g13dx: FL CL CPP AD

NAG CL Interface
g13dxc (uni_arma_roots)

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NAG Library Manual, Mark 27

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

G13 (Tsa) Chapter Contents

G13 (Tsa) Chapter Introduction

g13dx: FL CL CPP AD

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

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

2 Specification

#include <nag.h>

void	g13dxc (Integer k, Integer ip, const double par[], double rr[], double ri[], double rmod[], NagError *fail)

The function may be called by the names: g13dxc, nag_tsa_uni_arma_roots or nag_tsa_arma_roots.

3 Description

Consider the vector autoregressive moving average (VARMA) model

W_{t} - μ = ϕ_{1} (W_{t - 1} - μ) + ϕ_{2} (W_{t - 2} - μ) + \dots + ϕ_{p} (W_{t - p} - μ) + ε_{t} - θ_{1} ε_{t - 1} - θ_{2} ε_{t - 2} - \dots - θ_{q} ε_{t - q},

(1)

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.

ϕ_{1}, ϕ_{2}, \dots, ϕ_{p}

denotes a sequence of

k

k

matrices of autoregressive (AR) parameters and

θ_{1}, θ_{2}, \dots, θ_{q}

denotes a sequence of

k

k

matrices of moving average (MA) parameters.

μ

is a vector of length

k

containing the series means. Let

A (ϕ) = {[\begin{array}{r} ϕ_{1} & I & 0 & . & . & . & 0 \\ ϕ_{2} & 0 & I & 0 & . & . & 0 \\ . & . \\ . & . \\ . & . \\ ϕ_{p - 1} & 0 & . & . & . & 0 & I \\ ϕ_{p} & 0 & . & . & . & 0 & 0 \end{array}]}_{p k \times p k}

where

I

denotes the

k

k

identity matrix.

The model (1) is said to be stationary if the eigenvalues of

A (ϕ)

lie inside the unit circle. Similarly let

B (θ) = {[\begin{array}{r} θ_{1} & I & 0 & . & . & . & 0 \\ θ_{2} & 0 & I & 0 & . & . & 0 \\ . & . \\ . & . \\ . & . \\ θ_{q - 1} & 0 & . & . & . & 0 & I \\ θ_{q} & 0 & . & . & . & 0 & 0 \end{array}]}_{q k \times q k} .

Then the model is said to be invertible if the eigenvalues of

B (θ)

lie inside the unit circle.

g13dxc returns the

p k

eigenvalues of

A (ϕ)

(or the

q k

eigenvalues of

B (θ)

) 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

4 References

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

5 Arguments

1: $k$ – Integer Input: On entry: $k$ , the dimension of the multivariate time series.

Constraint: $k \geq 1$ .
2: $ip$ – Integer Input: On entry: the number of AR (or MA) parameter matrices, $p$ (or $q$ ).

Constraint: $ip \geq 1$ .
3: $par [ip \times k \times k]$ – const double Input: On entry: the AR (or MA) parameter matrices read in row by row in the order $ϕ_{1}, ϕ_{2}, \dots, ϕ_{p}$ (or $θ_{1}, θ_{2}, \dots, θ_{q}$ ). That is, $par [(l - 1) \times k \times k + (i - 1) \times k + j - 1]$ must be set equal to the $(i, j)$ th element of $ϕ_{l}$ , for $l = 1, 2, \dots, p$ (or the $(i, j)$ th element of $θ_{l}$ , for $l = 1, 2, \dots, q$ ).
4: $rr [k \times ip]$ – double Output: On exit: the real parts of the eigenvalues.
5: $ri [k \times ip]$ – double Output: On exit: the imaginary parts of the eigenvalues.
6: $rmod [k \times ip]$ – double Output: On exit: the moduli of the eigenvalues.
7: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_EIGENVALUES: An excessive number of iterations have been required to calculate the eigenvalues.
NE_INT: On entry, $ip = 〈value〉$ .
Constraint: $ip \geq 1$ .

On entry, $k = 〈value〉$ .
Constraint: $k \geq 1$ .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

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

8 Parallelism and Performance

g13dxc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

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.

9 Further Comments

The time taken is approximately proportional to

k p^{3}

(or

k q^{3}

10 Example

This example finds the eigenvalues of

A (ϕ)

where

k = 2

and

p = 1

and

ϕ_{1} = [\begin{array}{r} 0.802 & 0.065 \\ 0.000 & 0.575 \end{array}]

g13dx: FL CL CPP AD

NAG CL Interfaceg13dxc (uni_​arma_​roots)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g13dxc (uni_arma_roots)