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Are there ways in which $\text{Cov}(X,Y) = 1$ can hold and yet "something surprising" can still also be true?

For example: it is possible for the mean depth of a pool to be only $10 \text{ ft}$, yet small portions of the pool may nevertheless extend downward to any arbitrary length. In this sense, the mean statistic can be surprising.

Are there similar or analogous ways that perfect correlation can be surprising?

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    $\begingroup$ Since you write "perfect correlation" I assume $Cov$ is a typo and you mean $Corr$? $\endgroup$ Jun 10, 2016 at 1:41

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If you're referring to the actual correlation coefficient (i.e. calculated from population moments):

In a practical sense, no, but it depends what you mean by surprising! Let's assume:

  • $x$ and $y$ are random variables
  • For simplicity, that $E[x] = 0$ and $E[y] = 0$
  • $E[x^2]$ and $E[y^2]$ both exist and are greater than zero, and that $E[xy]$ exists. Hence correlation coefficient exists.

You might think that if $Corr(x,y)=1$ then $x$ and $y$ must be scalar multiples of each other. That's correct for a finite probability space, but not quite right in the more general case. Instead, you can show $\left(E[xy]\right)^2 = E[x^2]E[y^2]$, i.e. perfect correlation, if and only if there exists a scalar $\lambda$ such that: $$E[(x - \lambda y)^2] = 0$$

$x$ can be different from $\lambda y$ for some outcomes in the sample space, but the probability measure of those outcomes has to sum to zero.

To show this, observe the expectation operator defines an inner product, and the problem reduces to finding a condition for equality in the Cauchy-Schwarz inequality. See the comments of Dilip Sarwate here.

If the correlation coefficient is calculated from sample statistics:

If the correlation coefficient is an estimate, the estimate may not be correct. You may especially worry that rare events. Example:

  • In the data, a company's interest payments are perfectly correlated with the interest rates listed on the company's bonds. In the data, the correlation coefficient is one! But later, the company defaults! In actuality, the two weren't perfectly correlated, rather, the sample size was too small to observe the rare event of default.
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If you combine it with common sense (e.g. examining a plot) then it does mean something, but correlations' sensitivity to outliers make can make it deceiving in isolation, e.g. throwing a huge outlier into otherwise uncorrelated data will generate near perfect correlation. In R:

cor( c(rnorm(50),1e10), c(rnorm(50),1e10) ) 
[1] 1
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    $\begingroup$ Good point that high correlation between two series may be driven by just a few highly correlated outliers. It has to be fairly extreme though to make the correlation approach 1! $\endgroup$ Jun 11, 2016 at 17:44

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