NAG Library Routine Document

G13BCF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G13BCF calculates cross-correlations between two time series.

2 Specification

SUBROUTINE G13BCF (

X, Y, NXY, NL, S, R0, R, STAT, IFAIL)

INTEGER	NXY, NL, IFAIL
REAL (KIND=nag_wp)	X(NXY), Y(NXY), S, R0, R(NL), STAT

3 Description

Given two series

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

and

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

the routine 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 Parameters

1: X(NXY) – REAL (KIND=nag_wp) arrayInput: On entry: the $n$ values of the $x$ series.
2: Y(NXY) – REAL (KIND=nag_wp) arrayInput: 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 – REAL (KIND=nag_wp)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 – REAL (KIND=nag_wp)Output: On exit: the cross-correlation between the $x$ and $y$ series at lag zero.
7: R(NL) – REAL (KIND=nag_wp) arrayOutput: On exit: $R (l)$ contains the cross-correlations between the $x$ and $y$ series at lags $L$ , $r_{x y} (l)$ , for $l = 1, 2, \dots, L$ .
8: STAT – REAL (KIND=nag_wp)Output: On exit: the statistic for testing for absence of cross-correlation.
9: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . 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,	$NXY \leq 1$ ,
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.

7 Accuracy

All computations are believed to be stable.

8 Further Comments

n < 100

, or

L < 10 \log (n)

then the autocorrelations are calculated directly and the time taken by G13BCF 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 G13BCF internally allocates approximately

6 n

real elements.

9 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 Routine DocumentG13BCF