nag_rngs_arma_time_series (g05pac) generates a realisation of a univariate time series from an autoregressive moving average (ARMA) model. The realisation may be continued or a new realisation generated at subsequent calls to nag_rngs_arma_time_series (g05pac).

Let the vector , denote a time series which is assumed to follow an autoregressive moving average (ARMA) model of the form:

(1)

where , is a residual series of independent random perturbations assumed to be Normally distributed with zero mean and variance . The arguments , for , are called the autoregressive (AR) arguments, and , for , the moving average (MA) arguments. The arguments in the model are thus the    values, the    values, the mean  and the residual variance .

nag_rngs_arma_time_series (g05pac) 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 realisation of . On a successful exit, the recent history is updated and saved in the reference vector r so that nag_rngs_arma_time_series (g05pac) may be called again to generate a realisation of etc. See the description of the argument mode in Section [Parameters] for details.

One of the initialization functions g05kbc (nag_rngs_init_repeatable) (for a repeatable sequence if computed sequentially) or g05kcc (nag_rngs_init_nonrepeatable) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_arma_time_series (g05pac).

"NE_BAD_PARAM"

On entry, argument  had an illegal value.

"NE_INT"

 On entry, . Constraint: .

 On entry, . Constraint: .

On entry, . Constraint: .

On entry, . Constraint: .

"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 consult NAG for assistance.

"NE_REAL"

 On entry, . Constraint: .

"NE_STATIONARY_AR"

  phi does not define a stationary autoregressive process.

The errors in the initialization process should be very much smaller than the error term; see Tunnicliffe–Wilson (1979).

The time taken by nag_rngs_arma_time_series (g05pac) is essentially of order .

Note: g05kbc (nag_rngs_init_repeatable) and g05kcc (nag_rngs_init_nonrepeatable) must be used with care if this function is used as well. The reference vector, as mentioned before, contains a copy of the recent history of the series. This will not be altered properly by calls to any of the above functions. A call to g05kbc (nag_rngs_init_repeatable) or g05kcc (nag_rngs_init_nonrepeatable) should be followed by a call to nag_rngs_arma_time_series (g05pac) with  to re-initialize the time series reference vector in use. To maintain repeatability with g05kbc (nag_rngs_init_repeatable), the calls to nag_rngs_arma_time_series (g05pac) should be performed in the same order and at the same point or points in the simulation every time g05kbc (nag_rngs_init_repeatable) is used. When the generator state is saved and restored using the arguments igen and iseed, 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

and

 This is related to the form given in Section [Description] by: