NAG Library Function Document
nag_rand_arma (g05phc)
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:
where
${\epsilon}_{t}$, is a residual series of independent random perturbations assumed to be Normally distributed with zero mean and variance
${\sigma}^{2}$. The parameters
$\left\{{\varphi}_{i}\right\}$, for
$\mathit{i}=1,2,\dots ,p$, are called the autoregressive (AR) parameters, and
$\left\{{\theta}_{j}\right\}$, for
$\mathit{j}=1,2,\dots ,q$, the moving average (MA) parameters. The parameters in the model are thus the
$p$ $\varphi $ values, the
$q$ $\theta $ values, the mean
$\mu $ and the residual variance
${\sigma}^{2}$.
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 nonrepeatable 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:
$\mathbf{mode}$ – Nag_ModeRNGInput

On entry: a code for selecting the operation to be performed by the function.
 ${\mathbf{mode}}=\mathrm{Nag\_InitializeReference}$
 Set up reference vector only.
 ${\mathbf{mode}}=\mathrm{Nag\_GenerateFromReference}$
 Generate terms in the time series using reference vector set up in a prior call to nag_rand_arma (g05phc).
 ${\mathbf{mode}}=\mathrm{Nag\_InitializeAndGenerate}$
 Set up reference vector and generate terms in the time series.
Constraint:
${\mathbf{mode}}=\mathrm{Nag\_InitializeReference}$, $\mathrm{Nag\_GenerateFromReference}$ or $\mathrm{Nag\_InitializeAndGenerate}$.
 2:
$\mathbf{n}$ – IntegerInput

On entry: $n$, the number of observations to be generated.
Constraint:
${\mathbf{n}}\ge 0$.
 3:
$\mathbf{xmean}$ – doubleInput

On entry: the mean of the time series.
 4:
$\mathbf{ip}$ – IntegerInput

On entry: $p$, the number of autoregressive coefficients supplied.
Constraint:
${\mathbf{ip}}\ge 0$.
 5:
$\mathbf{phi}\left[{\mathbf{ip}}\right]$ – const doubleInput

On entry: the autoregressive coefficients of the model, ${\varphi}_{1},{\varphi}_{2},\dots ,{\varphi}_{p}$.
 6:
$\mathbf{iq}$ – IntegerInput

On entry: $q$, the number of moving average coefficients supplied.
Constraint:
${\mathbf{iq}}\ge 0$.
 7:
$\mathbf{theta}\left[{\mathbf{iq}}\right]$ – const doubleInput

On entry: the moving average coefficients of the model, ${\theta}_{1},{\theta}_{2},\dots ,{\theta}_{q}$.
 8:
$\mathbf{avar}$ – doubleInput

On entry: ${\sigma}^{2}$, the variance of the Normal perturbations.
Constraint:
${\mathbf{avar}}\ge 0.0$.
 9:
$\mathbf{r}\left[{\mathbf{lr}}\right]$ – doubleCommunication Array

On entry: if ${\mathbf{mode}}=\mathrm{Nag\_GenerateFromReference}$, the reference vector from the previous call to nag_rand_arma (g05phc).
On exit: the reference vector.
 10:
$\mathbf{lr}$ – IntegerInput

On entry: the dimension of the array
r.
Constraint:
${\mathbf{lr}}\ge {\mathbf{ip}}+{\mathbf{iq}}+6+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}+1\right)$.
 11:
$\mathbf{state}\left[\mathit{dim}\right]$ – IntegerCommunication Array

Note: the dimension,
$\mathit{dim}$, 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:
$\mathbf{var}$ – double *Output

On exit: the proportion of the variance of a term in the series that is due to the movingaverage (error) terms in the model. The smaller this is, the nearer is the model to nonstationarity.
 13:
$\mathbf{x}\left[{\mathbf{n}}\right]$ – doubleOutput

On exit: contains the next $n$ observations from the time series.
 14:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{ip}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ip}}\ge 0$.
On entry, ${\mathbf{iq}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{iq}}\ge 0$.
On entry,
lr is not large enough,
${\mathbf{lr}}=\u2329\mathit{\text{value}}\u232a$: minimum length required
$\text{}=\u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 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.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
 NE_INVALID_STATE

On entry,
state vector has been corrupted or not initialized.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
 NE_PREV_CALL

ip or
iq is not the same as when
r was set up in a previous call.
Previous value of
${\mathbf{ip}}=\u2329\mathit{\text{value}}\u232a$ and
${\mathbf{ip}}=\u2329\mathit{\text{value}}\u232a$.
Previous value of
${\mathbf{iq}}=\u2329\mathit{\text{value}}\u232a$ and
${\mathbf{iq}}=\u2329\mathit{\text{value}}\u232a$.
 NE_REAL

On entry, ${\mathbf{avar}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{avar}}\ge 0.0$.
 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
nag_rand_arma (g05phc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The time taken by nag_rand_arma (g05phc) is essentially of order ${\left({\mathbf{ip}}\right)}^{2}$.
Note: The reference vector,
r, contains a copy of the recent history of the series. If attempting to reinitialize the series by calling
nag_rand_init_repeatable (g05kfc) or
nag_rand_init_nonrepeatable (g05kgc) a call to
nag_rand_arma (g05phc) with
${\mathbf{mode}}=\mathrm{Nag\_InitializeReference}$ 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:
where
 ${x}_{n}$ is the observed value of the time series at time $n$,
 $\mathit{NA}$ is the number of autoregressive parameters, ${A}_{i}$,
 $\mathit{NB}$ 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:
 ${B}_{1}^{2}={\sigma}^{2}$,
 ${B}_{i+1}={\theta}_{i}\sigma ={\theta}_{i}{B}_{1}\text{, \hspace{1em}}i=1,2,\dots ,q$,
 $\mathit{NB}=q+1$,
 $E=\mu $,
 ${A}_{i}={\varphi}_{i}\text{, \hspace{1em}}i=1,2,\dots ,p$,
 $\mathit{NA}=p$.
10
Example
This example generates values for an autoregressive model given by
where
${\epsilon}_{t}$ 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.
10.1
Program Text
Program Text (g05phce.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (g05phce.r)