having zero mean and a constant variance-covariance matrix. The components of εt are also assumed to be uncorrelated at non-simultaneous lags. ϕ1,ϕ2,,ϕp denotes a sequence of k by k matrices of autoregressive (AR) parameters and θ1,θ2,,θq denotes a sequence of k by k matrices of moving average (MA) parameters. μ 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 Aϕ 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 Bθ lie inside the unit circle.
nag_tsa_arma_roots (g13dxc) returns the pk eigenvalues of Aϕ (or the qk eigenvalues of Bθ) 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.

4  References

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

5  Arguments

1:     kIntegerInput
On entry: k, the dimension of the multivariate time series.
Constraint: k1.
2:     ipIntegerInput
On entry: the number of AR (or MA) parameter matrices, p (or q).
Constraint: ip1.
3:     par[ip×k×k]const doubleInput
On entry: the AR (or MA) parameter matrices read in row by row in the order ϕ1,ϕ2,,ϕp (or θ1,θ2,,θq). That is, par[l-1×k×k+i-1×k+j-1] must be set equal to the i,jth element of ϕl, for l=1,2,,p (or the i,jth element of θl, for l=1,2,,q).
4:     rr[k×ip]doubleOutput
On exit: the real parts of the eigenvalues.
5:     ri[k×ip]doubleOutput
On exit: the imaginary parts of the eigenvalues.
6:     rmod[k×ip]doubleOutput
On exit: the moduli of the eigenvalues.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
An excessive number of iterations have been required to calculate the eigenvalues.
On entry, ip=value.
Constraint: ip1.
On entry, k=value.
Constraint: k1.
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.

7  Accuracy

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

8  Parallelism 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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The time taken is approximately proportional to kp3 (or kq3).

10  Example

This example finds the eigenvalues of Aϕ where k=2 and p=1 and ϕ1= 0.802 0.065 0.000 0.575 .

10.1  Program Text

Program Text (g13dxce.c)

10.2  Program Data

Program Data (g13dxce.d)

10.3  Program Results

Program Results (g13dxce.r)

nag_tsa_arma_roots (g13dxc) (PDF version)
g13 Chapter Contents
g13 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014