I am working with Recentered Influence Functions (RIF) to estimate regressions in distribution. We have the following regression
$RIF(F_y, \nu (F_y)) = \beta_0 + \beta_1X + \varepsilon$
where $\nu(F_y)$ is some functional
$\begin{array}{r}\operatorname{RIF}\left(y ; \nu, F_Y\right)= \\ \nu\left(F_Y\right)+\int I F\left(y ; \nu, F_Y\right) \cdot d F_Y(y) \\ \nu\left(F_Y\right)+\operatorname{IF}(y ; \nu)\end{array}$
In the paper by Firpo, Fortin, and Lemieux (2009), they derive the expectation property of the RIF and its interpretation. But I would like to know how the variance of the RIF is interpreted in the context of regression. The goal is to understand the explained variance of some predictors with respect to the variance of the RIF.
This problem is tangentially addressed in the following question, but focused on the interpretation of the $\beta$ in the context of Influence Functions (IF). Link
Any help is appreciated!
