What is a "log-F" distribution? Recently I have encountered the Wiener-Granger causality test. The statistic which is used is the logarithm of the ratio of two residual variances. It is well known that the ratio of two such (independent) Chi-squared random variables is F - distributed with parameters df1 (nominator) and df2 (denominator). What is the distribution of the log of a random variable which is F - distributed? Any reference would be highly appreciated.
 A: It is a Type IV Generalized Logistic distribution.

Begin with the pdf of an $F(\nu_1, \nu_2)$ distribution ($\nu_1$ is "df1" and $\nu_2$ is "df2"), written $f(x)$.  The pdf for the logarithm $y = \log(x)$ will be the coefficient of $dy$ in
$$f(\exp(y))\, |\mathrm d \exp(y)| =\frac{1}{B\left(\frac{\nu _1}{2},\frac{\nu _2}{2}\right)} \left(\frac{\nu_1}{\nu_2 + \nu_1 \exp(y)}\right)^{\nu_1/2}  \left(\frac{\nu_2}{\nu_2 + \nu_1 \exp(y)}\right)^{\nu_2/2} \mathrm d y.$$
Letting $\mu = \log(\nu_1) - \log(\nu_2)$ exhibits the pdf in the form
$$\frac{1}{B\left(\frac{\nu _1}{2},\frac{\nu _2}{2}\right)}\frac{\exp(-\nu_2(y-\mu))}{\left(1 + \exp(-(y-\mu))\right)^{(\nu_1+\nu_2)/2}} $$
which (upon setting $\alpha = \nu_1/2$ and $\beta=\nu_2/2$) is recognizable as being derived from
$$\frac{\exp(-\beta y)}{\left(1 + \exp(-y)\right)^{\alpha+\beta}}$$
via a shift of location to $\mu = 2\log(\alpha) - 2\log(\beta)$ and normalization to unit probability.

Incidentally, if we let $l(y) = 1/(1 + \exp(-y))$ be the logistic transformation (whose graph is a "sigmoid" that maps the real numbers to the interval $(0,1)$), this PDF can be presented in a more symmetric form 
$$l(y)^\alpha \left(1 - l(y)\right)^\beta$$
which is reminiscent of the Beta distribution (and will be converted into it via $l$).  Thus $\alpha$ and $\beta$ control the mean and variance of the distribution in a familiar way.
