# Correlation of non-stationary time series

I am looking at finding correlations between house price time series and the time series of multiple indicators in an area. For example

These two trends clearly show a sort of strong negative correlation.

Other indicators in this example could be income (expected positive correlation), crime (expected weak negative correlation), number of pizzas I've eaten that month (expected zero correlation).

I've read that cross-correlation if the method used to find a correlation between stationary time series but these are clearly non-stationary.

This is where I'm getting confused. Is it correct to detrend e.g. take the residual part of this plot:

and then perform cross-correlation on this, providing that it is stationary enough?

I'm struggling to believe this as I feel that if we do this, we're ignoring the key part of the information which is the overall trend of the two-series through time.

I feel that perhaps a better option is to take the trend and then perform e.g. first order differencing on it. And then hopefully providing that both of the differenced time series are stationary enough, performing cross correlation on that.

Which of these options, if either is correct?

• The discussion in this post makes it doubtful that your objective is to find correlations among time series, because it looks like you're perfectly willing to modify those series. Could you articulate your ultimate objective explicitly?
– whuber
Commented Aug 10, 2018 at 16:43
• hi @whuber the ultimate objective is to know, for each area, how each indicator correlates with house price (and present pretty table to the stakeholder who has employed me to do this work) so I would guess that the thing in my post that is actually doubtful is my knowledge of how to proceed rather than my objective! It is this unwillingness to modify these series that led me to question that detrending was the correct way to go. Any help / advice would be greatly appreciated Commented Aug 10, 2018 at 16:58
• I'm sorry; that makes no sense, because you can always compute a Pearson correlation coefficient between two matched non-constant sequences of data. It's just a mathematical formula and it doesn't care about any "non-linear attributes" of the numbers.
– whuber
Commented Aug 12, 2018 at 0:53
• Hi whuber, do you know of a way to calculate a meaningful value that will help to indicate if two time series are positively correlated, negatively correlated or not correlated? I ultimately want to say, if e.g. unemployment goes up then house prices are likely going down. Commented Aug 13, 2018 at 8:08
• The answers here (and the upvotes) are reflecting well why people misunderstand this. The question is quite simple: how to mathematically show the correlation between the clearly correlated time series? I'm still yet to see a comprehensive practical answer that helps educate us, as opposed to re-stating how badly formulated the question is... Commented Sep 13, 2022 at 23:28

The point is that comparing two (or more) non-stationary processes that move together, two situations are possible:

1. They are both caused by a third (or more) process,
2. The coincidence of their trends is random (spurious correlation).

There are statistical tests to test the corresponding hypothesis (Engle-Granger Two-Step Method, Johansen Test).

Short and simple explanation

Find correlation between two time series. Theory and practice (R) is a good place to start your education. Note the discussion that points to the flaw of interpreting ( not computing ! ) correlation coefficients when you have auto-correlated data ...as you do .

This problem was recognized for time series as early as 1926 by Yule in his presidential address to the Royal Statistical Society and nearly 100 years later we have Google https://www.google.com/trends/correlate/tutorial and tons of others promoting the erroneous interpretation ( i.e. using standard significance testing !) of time series correlations.

• Thanks for the reference material, I'll study it properly when I'm back in work on Monday. From the rest of what you're saying, am I to interpret it as the calculated correlation coefficients will be useless to me? As this seems contradictory to the comments left by Whuber above who seems to imply that it's trivially easy to do so. Commented Aug 10, 2018 at 20:00
• The distribution is affected by the autocorrelation in X and Y thus tests of statistical significance are affected. Ease of computation doesn't imply usability. Commented Aug 10, 2018 at 21:39
• An important clarification ....The distribution is affected by the auto-correlation in X and Y thus "standard" tests of statistical significance are affected. Ease of computation doesn't imply usability. Commented Aug 11, 2018 at 12:13