# How to interpret difference in correlation coefficient between 'absolute value' correl and 'returns' correl?

Been really bewildered by this situation - I'm running a correlation between 'A' and 'B'; and 'C' and 'D'. I've encountered 2 scenarios which I can't really explain:

1) High correlation between absolute values of 'A' and 'B' but low correlation between 'change in A' and 'change in B'.

2) Negative correlation between absolute values of 'C' and 'D', but relatively strong positive correlation between 'change in C' and 'change in D'.

AFAIK - Scenario (1) implies that the deviations from the mean absolute value move in the same direction, but deviations from the mean changes don't move together as well. But what does this truly mean? How can I interpret this in a lay-man's sense?

No real idea about scenario (2)...

Edit: Some details to make the question clearer:

A and B are both represent time-series data (monthly) for 36 periods (3 years).

A = 100, 120, 140,...., 280, 300.

B = 10, 20, 30,...., 90, 110.

The correlation between the absolute values is relatively high. But, it may not always be accurate to correlate absolute values, so I want to understand whether % change in A and % change in B are correlated (I remember while working with stocks, we were always told to use returns rather than prices).

Change in A (only 35 data points) = 20%, 16.67%,..., 7.1%

Change in B (only 35 data points) = 100%, 50%,..., 22%.

Satterplot to visualise - https://imgur.com/a/W0apDm1

• Can you plot the scatter plot between pairs of variables? And what is 'Change in A'? – Simone Apr 26 '18 at 6:44
• Hi, thanks for your answer. By 'change in A' I mean % difference. So in Period 1 if A=100 and in Period 2, A=110, 'change in 'A' would be 10%. Similar to a correlation between prices and returns, I guess. RE: Scatterplot - of all the absolute data or of the changes? – NPJobs Apr 26 '18 at 6:49
• So each variable is a time series? Regarding the scatter plot, eg for 1 I would plot both a scatter plot between A and B and one between their change. The Pearson correlation is just a summary value of a scatter plot and it does not tell you everything. Maybe when you see the scatter plot you see the correlation is not that high. What are the numbers though? – Simone Apr 26 '18 at 7:05
• (1) could be typical if A and B are unrelated random walks. Correlation between absolute values would be ill-defined there. – Richard Hardy Apr 26 '18 at 7:32
• @Simone - Here is a scatterplot of both absolute values and returns - imgur.com/a/W0apDm1. Yes, each variable is a time-series in essence. – NPJobs Apr 26 '18 at 7:49

It seems to me that the time-series are loading on a common trend somehow, and this is generating this behavior in the correlations.

If your time series A and B share a common trend, their absolute values will be highly correlated (decreasing with the variance of the individual series' noise term). However, once you differentiate the series, the only correlation that remains is in the noise term, which will be low if the noise terms are iid. This should expain (1). A simple code in R shows what I mean:

x <- seq(1,10,length.out=1000) + rnorm(1000)

y <- seq(1,10,length.out=1000) + rnorm(1000)

cor(x,y) # 0.8664099

dx <- diff(x)

dy <- diff(y)

cor(dx,dy) #-0.005354925


(2) is a little harder (and maybe farfetched, but I dont know your data and all I am saying is that this is a plausible explanation - its likelihood depends on your data) If D,C load on a trend with opposing signs but they load on a correlated noise term with the same sign, then this pattern could be observed as well. A more realistic pattern would be if they would each have a random iid noise term. This would generate high correlations in returns - due to the correlated noise term and low correlation in absolute value - due to the trend with opposing signs. More code to exemplify:

rnd <- rnorm(1000)

z <- -seq(1,10,length.out=1000) + rnd

w <-  seq(1,10,length.out=1000) + 0.5*rnd + rnorm(1000)

cor(z,w) # -0.8968412

dw <- diff(w)

dz <- diff(z)

cor(dw,dz) # 0.4667939


You asked for an explanation - I gave an example of how it can happen. Not sure if it answers your question, but I hope it helps you pinning down what may be the case.

• Thanks a lot for your reply! Yes, I understand your point regarding scenario 1 - from a purely technical perspective. But I'm still unable to condense it down into an intuitive understanding. Can it be inferred that A and B do not impact each other but rather there is a common trend/another variable responsible for their movement in the same direction? – NPJobs Apr 26 '18 at 10:38
• Yes, something like that! correlation measures the degree of (linear) co-movement between series. If the series seem to move together on levels but on returns they seem uncorrelated, it may be the case that there is something that moves both series on the levels, but does not affect their returns - i.e a trend or a variable that affects only their mutual trend. Alternatively, this could be evidence of spurious correlation. – user30978 Apr 26 '18 at 10:58