Integer, Intent (In)	::	nl, nna(3), nwds, iwa
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	isf(2)
Real (Kind=nag_wp), Intent (In)	::	r0, r(nl), s
Real (Kind=nag_wp), Intent (Out)	::	wa(iwa), wds(nwds)

C Header Interface

#include nagmk26.h

void	g13bdf_ ( const double r0, const double r[], const Integer nl, const Integer nna[], const double s, const Integer nwds, double wa[], const Integer iwa, double wds[], Integer isf[], Integer ifail)

3

Description

g13bdf calculates estimates of parameters

δ_{1}, δ_{2}, \dots, δ_{p}

ω_{0}, ω_{1}, \dots, ω_{q}

in the transfer function model

y_{t} = δ_{1} y_{t - 1} + δ_{2} y_{t - 2} + \dots + δ_{p} y_{t - p} + ω_{0} x_{t - b} - ω_{1} x_{t - b - 1} - \dots - ω_{q} x_{t - b - q}

given cross-correlations between the series

x_{t}

and lagged values of

y_{t}

r_{x y} (l), l = 0, 1, \dots, L

and the ratio of standard deviations

s_{y} / s_{x}

, as supplied by g13bcf.

It is assumed that the series

x_{t}

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

Arguments

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 arguments 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 argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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	$iwa < nna (3) \times (nna (3) + 1)$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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

Parallelism and Performance

g13bdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g13bdf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

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}

10

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 Document

g13bdf (multi_transf_prelim)

▸▿ 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

2

Specification