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 g13bac, 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: $r0$ – double 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: $r [nl]$ – const double Input

On entry: the cross-correlations between the two series at lags

1

L

r_{x y} (l)

, for

l = 1, 2, \dots, L

Constraint:

- 1.0 \leq r [i] \leq 1.0

, for

i = 0, 1, \dots, nl - 1

3: $nl$ – Integer Input

On entry:

L

, the number of lagged cross-correlations in the array r.

Constraint:

nl \geq \max (transfv . nag_b + transfv . nag_q + transfv . nag_p, 1)

4: $transfv$ – Nag_TransfOrder * Input

On entry: the orders of the transfer function model where the triplet (transfv. nag_b, transfv. nag_q, transfv. nag_p) corresponds to the triplet

(b, q, p)

as described in Section 2.3.1 in the G13 Chapter Introduction.

Constraints:

$transfv . nag_b \geq 0$ ;
$transfv . nag_q \geq 0$ ;
$transfv . nag_p \geq 0$ .

5: $s$ – double 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: $wds [\dim]$ – double Output

Note: the dimension, dim, of the array wds must be at least

transfv . nag_q + transfv . nag_p + 1

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

7: $isf [2]$ – Integer 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

−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.

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_CONSTRAINT: Constraint: $transfv . nag_b \geq 0$ .

Constraint: $transfv . nag_p \geq 0$ .

Constraint: $transfv . nag_q \geq 0$ .
NE_INT_4: On entry, $nl = ⟨ value ⟩$ , $transfv . nag_b = ⟨ value ⟩$ , $transfv . nag_q = ⟨ value ⟩$ and $transfv . nag_p = ⟨ value ⟩$ .
Constraint: $nl \geq \max (transfv . nag_b + transfv . nag_q + transfv . nag_p, 1)$ .
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 contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL: On entry, $r0 = ⟨ value ⟩$ .
Constraint: $- 1.0 \leq r0 \leq 1.0$ .

On entry, $s = ⟨ value ⟩$ .
Constraint: $s > 0.0$ .
NE_REAL_ARRAY_ELEM_CONS: On entry, $i = ⟨ value ⟩$ and $r [i - 1] = ⟨ value ⟩$ .
Constraint: $- 1.0 \leq r [i] \leq 1.0$ .

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

Background information to multithreading can be found in the Multithreading documentation.

g13bdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g13bdc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by g13bdc is roughly proportional to

{transfv . nag_q + transfv . nag_p + 1}^{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 PY MB

NAG CL Interfaceg13bdc (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 CL Interface
g13bdc (multi_transf_prelim)