I have performed cross-correlation between two autocorrelated time series, in this case monthly net recharge versus each of 12 monthly groundwater level series. I am interested in finding the lag of maximum correlation between net recharge and each groundwater level series.
I used two prewhitening techniques that make use of the residuals from a high-order autoregressive model of the series. The function prewhiten(x,y) gives the CCF of the residuals of a high-order AR model of x. I did this using the function "prewhiten" from the package "forecast", saved the resulting AR(p) models, checked the residuals by Ljung-Box to verify lack of autocorrelation (see following image).
The equal footing approach correlates the residuals of each model for both series being correlated. In the systems approach, a model is estimated for only one series which can be considered the 'system input' (recharge in this case). The same model is used to filter the 'output' series (GW levels) and the residuals of the first series and the values of the filtered second series are cross-correlated. (If this explanation is wrong or misguided, please advise).
In the images below are the CCF results of both approaches, as well as the CCF of the raw series. I'm looking for help interpreting these results.
- Are elements of the CCFs of the raw series indicative of autocorrelation in the input series?
- What can we conclude from the prewhitened results? Do the results complement each other or conflict? Is one method more appropriate than the other and why?
- How can we demonstrate that prewhitening with a high-order AR process preserved the relevant information from the raw series? (in other words, Why are we sure that the CCF of the residuals still tells us something about the time lag between recharge and groundwater levels?)
Not enough rep to post all the images so please see the prewhitened CCFs here