NAG Library Function Document

nag_arma_time_series (g05hac)

+− 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

1 Purpose

nag_arma_time_series (g05hac) generates an autoregressive moving average (ARMA) time series with Normally distributed errors (or residuals). It initializes the series to a stationary position and sets up a reference vector enabling the function to be called repeatedly, adding terms to the previous series at each call.

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_arma_time_series (Nag_Boolean start, Integer p, Integer q, const double phi[], const double theta[], double mean, double vara, Integer n, double w[], double ref[], NagError *fail)

3 Description

An ARMA model, denoted by ARMA

(p, q)

, is a mixture of an autoregressive process of order

p

(AR) and a moving average (MA) process of order

q

and can be written as

(x_{n} - μ) = ϕ_{1} (x_{n - 1} - μ) + \dots + ϕ_{p} (x_{n - p} - μ) + a_{n} - θ_{1} a_{n - 1} \dots - θ_{q} a_{n - q}

where

x_{n}

are the realization of the series,

μ

is the mean of the series and

a_{n}

are the errors (or residuals, also often called the white noise) which are independently distributed as normal with mean zero and variance

σ^{2}

. The arguments

ϕ_{i}

are the autoregressive arguments and the arguments

θ_{i}

are the moving average arguments.

The function sets up initial values corresponding to a stationary position using the method described by Tunnicliffe–Wilson (1979). It generates

n

terms of the time series by first calculating the next term in the autoregressive series and then applying the moving-average summation and storing the result.

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: start – Nag_BooleanInput: On entry: start must be Nag_TRUE if a new series is to begin, if start is Nag_FALSE a previously generated series will be continued. If start is Nag_FALSE then the scalar arguments p, q, mean and vara and the contents of the array arguments, phi and theta must not be changed.
2: p – IntegerInput: On entry: the number of autoregressive coefficients supplied.
Constraint: $p \geq 0$ .
3: q – IntegerInput: On entry: the number of moving-average coefficients supplied.
Constraint: $q \geq 0$ .
4: phi[p] – const doubleInput: On entry: the autoregressive coefficients of the model, if any, $phi [i - 1]$ must contain $ϕ_{i}$ , for $i = 1, 2, \dots, p$ .
5: theta[q] – const doubleInput: On entry: the moving-average coefficients of the model, if any, $theta [i - 1]$ must contain $θ_{i}$ , for $i = 1, 2, \dots, q$ .
6: mean – doubleInput: On entry: the mean of the time series.
7: vara – doubleInput: On entry: the variance of the errors, $σ^{2}$ .
Constraint: $vara > 0.0$ .
8: n – IntegerInput: On entry: the number of observations to be generated.
Constraint: $n \geq 1$ .
9: w[n] – doubleOutput: On exit: the realization of the time series.
10: ref[ $5 \times \max (p, q) + 7$ ] – doubleOutput: On exit: the reference vector and the recent history of the series.
11: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_INT_ARG_LT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .; On entry, $p = 〈value〉$ .
Constraint: $p \geq 0$ .; On entry, $q = 〈value〉$ .
Constraint: $q \geq 0$ .
NE_REAL_ARG_LE: On entry, vara must not be less than or equal to 0.0: $vara = 〈value〉$ .
NE_REF_VEC: The reference vector set up by the previous call of this function has become corrupt.
NE_START_P_Q: The function has been called either with $start = Nag_FALSE$ the first time or at least one of p or q has been changed in a subsequent call with $start = Nag_FALSE$ .
NE_STATIONARITY: The input series does not constitute a stationary time-series model.

7 Accuracy

Not applicable.

8 Further Comments

None.

9 Example

The program below shows two calls of nag_arma_time_series (g05hac). In the first call an ARMA series is generated. In the second call terms are added to the already existing series.