are samples of series whose true autocorrelation functions are zero, then, under the null hypothesis that the true cross-correlations between the series are zero, stat has a

χ^{2}

-distribution with

L

degrees of freedom. Values of stat in the upper tail of this distribution provide evidence against the null hypothesis.

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: $x (nxy)$ – double array: The $n$ values of the $x$ series.
2: $y (nxy)$ – double array: The $n$ values of the $y$ series.
3: $nl$ – int64int32nag_int scalar: $L$ , the maximum lag for calculating cross-correlations.

Constraint: $1 \leq nl < nxy$ .

Optional Input Parameters

1: $nxy$ – int64int32nag_int scalar: Default: the dimension of the arrays x, y. (An error is raised if these dimensions are not equal.)
$n$ , the length of the time series.

Constraint: $nxy \geq 2$ .

Output Parameters

1: $s$ – double scalar: The ratio of the standard deviation of the $y$ series to the standard deviation of the $x$ series, $s_{y} / s_{x}$ .
2: $r0$ – double scalar: The cross-correlation between the $x$ and $y$ series at lag zero.
3: $r (nl)$ – double array: $r (l)$ contains the cross-correlations between the $x$ and $y$ series at lags $L$ , $r_{x y} (l)$ , for $l = 1, 2, \dots, L$ .
4: $stat$ – double scalar: The statistic for testing for absence of cross-correlation.
5: $ifail$ – 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,	$nxy \leq 1$ ,
or	$nl < 1$ ,
or	$nl \geq nxy$ .

$ifail = 2$: One or both of the $x$ and $y$ series have zero variance and hence cross-correlations cannot be calculated.

$ifail = - 99$: 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

All computations are believed to be stable.

Further Comments

n < 100

, or

L < 10 \log (n)

then the autocorrelations are calculated directly and the time taken by nag_tsa_multi_xcorr (g13bc) is approximately proportional to

n L

, otherwise the autocorrelations are calculated by utilizing fast Fourier transforms (FFTs) and the time taken is approximately proportional to

n \log (n)

. If FFTs are used then nag_tsa_multi_xcorr (g13bc) internally allocates approximately

6 n

real elements.

Example

This example reads two time series of length

20

. It calculates and prints the cross-correlations up to lag

15

for the first series leading the second series and then for the second series leading the first series.

Open in the MATLAB editor: g13bc_example

function g13bc_example


fprintf('g13bc example results\n\n');

% Series
x = [0.02;  0.05;  0.08;  0.03; -0.05;  0.11; -0.01; -0.08; -0.08; -0.11;
    -0.18; -0.19; -0.09;  0.03;  0.10;  0.15; -0.14;  0.07;  0.09;  0.16];
y = [3.18;  3.21;  3.26;  3.25;  3.08;  3.01;  3.06;  3.17;  3.12;  3.04;
     3.26;  3.45;  3.33;  3.70;  3.31;  3.81;  3.33;  2.96;  3.28;  3.10];

% Number of lags
nl = int64(15);

% Cross correlations between x and y
[sxy, r0xy, rxy, statxy, ifail] = g13bc( ...
                                 x, y, nl);
% Cross correlations between y and x
[syx, r0yx, ryx, statyx, ifail] = g13bc( ...
                                 y, x, nl);

% Display results
fprintf('%34s%15s\n', 'Between', 'Between');
fprintf('%34s%15s\n', 'x and y', 'y and x');
fmt1 = '\n%24s%10.4f%15.4f\n';
fprintf(fmt1, 'Standard deviation ratio', sxy, syx);
fprintf('\nCross correlation at lag');
fprintf(fmt1, '0', r0xy, r0yx);
ivar = double([1:nl]');
fprintf('%24d%10.4f%15.4f\n', [ivar rxy ryx]')
fprintf(fmt1,'Test statistic          ', statxy, statyx);

g13bc example results

                           Between        Between
                           x and y        y and x

Standard deviation ratio    2.0053         0.4987

Cross correlation at lag
                       0    0.0568         0.0568
                       1    0.0438        -0.0151
                       2   -0.3762         0.3955
                       3   -0.4864         0.3417
                       4   -0.6294         0.5486
                       5   -0.3871         0.2291
                       6   -0.1690         0.3190
                       7   -0.0678         0.1980
                       8    0.0962         0.0438
                       9    0.0788        -0.1428
                      10    0.2910        -0.1376
                      11    0.0950        -0.0387
                      12    0.0547        -0.0380
                      13    0.1855        -0.1551
                      14    0.0243        -0.1536
                      15    0.0034        -0.0696

Test statistic             22.1269        17.2917

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

Chapter Contents

Chapter Introduction

NAG Toolbox