How to statistically compare two time series?

Question

I have two time series, shown in the plot below:

The plot is showing the full detail of both time series, but I can easily reduce it to just the coincident observations if needed.

My question is: What statistical methods can I use to assess the differences between the time series?

I know this is a fairly broad and vague question, but I can't seem to find much introductory material on this anywhere. As I can see it, there are two distinct things to assess:

1. Are the values the same?

2. Are the trends the same?

What sort of statistical tests would you suggest looking at to assess these questions? For question 1 I can obviously assess the means of the different datasets and look for significant differences in distributions, but is there a way of doing this that takes into account the time-series nature of the data?

For question 2 - is there something like the Mann-Kendall tests that looks for the similarity between two trends? I could do the Mann-Kendall test for both datasets and compare, but I don't know if that is a valid way to do things, or whether there is a better way?

I'm doing all of this in R, so if tests you suggest have a R package then please let me know.

Answer 1

As others have stated, you need to have a common frequency of measurement ( i.e. the time between observations ). With that in place I would identify a common model that would reasonably describe each series separately. This might be an ARIMA model or a multiply-trended Regression Model with possible Level Shifts or a composite model integrating both memory (ARIMA) and dummy variables. This common model could be estimated globally and separately for each of the two series and then one could construct an F Test to hest the hypothesis of a common set of parameters.

Answer 2

Consider the grangertest() in the lmtest library.

It is a test to see if one time series is useful in forecasting another.

A couple references to get you started:

http://monogan.myweb.uga.edu/teaching/ts/13granger.pdf

http://en.wikipedia.org/wiki/Granger_causality