NAG Library Function Document

nag_tsa_cross_corr (g13bcc)

+− 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

nag_tsa_cross_corr (g13bcc) calculates cross-correlations between two time series.

2 Specification

#include <nag.h>

#include <nagg13.h>

void	nag_tsa_cross_corr (const double x[], const double y[], Integer nxy, Integer nl, double s, double r0, double r[], double stat, NagError fail)

3 Description

Given two series

x_{1}, x_{2}, \dots, x_{n}

and

y_{1}, y_{2}, \dots, y_{n}

the function calculates the cross-correlations between

x_{t}

and lagged values of

y_{t}

r_{x y} (l) = \frac{\sum_{t = 1}^{n - l} (x_{t} - \bar{x}) (y_{t + l} - \bar{y})}{n s_{x} s_{y}}, l = 0, 1, \dots, L

where

\bar{x} = \frac{\sum_{t = 1}^{n} x_{t}}{n}

s_{x}^{2} = \frac{\sum_{t = 1}^{n} {(x_{t} - \bar{x})}^{2}}{n}

and similarly for

y

The ratio of standard deviations

s_{y} / s_{x}

is also returned, and a portmanteau statistic is calculated:

stat = n \sum_{l = 1}^{L} r_{x y} {(l)}^{2} .

Provided

n

is large,

L

much less than

n

, and both

x_{t}, y_{t}

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.

4 References

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

5 Arguments

1: x[nxy] – const doubleInput: On entry: the $n$ values of the $x$ series.
2: y[nxy] – const doubleInput: On entry: the $n$ values of the $y$ series.
3: nxy – IntegerInput: On entry: $n$ , the length of the time series.
Constraint: $nxy \geq 2$ .
4: nl – IntegerInput: On entry: $L$ , the maximum lag for calculating cross-correlations.
Constraint: $1 \leq nl < nxy$ .
5: s – double *Output: On exit: the ratio of the standard deviation of the $y$ series to the standard deviation of the $x$ series, $s_{y} / s_{x}$ .
6: r0 – double *Output: On exit: the cross-correlation between the $x$ and $y$ series at lag zero.
7: r[nl] – doubleOutput: On exit: $r [l - 1]$ contains the cross-correlations between the $x$ and $y$ series at lags $L$ , $r_{x y} (l)$ , for $l = 1, 2, \dots, L$ .
8: stat – double *Output: On exit: the statistic for testing for absence of cross-correlation.
9: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $nl = ⟨value⟩$ .
Constraint: $nl \geq 1$ .
On entry, $nxy = ⟨value⟩$ .
Constraint: $nxy > 1$ .
NE_INT_2: On entry, $nl \geq nxy$ : $nl = ⟨value⟩$ and $nxy = ⟨value⟩$ .
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.
NE_ZERO_VARIANCE: One or both of the $x$ and $y$ series have zero variance.

7 Accuracy

All computations are believed to be stable.

8 Parallelism and Performance

Not applicable.

9 Further Comments

n < 100

, or

L < 10 \log (n)

then the autocorrelations are calculated directly and the time taken by nag_tsa_cross_corr (g13bcc) 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_cross_corr (g13bcc) internally allocates approximately

6 n

real elements.

10 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.

NAG Library Function Documentnag_tsa_cross_corr (g13bcc)