Pearson correlation, rank correlation vs copulas

[ActPass]

Can anyone please explain what are the main differences among Pearson correlation, rank correlation and copulas, apart from Pearson is suitable for the variables with linear/Normal assumptions? When is suitable to use each?

Aren't they all about describing the correlation between variables?

 

[ActPass]

Thanks for the response.

My initial question actually arose from the situation where we are usually asked to explicitly use Pearson correlation, rank correlation or a certain form of copulas to perform some calculation.

What if we are given two set of data, say bond index and stock index, when would you use Pearson, rank and copulas? We are not given other information other than the two raw data.

And is it possible to relate one measure to another?

Any idea?

 

[David Wilmot]

If you want a summary statistic giving you some information about the dependency between the two data sets then Pearson's linear correlation coefficient or a measure of rank correlation meet that requirement.

The former only quantifies a particular form of dependency. namely linear dependency. A value of zero only implies independent if there is no non-linear dependency present. This would be true if the two are related by an elliptical dependency structure (e.g. Gaussian/Normal).

So, Pearson's correlation coefficient is limited in its application.

Rank correlation coefficients measure broader forms of dependency (i.e. non-linear). They are also a single summary statistic quantifying the degree of dependency - they give no information about the structure of that relationship.

If you want more information about the dependency structure then one way of describing that structure is by using a copula.

So, maybe we should be careful about using the word correlation? It is generally assumed that it means Pearson's linear correlation! It is perhaps safer to use the term dependency when meaning broader forms of inter-relationship.

 

[ActPass]

Thanks David. The explanation is helpful.

 

[Avi0013]

I believe Sweeting uses "concordance" to reflect dependance which may not be either linear or rank correlation, but I don't have the book in front of me now.

 

[SpeakLife!]

If you want a summary statistic giving you some information about the dependency between the two data sets then Pearson's linear correlation coefficient or a measure of rank correlation meet that requirement.

The former only quantifies a particular form of dependency. namely linear dependency. A value of zero only implies independent if there is no non-linear dependency present. This would be true if the two are related by an elliptical dependency structure (e.g. Gaussian/Normal).

So, Pearson's correlation coefficient is limited in its application.

Rank correlation coefficients measure broader forms of dependency (i.e. non-linear). They are also a single summary statistic quantifying the degree of dependency - they give no information about the structure of that relationship.

If you want more information about the dependency structure then one way of describing that structure is by using a copula.

So, maybe we should be careful about using the word correlation? It is generally assumed that it means Pearson's linear correlation! It is perhaps safer to use the term dependency when meaning broader forms of inter-relationship.

A response to a question written in true exam fashion. Very nice (and very helpful, too)!