nag_tsa_arma_roots (g13dxc) : NAG Library, Mark 23

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.

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.

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 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

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 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_tsa_arma_roots (g13dxc)