Time Series Analysis - Crosscorrelations
The crosscorrelation coefficient rxy(k) represents the correlation between two series X and Y, where X is lagged by k observations (see also, Time Series Analysis Index). Note that the crosscorrelation function is not symmetrical about the lag of 0, that is, different correlations will emerge depending on whether the x-axis is shifted forward or backward. The crosscorrelation coefficient is computed following the standard formulas, as described in most time series references (e.g., Box & Jenkins, 1976):
| rxy(k) = cxy(k) / sxsy | for k = 0, ±1, ±2, ... |
where
| cxy(k) | cross covariance coefficient at lag k |
| sx | standard deviation of series X = Ö(1/n)*Σ(Xt - X-bar)2 |
| sy | standard deviation of series Y = Ö(1/n)*Σ(Yt - Y-bar)2 |
Estimation of the cross covariance function is given by
| cxy(k) = (1/N)*Σ[(Xt - X-bar)*(Yt+m - Y-bar)] | for t = 1 to N - l; m = 0 to k |
| cxy(k) = (1/N)*Σ[(Yt - Y-bar)*(Xt+m - X-bar)] | for t = 1 to N + l; m = -1 to -k |
where
| cxy(k) | cross covariance coefficient at lag k |
| X-bar and Y-bar | are the means of the respective two series |
| t | time |
Standard error of rxy(k). Under the assumption that X and Y are independent, and that one of the series consists only of white noise, then the standard error of the crosscorrelation is approximately equal to:
StdErr(rxy(k)) = Ö[1/(N - k)]
where
| N | number of observations in the series |