NAG Library Routine Document

G13DXF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G13DXF calculates the zeros of a vector autoregressive (or moving average) operator. This routine is likely to be used in conjunction with G05PJF, G13ASF, G13DDF or G13DSF.

2 Specification

SUBROUTINE G13DXF (

K, IP, PAR, RR, RI, RMOD, WORK, IWORK, IFAIL)

INTEGER	K, IP, IWORK(K*IP), IFAIL
REAL (KIND=nag_wp)	PAR(IPKK), RR(KIP), RI(KIP), RMOD(KIP), WORK(KKIPIP)

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.

G13DXF 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 Parameters

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( $IP \times K \times K$ ) – REAL (KIND=nag_wp) arrayInput: 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, $PAR ((l - 1) \times k \times k + (i - 1) \times k + j)$ 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( $K \times IP$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: the real parts of the eigenvalues.
5: RI( $K \times IP$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: the imaginary parts of the eigenvalues.
6: RMOD( $K \times IP$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: the moduli of the eigenvalues.
7: WORK( $K \times K \times IP \times IP$ ) – REAL (KIND=nag_wp) arrayWorkspace
8: IWORK( $K \times IP$ ) – INTEGER arrayWorkspace
9: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$K < 1$ ,
or	$IP < 1$ .

$IFAIL = 2$: An excessive number of iterations are needed to evaluate the eigenvalues of $A (ϕ)$ (or $B (θ)$ ). This is an unlikely exit. All output parameters are undefined.

7 Accuracy

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

8 Further Comments

The time taken is approximately proportional to

k p^{3}

(or

k q^{3}

9 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 Routine DocumentG13DXF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G13DXF