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I am writing a paper on comparing different business cycles and have calculated bilateral correlation coefficients between individual countries and the overall aggregate. I have read in some of the academic literature that the Fisher transformation is necessary in order to enable a comparison of the means. If someone could explain, if I were to change my cc's with the Fisher transformation, what I would be doing to them and if it would help with comparison?

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The Fisher transformation https://en.wikipedia.org/wiki/Fisher_transformation of an estimated correlation coefficient $r$ is $$ z= \frac12 \ln\left(\frac{1+r}{1-r}\right). $$ It is an approximate variance-stabilizing transform, so that its variance which is about $\frac{1}{N-3}$, where $N$ is the sample size, does not depend on the true underlying value of the correlation coefficient. This can be used to construct a confidence interval for the correlation coefficient $\rho$.

A modern alternative would be to use the bootstrap. One of the advantages of the bootstrap, according to Efron, is that it can "find" a variance-stabilizing transform like that above "automatically".

To construct the confidence interval, use the approximation, for sufficiently large $N$, that $$ Z \stackrel{\text{a}}{\sim} \text{N}\left(\frac12\ln\left(\frac{1+\rho}{1-\rho}\right),\frac{1}{N-3}\right), $$ to find a confidence interval (on the $z$ scale) of the form $(z-q\frac1{\sqrt{N-3}},z+q\frac1{\sqrt{N-3}})$ where $q$ is the appropriate normal quantile, and invert it by using the inverse function $g$ of the Fisher transform, $$ g(z)=\frac{e^{2z}-1}{e^{2z}+1}, $$ thus obtaining the confidence interval for the correlation coefficient.

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    $\begingroup$ See stata-journal.com/sjpdf.html?articlenum=pr0041 for a Stata-flavoured tutorial at greater length, which includes comparison with the bootstrap. People using other software should find it fairly easy to hum or skip their way through the Stata content or translate into their own favourite language. (Detail: A gap at the time that Stata lacked sinh() and cosh() functions has long since been filled.) $\endgroup$
    – Nick Cox
    Apr 7, 2017 at 11:32
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    $\begingroup$ (-1) Doesn't explain "in layman's terms." $\endgroup$
    – rolando2
    Apr 7, 2017 at 11:42

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