What is the easiest way / method to compute the correlation between two time series that are exactly the same size? I thought of multiplying $(x[t]-\mu_x)$ and $(y[t] – \mu_y)$, and adding up the multiplication. So if this single number was positive, can we say these two series are correlated? I can think of some examples however where a linearly another exponentially growing time series would have no relation to eachother, but the computation above would report they were correlated.
Any thoughts?
Best Answer
Macro's point is correct the proper way to compare for relationships between time series is by the cross-correlation function (assuming stationarity). Having the same length is not essential. The cross correlation at lag 0 just computes a correlation like doing the Pearson correlation estimate pairing the data at the identical time points. If they do have the same length as you are assuming, you will have exact T pairs where T is the number of time points for each series. Lag 1 cross correlation matches time t from series 1 with time t+1 in series 2. Note that here even though the series are the same length you only have T-2 pair as one point in the first series has no match in the second and one other point in the second series will not have a match from the first. Given these two series you can estimate the cross-correlation at several lags . If any of the cross correlations is statistically significantly different from 0 it will indicate a correlation between the two series.