NAG Library Function Document

nag_tsa_arma_roots (g13dxc)

+− 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

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

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.

nag_tsa_arma_roots (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 one.

4 References

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

5 Arguments

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

Constraint: $k \geq 1$ .
2: ip – IntegerInput: 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 doubleInput: 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[ $k \times ip$ ] – doubleOutput: On exit: the real parts of the eigenvalues.
5: ri[ $k \times ip$ ] – doubleOutput: On exit: the imaginary parts of the eigenvalues.
6: rmod[ $k \times ip$ ] – doubleOutput: On exit: the moduli of the eigenvalues.
7: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.

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

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}]

NAG Library Function Documentnag_tsa_arma_roots (g13dxc)

+− 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 Library Function Document

nag_tsa_arma_roots (g13dxc)