NAG Library Routine Document

G05PJF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

G05PJF generates a realization of a multivariate time series from a vector autoregressive moving average (VARMA) model. The realization may be continued or a new realization generated at subsequent calls to G05PJF.

2 Specification

SUBROUTINE G05PJF (

MODE, N, K, XMEAN, IP, PHI, IQ, THETA, VAR, LDVAR, R, LR, STATE, X, LDX, IFAIL)

INTEGER	MODE, N, K, IP, IQ, LDVAR, LR, STATE(*), LDX, IFAIL
REAL (KIND=nag_wp)	XMEAN(K), PHI(KKIP), THETA(KKIQ), VAR(LDVAR,K), R(LR), X(LDX,N)

3 Description

Let the vector

X_{t} = {(x_{1 t}, x_{2 t}, \dots, x_{k t})}^{T}

, denote a

k

-dimensional time series which is assumed to follow a vector autoregressive moving average (VARMA) model of the form:

\begin{matrix} X_{t} - μ = & ϕ_{1} (X_{t - 1} - μ) + ϕ_{2} (X_{t - 2} - μ) + \dots + ϕ_{p} (X_{t - p} - μ) + \\ ε_{t} - θ_{1} ε_{t - 1} - θ_{2} ε_{t - 2} - \dots - θ_{q} ε_{t - q} \end{matrix}

(1)

where

ε_{t} = {(ε_{1 t}, ε_{2 t}, \dots, ε_{k t})}^{T}

, is a vector of

k

residual series assumed to be Normally distributed with zero mean and covariance matrix

Σ

. The components of

ε_{t}

are assumed to be uncorrelated at non-simultaneous lags. The

ϕ_{i}

's and

θ_{j}

's are

k

k

matrices of parameters.

\{ϕ_{i}\}

, for

i = 1, 2, \dots, p

, are called the autoregressive (AR) parameter matrices, and

\{θ_{j}\}

, for

j = 1, 2, \dots, q

, the moving average (MA) parameter matrices. The parameters in the model are thus the

p

k

k

ϕ

-matrices, the

q

k

k

θ

-matrices, the mean vector

μ

and the residual error covariance matrix

Σ

. Let

A (ϕ) = {[\begin{matrix} ϕ_{1} & I & 0 & . & . & . & 0 \\ ϕ_{2} & 0 & I & 0 & . & . & 0 \\ . & . \\ . & . \\ . & . \\ ϕ_{p - 1} & 0 & . & . & . & 0 & I \\ ϕ_{p} & 0 & . & . & . & 0 & 0 \end{matrix}]}_{p k \times p k} and B (θ) = {[\begin{matrix} θ_{1} & I & 0 & . & . & . & 0 \\ θ_{2} & 0 & I & 0 & . & . & 0 \\ . & . \\ . & . \\ . & . \\ θ_{q - 1} & 0 & . & . & . & 0 & I \\ θ_{q} & 0 & . & . & . & 0 & 0 \end{matrix}]}_{q k \times q k}

where

I

denotes the

k

k

identity matrix.

The model (1) must be both stationary and invertible. The model is said to be stationary if the eigenvalues of

A (ϕ)

lie inside the unit circle and invertible if the eigenvalues of

B (θ)

lie inside the unit circle.

For

k \geq 6

the VARMA model (1) is recast into state space form and a realization of the state vector at time zero computed. For all other cases the routine computes a realization of the pre-observed vectors

X_{0}, X_{- 1}, \dots, X_{1 - p}

ε_{0}, ε_{- 1}, \dots, ε_{1 - q}

, from (1), see Shea (1988). This realization is then used to generate a sequence of successive time series observations. Note that special action is taken for pure MA models, that is for

p = 0

At your request a new realization of the time series may be generated more efficiently using the information in a reference vector created during a previous call to G05PJF. See the description of the parameter MODE in Section 5 for details.

The routine returns a realization of

X_{1}, X_{2}, \dots, X_{n}

. On a successful exit, the recent history is updated and saved in the array R so that G05PJF may be called again to generate a realization of

X_{n + 1}, X_{n + 2}, \dots

, etc. See the description of the parameter MODE in Section 5 for details.

Further computational details are given in Shea (1988). Note, however, that G05PJF uses a spectral decomposition rather than a Cholesky factorization to generate the multivariate Normals. Although this method involves more multiplications than the Cholesky factorization method and is thus slightly slower it is more stable when faced with ill-conditioned covariance matrices. A method of assigning the AR and MA coefficient matrices so that the stationarity and invertibility conditions are satisfied is described in Barone (1987).

One of the initialization routines G05KFF (for a repeatable sequence if computed sequentially) or G05KGF (for a non-repeatable sequence) must be called prior to the first call to G05PJF.

4 References

Barone P (1987) A method for generating independent realisations of a multivariate normal stationary and invertible ARMA

(p, q)

process J. Time Ser. Anal. 8 125–130

Shea B L (1988) A note on the generation of independent realisations of a vector autoregressive moving average process J. Time Ser. Anal. 9 403–410

5 Parameters

1: $MODE$ – INTEGERInput

On entry: a code for selecting the operation to be performed by the routine.

$MODE = 0$: Set up reference vector and compute a realization of the recent history.
$MODE = 1$: Generate terms in the time series using reference vector set up in a prior call to G05PJF.
$MODE = 2$: Combine the operations of $MODE = 0$ and $1$ .
$MODE = 3$: A new realization of the recent history is computed using information stored in the reference vector, and the following sequence of time series values are generated.

MODE = 1

3

, then you must ensure that the reference vector R and the values of K, IP, IQ, XMEAN, PHI, THETA, VAR and LDVAR have not been changed between calls to G05PJF.

Constraint:

MODE = 0

1

2

3

2: $N$ – INTEGERInput

On entry:

n

, the number of observations to be generated.

Constraint:

N \geq 0

3: $K$ – INTEGERInput

On entry:

k

, the dimension of the multivariate time series.

Constraint:

K \geq 1

4: $XMEAN (K)$ – REAL (KIND=nag_wp) arrayInput

On entry:

μ

, the vector of means of the multivariate time series.

5: $IP$ – INTEGERInput

On entry:

p

, the number of autoregressive parameter matrices.

Constraint:

IP \geq 0

6: $PHI (K \times K \times IP)$ – REAL (KIND=nag_wp) arrayInput

On entry: must contain the elements of the

IP \times K \times K

autoregressive parameter matrices of the model,

ϕ_{1}, ϕ_{2}, \dots, ϕ_{p}

. If PHI is considered as a three-dimensional array, dimensioned as

PHI (K, K, IP)

, then the

(i, j)

th element of

ϕ_{l}

would be stored in

PHI (i, j, l)

; that is,

PHI ((l - 1) \times k \times k + (j - 1) \times k + i)

must be set equal to the

(i, j)

th element of

ϕ_{l}

, for

l = 1, 2, \dots, p

i = 1, 2, \dots, k

and

j = 1, 2, \dots, k

Constraint: the elements of PHI must satisfy the stationarity condition.

7: $IQ$ – INTEGERInput

On entry:

q

, the number of moving average parameter matrices.

Constraint:

IQ \geq 0

8: $THETA (K \times K \times IQ)$ – REAL (KIND=nag_wp) arrayInput

On entry: must contain the elements of the

IQ \times K \times K

moving average parameter matrices of the model,

θ_{1}, θ_{2}, \dots, θ_{q}

. If THETA is considered as a three-dimensional array, dimensioned as THETA(K,K,IQ), then the

(i, j)

th element of

θ_{l}

would be stored in

THETA (i, j, l)

; that is,

THETA ((l - 1) \times k \times k + (j - 1) \times k + i)

must be set equal to the

(i, j)

th element of

θ_{l}

, for

l = 1, 2, \dots, q

i = 1, 2, \dots, k

and

j = 1, 2, \dots, k

Constraint: the elements of THETA must be within the invertibility region.

9: $VAR (LDVAR, K)$ – REAL (KIND=nag_wp) arrayInput

On entry:

VAR (i, j)

must contain the (

i, j

)th element of

Σ

, for

i = 1, 2, \dots, K

and

j = 1, 2, \dots, K

. Only the lower triangle is required.

Constraint: the elements of VAR must be such that

Σ

is positive semidefinite.

10: $LDVAR$ – INTEGERInput

On entry: the first dimension of the array VAR as declared in the (sub)program from which G05PJF is called.

Constraint:

LDVAR \geq K

11: $R (LR)$ – REAL (KIND=nag_wp) arrayCommunication Array

On entry: if

MODE = 1

3

, the array R as output from the previous call to G05PJF must be input without any change.

MODE = 0

2

, the contents of R need not be set.

On exit: information required for any subsequent calls to the routine with

MODE = 1

3

. See Section 9.

12: $LR$ – INTEGERInput

On entry: the dimension of the array R as declared in the (sub)program from which G05PJF is called.

Constraints:

if $K \geq 6$ , $LR \geq (5 r^{2} + 1) \times K^{2} + (4 r + 3) \times K + 4$ ;
if $K < 6$ , $LR \geq ({(IP + IQ)}^{2} + 1) \times K^{2} + (4 \times (IP + IQ) + 3) \times K + \max \{K r (K r + 2), K^{2} {(IP + IQ)}^{2} + l (l + 3) + K^{2} (IQ + 1)\} + 4$ .

Where

r = \max (IP, IQ)

and if

IP = 0

l = K (K + 1) / 2

, or if

IP \geq 1

l = K (K + 1) / 2 + (IP - 1) K^{2}

See Section 9 for some examples of the required size of the array R.

13: $STATE (*)$ – INTEGER arrayCommunication Array

Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

14: $X (LDX, N)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

X (i, t)

will contain a realization of the

i

th component of

X_{t}

, for

i = 1, 2, \dots, k

and

t = 1, 2, \dots, n

15: $LDX$ – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G05PJF is called.

Constraint:

LDX \geq K

16: $IFAIL$ – 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, $MODE = 〈value〉$ .
Constraint: $MODE = 0$ , $1$ , $2$ or $3$ .

$IFAIL = 2$: On entry, $N = 〈value〉$ .
Constraint: $N \geq 0$ .

$IFAIL = 3$: On entry, $K = 〈value〉$ .
Constraint: $K \geq 1$ .

$IFAIL = 5$: On entry, $IP = 〈value〉$ .
Constraint: $IP \geq 0$ .

$IFAIL = 6$: On entry, the AR parameters are outside the stationarity region.

$IFAIL = 7$: On entry, $IQ = 〈value〉$ .
Constraint: $IQ \geq 0$ .

$IFAIL = 8$: On entry, the moving average parameter matrices are such that the model is non-invertible.

$IFAIL = 9$: On entry, the covariance matrix VAR is not positive semidefinite to machine precision.

$IFAIL = 10$: On entry, $LDVAR = 〈value〉$ and $K = 〈value〉$ .
Constraint: $LDVAR \geq K$ .

$IFAIL = 11$: K is not the same as when R was set up in a previous call.
Previous value of $K = 〈value〉$ and $K = 〈value〉$ .

$IFAIL = 12$: On entry, LR is not large enough, $LR = 〈value〉$ : minimum length required $= 〈value〉$ .

$IFAIL = 13$: On entry, STATE vector has been corrupted or not initialized.

$IFAIL = 15$: On entry, $LDX = 〈value〉$ and $K = 〈value〉$ .
Constraint: $LDX \geq K$ .

$IFAIL = 20$: An excessive number of iterations were required by the NAG routine used to evaluate the eigenvalues of the matrices used to test for stationarity or invertibility.

$IFAIL = 21$: The reference vector cannot be computed because the AR parameters are too close to the boundary of the stationarity region.

$IFAIL = 22$: An excessive number of iterations were required by the NAG routine used to evaluate the eigenvalues of the covariance matrix.

$IFAIL = 23$: An excessive number of iterations were required by the NAG routine used to evaluate the eigenvalues stored in the reference vector.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

The accuracy is limited by the matrix computations performed, and this is dependent on the condition of the parameter and covariance matrices.

8 Parallelism and Performance

G05PJF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

G05PJF 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

Note that, in reference to

IFAIL = 8

, G05PJF will permit moving average parameters on the boundary of the invertibility region.

The elements of R contain amongst other information details of the spectral decompositions which are used to generate future multivariate Normals. Note that these eigenvectors may not be unique on different machines. For example the eigenvectors corresponding to multiple eigenvalues may be permuted. Although an effort is made to ensure that the eigenvectors have the same sign on all machines, differences in the signs may theoretically still occur.

The following table gives some examples of the required size of the array R, specified by the parameter LR, for

k = 1, 2

3

, and for various values of

p

and

q

				$q$

		0	1	2	3

		13	20	31	46
	0	36	56	92	144
		85	124	199	310

		19	30	45	64
	1	52	88	140	208
		115	190	301	448
p
		35	50	69	92
	2	136	188	256	340
		397	508	655	838

		57	76	99	126
	3	268	336	420	520
		877	1024	1207	1426

Note that G13DXF may be used to check whether a VARMA model is stationary and invertible.

The time taken depends on the values of

p

q

and especially

n

and

k

10 Example

This program generates two realizations, each of length

48

, from the bivariate AR(1) model

X_{t} - μ = ϕ_{1} (X_{t - 1} - μ) + ε_{t}

with

ϕ_{1} = [\begin{matrix} 0.80 & 0.07 \\ 0.00 & 0.58 \end{matrix}],

μ = [\begin{matrix} 5.00 \\ 9.00 \end{matrix}],

and

Σ = [\begin{matrix} 2.97 & 0 \\ 0.64 & 5.38 \end{matrix}] .

The pseudorandom number generator is initialized by a call to G05KFF. Then, in the first call to G05PJF,

MODE = 2

in order to set up the reference vector before generating the first realization. In the subsequent call

MODE = 3

and a new recent history is generated and used to generate the second realization.

NAG Library Routine DocumentG05PJF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

G05PJF