NAG Library Function Document

nag_rand_arma (g05phc)

+− 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_rand_arma (g05phc) generates a realization of a univariate time series from an autoregressive moving average (ARMA) model. The realization may be continued or a new realization generated at subsequent calls to nag_rand_arma (g05phc).

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rand_arma (Nag_ModeRNG mode, Integer n, double xmean, Integer ip, const double phi[], Integer iq, const double theta[], double avar, double r[], Integer lr, Integer state[], double var, double x[], NagError fail)

3 Description

Let the vector

x_{t}

, denote a time series which is assumed to follow an autoregressive moving average (ARMA) 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}

where

ε_{t}

, is a residual series of independent random perturbations assumed to be Normally distributed with zero mean and variance

σ^{2}

. The parameters

\{ϕ_{i}\}

, for

i = 1, 2, \dots, p

, are called the autoregressive (AR) parameters, and

\{θ_{j}\}

, for

j = 1, 2, \dots, q

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

p

ϕ

values, the

q

θ

values, the mean

μ

and the residual variance

σ^{2}

nag_rand_arma (g05phc) sets up a reference vector containing initial values corresponding to a stationary position using the method described in Tunnicliffe–Wilson (1979). The function can then return a realization of

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

. On a successful exit, the recent history is updated and saved in the reference vector r so that nag_rand_arma (g05phc) may be called again to generate a realization of

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

, etc. See the description of the argument mode in Section 5 for details.

One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_arma (g05phc).

4 References

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

Tunnicliffe–Wilson G (1979) Some efficient computational procedures for high order ARMA models J. Statist. Comput. Simulation 8 301–309

5 Arguments

1: mode – Nag_ModeRNGInput

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

$mode = Nag_InitializeReference$: Set up reference vector only.
$mode = Nag_GenerateFromReference$: Generate terms in the time series using reference vector set up in a prior call to nag_rand_arma (g05phc).
$mode = Nag_InitializeAndGenerate$: Set up reference vector and generate terms in the time series.

Constraint:

mode = Nag_InitializeReference

Nag_GenerateFromReference

Nag_InitializeAndGenerate

2: n – IntegerInput

On entry:

n

, the number of observations to be generated.

Constraint:

n \geq 0

3: xmean – doubleInput

On entry: the mean of the time series.

4: ip – IntegerInput

On entry:

p

, the number of autoregressive coefficients supplied.

Constraint:

ip \geq 0

5: phi[ip] – const doubleInput

On entry: the autoregressive coefficients of the model,

ϕ_{1}, ϕ_{2}, \dots, ϕ_{p}

6: iq – IntegerInput

On entry:

q

, the number of moving average coefficients supplied.

Constraint:

iq \geq 0

7: theta[iq] – const doubleInput

On entry: the moving average coefficients of the model,

θ_{1}, θ_{2}, \dots, θ_{q}

8: avar – doubleInput

On entry:

σ^{2}

, the variance of the Normal perturbations.

Constraint:

avar \geq 0.0

9: r[lr] – doubleCommunication Array

On entry: if

mode = Nag_GenerateFromReference

, the reference vector from the previous call to nag_rand_arma (g05phc).

On exit: the reference vector.

10: lr – IntegerInput

On entry: the dimension of the array r.

Constraint:

lr \geq ip + iq + 6 + \max (ip, iq + 1)

11: state[ $\dim$ ] – IntegerCommunication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).

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

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

12: var – double *Output

On exit: the proportion of the variance of a term in the series that is due to the moving-average (error) terms in the model. The smaller this is, the nearer is the model to non-stationarity.

13: x[n] – doubleOutput

On exit: contains the next

n

observations from the time series.

14: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $ip = ⟨value⟩$ .
Constraint: $ip \geq 0$ .
On entry, $iq = ⟨value⟩$ .
Constraint: $iq \geq 0$ .
On entry, lr is not large enough, $lr = ⟨value⟩$ : minimum length required $= ⟨value⟩$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
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.
NE_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
NE_PREV_CALL: ip or iq is not the same as when r was set up in a previous call.
Previous value of $ip = ⟨value⟩$ and $ip = ⟨value⟩$ .
Previous value of $iq = ⟨value⟩$ and $iq = ⟨value⟩$ .
NE_REAL: On entry, $avar = ⟨value⟩$ .
Constraint: $avar \geq 0.0$ .
NE_REF_VEC: Reference vector r has been corrupted or not initialized correctly.
NE_STATIONARY_AR: On entry, the AR parameters are outside the stationarity region.

7 Accuracy

Any errors in the reference vector's initial values should be very much smaller than the error term; see Tunnicliffe–Wilson (1979).

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_rand_arma (g05phc) is essentially of order

{(ip)}^{2}

Note: The reference vector, r, contains a copy of the recent history of the series. If attempting to re-initialize the series by calling nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc) a call to nag_rand_arma (g05phc) with

mode = Nag_InitializeReference

must also be made. In the repeatable case the calls to nag_rand_arma (g05phc) should be performed in the same order (at the same point(s) in simulation) every time nag_rand_init_repeatable (g05kfc) is used. When the generator state is saved and restored using the argument state, the time series reference vector must be saved and restored as well.

The ARMA model for a time series can also be written as:

(x_{n} - E) = A_{1} (x_{n - 1} - E) + \dots + A_{NA} (x_{n - NA} - E) + B_{1} a_{n} + \dots + B_{NB} a_{n - NB + 1}

where

$x_{n}$ is the observed value of the time series at time $n$ ,
$NA$ is the number of autoregressive parameters, $A_{i}$ ,
$NB$ is the number of moving average parameters, $B_{i}$ ,
$E$ is the mean of the time series,

and

$a_{t}$ is a series of independent random Standard Normal perturbations.

This is related to the form given in Section 3 by:

$B_{1}^{2} = σ^{2}$ ,
$B_{i + 1} = - θ_{i} σ = - θ_{i} B_{1}, i = 1, 2, \dots, q$ ,
$NB = q + 1$ ,
$E = μ$ ,
$A_{i} = ϕ_{i}, i = 1, 2, \dots, p$ ,
$NA = p$ .

10 Example

This example generates values for an autoregressive model given by

x_{t} = 0.4 x_{t - 1} + 0.2 x_{t - 2} + ε_{t}

where

ε_{t}

is a series of independent random Normal perturbations with variance

1.0

. The random number generators are initialized by nag_rand_init_repeatable (g05kfc) and then nag_rand_arma (g05phc) is called to initialize a reference vector and generate a sample of ten observations.