# How can correlation be 0 in % terms but 0.5 when measured in dollars?

I am trying to see if there is a causal relationship between Marketing Spend and Revenue on a monthly basis for the Jan to July 2015 period.

I calculated the percentage change in Spend and the % change in Revenue and found the correlation to be 0.

However, when I calculated the correlation between spend and revenue (both in terms of dollars) the correlation is 0.5.

What does this mean?

Look at the autocorrelation of dollar series and % changes, you'll see a very similar picture for both series: % changes will have little autocorrelation and level series will be strongly correlated.

It's a feature of random walk like processes:$$x_t=c+x_{t-1}+\varepsilon_t$$

Autocorelation of changes is zero: $$cov[x_t-x_{t-1},x_{t-1}-x_{t-2}]=cov[\varepsilon_t,\varepsilon_{t-1}]=0$$

and the same for levels is not zero, it's growing with time: $$cov[x_t,x_{t-1}]=var[x_{t-1}]=var[x_0] +(t-1)\sigma^2_\varepsilon$$

So, you are probably having a classic case of spurious correlation in levels.

• Great, so could I use this a test than? For example, I have for cohort A, the correlation is 0 for % changes, but 0.5 for dollars. For cohort B however, the correlation is 0.26 for % changes and 0.72 for dollar changes. Can we say that cohort A's revenue has no correlation with spend, while cohort B, does? Which basis provides the best estimate of correlation, % Changes or Dollars? – Hidden Markov Model Aug 24 '15 at 20:35
• @HiddenMarkovModel, generally, you don't look for correlation of non-stationary series directly, like you did with levels. You at least difference them. However, that's not the only issue. In your case the relationship is probably two-way: spending impacts revenue, and also depends on revenue. So, you may want to look at cointegration. – Aksakal Aug 24 '15 at 20:46