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 nag_tsa_multi_filter_arima (g13ba), 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.

References

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

Parameters

Compulsory Input Parameters

1: $r0$ – double scalar: The cross-correlation between the two series at lag $0$ , $r_{x y} (0)$ .

Constraint: $- 1.0 \leq r0 \leq 1.0$ .
2: $r (nl)$ – double array: 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: $nna (3)$ – int64int32nag_int array: 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 $i = 1, 2, 3$ .
4: $s$ – double scalar: The ratio of the standard deviation of the $y$ series to that of the $x$ series, $s_{y} / s_{x}$ .

Constraint: $s > 0.0$ .

Optional Input Parameters

1: $nl$ – int64int32nag_int scalar: Default: the dimension of the array r.
$L$ , the number of lagged cross-correlations in the array r.

Constraint: $nl \geq \max (nna (1) + nna (2) + nna (3), 1)$ .

Output Parameters

1: $wds (nwds)$ – double array: $nwds = nna (2) + nna (3) + 1$ .
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 $0.0$ .
2: $isf (2)$ – int64int32nag_int array: 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.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

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.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

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

Further Comments

nna (3) > 0

,a local workspace array of fixed length is allocated internally by nag_tsa_multi_transf_prelim (g13bd). The total size of this array amounts to

nna (3)

integer elements and

nna (3) \times (nna (3) + 1)

double elements.

The time taken by nag_tsa_multi_transf_prelim (g13bd) is roughly proportional to

{nwds}^{3}

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.

Open in the MATLAB editor: g13bd_example

function g13bd_example


fprintf('g13bd example results\n\n');

% Cross-correlation at lag 0
r0  = -0.0155;
% Other cross-correlations
r   = [0.0339; -0.0374; -0.2895; -0.3430; -0.4518; -0.2787];

% transfer function model orders
nna = [int64(3);2;1];

%Standard deviation ratio
s   = 1.9256;

% Calculate parameter estimates
[wds, isf, ifail] = g13bd( ...
                           r0, r, nna, s);

% Display results
fprintf('Success/failure indicator       = %4d%4d\n\n', isf(1:2));
fprintf('Transfer function model B, Q, P = %4d%4d%4d\n\n', nna(1:3));
disp('Parameter initial estimates');
disp(wds');

g13bd example results

Success/failure indicator       =    1   1

Transfer function model B, Q, P =    3   2   1

Parameter initial estimates
   -0.5575    0.3166    0.4626    0.6169

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox