NAG Library Routine Document

G13BDF

used to calculate the cross-correlations is a sample from a time series with true autocorrelations of zero. Otherwise the cross-correlations between the series

b_{t}

and

a_{t}

, as defined in the description of G13BAF, should be used in place of those between

y_{t}

and

x_{t}

The estimates are obtained by solving for

δ_{1}, δ_{2}, \dots, δ_{p}

the equations

r_{x y} (b + q + j) = δ_{1} r_{x y} (b + q + j - 1) + \dots + δ_{p} r_{x y} (b + q + j - p), j = 1, 2, \dots, p

then calculating

ω_{i} = \pm (s_{y} / s_{x}) [r_{x y} (b + i) - δ_{1} r_{x y} (b + i - 1) - \dots - δ_{p} r_{x y} (b + i - p)], i = 0, 1, \dots, q

where the ‘

+

’ is used for

ω_{0}

and ‘

-

’ for

ω_{i}

i > 0

Any value of

r_{x y} (l)

arising in these equations for

l < b

is taken as zero. The parameters

δ_{1}, δ_{2}, \dots, δ_{p}

are checked as to whether they satisfy the stability criterion.

4 References

Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day

5 Parameters

1: R0 – REAL (KIND=nag_wp)Input: On entry: the cross-correlation between the two series at lag $0$ , $r_{x y} (0)$ .
Constraint: $- 1.0 \leq R0 \leq 1.0$ .
2: R(NL) – REAL (KIND=nag_wp) arrayInput: On entry: the cross-correlations between the two series at lags $1$ to $L$ , $r_{x y} (l)$ , for $l = 1, 2, \dots, L$ .
Constraint: $- 1.0 \leq R (i) \leq 1.0$ , for $i = 1, 2, \dots, NL$ .
3: NL – INTEGERInput: On entry: $L$ , the number of lagged cross-correlations in the array R.

Constraint: $NL \geq \max (NNA (1) + NNA (2) + NNA (3), 1)$ .
4: NNA( $3$ ) – INTEGER arrayInput: On entry: the transfer function model orders in the standard form $b, q, p$ (i.e., delay time, number of moving-average MA-like followed by number of autoregressive AR-like parameters).
Constraint: $NNA (i) \geq 0$ , for $i = 1, 2, 3$ .
5: S – REAL (KIND=nag_wp)Input: On entry: the ratio of the standard deviation of the $y$ series to that of the $x$ series, $s_{y} / s_{x}$ .
Constraint: $S > 0.0$ .
6: NWDS – INTEGERInput: On entry: the exact number of parameters in the transfer function model.
Constraint: $NWDS = NNA (2) + NNA (3) + 1$ .
7: WA(IWA) – REAL (KIND=nag_wp) arrayWorkspace
8: IWA – INTEGERInput: On entry: the dimension of the array WA as declared in the (sub)program from which G13BDF is called.
Constraint: $IWA \geq NNA (3) \times (NNA (3) + 1)$ .
9: WDS(NWDS) – REAL (KIND=nag_wp) arrayOutput: On exit: the preliminary estimates of the parameters of the transfer function model in the order of $q + 1$ MA-like parameters followed by the $p$ AR-like parameters. If the estimation of either type of parameter fails then these parameters are set to $0.0$ .
10: ISF( $2$ ) – INTEGER arrayOutput: On exit: indicators of the success of the estimation of MA-like and AR-like parameters respectively. A value $0$ indicates that there are no parameters of that type to be estimated. A value of $1$ or $- 1$ indicates that there are parameters of that type in the model and the estimation of that type has been successful or unsuccessful respectively. Note that there is always at least one MA-like parameter in the model.
11: 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,	$NNA (i) < 0$ , for $i = 1, 2, 3$ ,
or	$NL < \max (NNA (1) + NNA (2) + NNA (3), 1)$ ,
or	$R0 < - 1.0$ or $R0 > 1.0$ ,
or	$R (i) < - 1.0$ or $R (i) > 1.0$ , for some $i = 1, 2, \dots, NL$ ,
or	$S \leq 0.0$ ,
or	$NWDS \neq NNA (2) + NNA (3) + 1$ ,
or	$IWA < NNA (3) \times (NNA (3) + 1)$ .

$IFAIL = - 999$: Internal memory allocation failed.

7 Accuracy

Equations used in the computations may become unstable, in which case results are reset to zero with array ISF values set accordingly.

8 Further Comments

NNA (3) > 0

,a local workspace array of fixed length is allocated internally by G13BDF. The total size of this array amounts to

NNA (3)

integer elements and

NNA (3) \times (NNA (3) + 1)

real elements.

The time taken by G13BDF is roughly proportional to

{NWDS}^{3}

9 Example

This example reads the cross-correlations between two series at lags

0

6

. It then reads a

(3, 2, 1)

transfer function model and calculates and prints the preliminary estimates of the parameters of the model.

NAG Library Routine DocumentG13BDF

+− Contents

1 Purpose

2 Specification

3 Description