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 <nag.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)

The routine may be called by the names g13bdf or nagf_tsa_multi_transf_prelim.

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) array Input: 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: $nl$ – Integer Input: 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 array Input: 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: $nwds$ – Integer Input: 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) array Output
8: $iwa$ – Integer Input: These arguments are no longer accessed by g13bdf. Workspace is provided internally by dynamic allocation instead.
9: $wds (nwds)$ – Real (Kind=nag_wp) array Output: 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 array Output: 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: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $- 1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $- 1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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, $i = ⟨ value ⟩$ and $nna (i) = ⟨ value ⟩$ .
Constraint: $nna (i) \geq 0$ .

On entry, $i = ⟨ value ⟩$ and $r (i) = ⟨ value ⟩$ .
Constraint: $- 1.0 \leq r (i) \leq 1.0$ .

On entry, $nl = ⟨ value ⟩$ , $nna (1) = ⟨ value ⟩$ , $nna (2) = ⟨ value ⟩$ and $nna (3) = ⟨ value ⟩$ .
Constraint: $nl \geq \max (nna (1) + nna (2) + nna (3), 1)$ .

On entry, $nwds = ⟨ value ⟩$ , $nna (2) = ⟨ value ⟩$ and $nna (3) = ⟨ value ⟩$ .
Constraint: $nwds = nna (2) + nna (3) + 1$ .

On entry, $r0 = ⟨ value ⟩$ .
Constraint: $- 1.0 \leq r0 \leq 1.0$ .

On entry, $s = ⟨ value ⟩$ .
Constraint: $s > 0.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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.

g13bd: FL CL CPP AD

NAG FL Interfaceg13bdf (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

NAG FL Interface
g13bdf (multi_transf_prelim)