g13dx calculates the zeros of a vector autoregressive (or moving average) operator. This method is likely to be used in conjunction with g05pj g13as g13dd (G13DSF not in this release).

Syntax

C#
public static void g13dx( int k, int ip, double[] par, double[] rr, double[] ri, double[] rmod, out int ifail )

Visual Basic
Public Shared Sub g13dx ( _ k As Integer, _ ip As Integer, _ par As Double(), _ rr As Double(), _ ri As Double(), _ rmod As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual C++
public: static void g13dx( int k, int ip, array<double>^ par, array<double>^ rr, array<double>^ ri, array<double>^ rmod, [OutAttribute] int% ifail )

F#
static member g13dx : k : int * ip : int * par : float[] * rr : float[] * ri : float[] * rmod : float[] * ifail : int byref -> unit

Parameters

k: Type: System..::..Int32
On entry: $k$ , the dimension of the multivariate time series.

Constraint: $k \geq 1$ .

ip: Type: System..::..Int32
On entry: the number of AR (or MA) parameter matrices, $p$ (or $q$ ).

Constraint: $ip \geq 1$ .

par: Type: array<System..::..Double>[]()[][]
An array of size [ $ip \times k \times k$ ]
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 $l = 1, 2, \dots, q$ ).

rr: Type: array<System..::..Double>[]()[][]
An array of size [ $k \times ip$ ]
On exit: the real parts of the eigenvalues.

ri: Type: array<System..::..Double>[]()[][]
An array of size [ $k \times ip$ ]
On exit: the imaginary parts of the eigenvalues.

rmod: Type: array<System..::..Double>[]()[][]
An array of size [ $k \times ip$ ]
On exit: the moduli of the eigenvalues.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

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.

g13dx 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.

References

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

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

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.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

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

Parallelism and Performance

None.

Further Comments

The time taken is approximately proportional to

k p^{3}

(or

k q^{3}

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}]

Example program (C#): g13dxe.cs

Example program data: g13dxe.d

Example program results: g13dxe.r

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also