Compute the inverse of a conditional quantile regression output

Short Clarification :

This question was asked at the Cross Validated SE (Question at CV) but one highlighted in the comments, that this might be more applicable to this SE due to its economic topic. Feel free to migrate the question (if possible) to this SE. On the other hand, there is a related question at CV SE which focuses just on the estimation technique. Feel free to go there if some things in this question need clarification, which I hope is not the case. (Question about Control Variate QR )

Please excuse me for this long introduction.

Brunello et al (2009) show that extended compulsory schooling leads to increased wages respectivly to the individual gender. Their empirical model first uses quantile regression to show the impact of compulsory schooling years (ycomp,
defined as the instrument variable z) on actual years of education (s). Afterwards they subtract those fitted values of the regression from s to get the ability of a person of a specific quantile.(2)

They claim that their model is exactly identified to do so.(3)

In the end they come up with a quantile regression aproach which is augmented by the control variate computed in (2). But if I understand them correctly they compute the inverse of the $tau$ – quantiles of distribution $a$ and $u$. (4)

If I got that right, could somenone help me to show how this is done? I suspect some kind of Monte-Carlo Method, e.g. importance sampling, but I’m unsure. A solution with R-code is appreciated but not necessary.

Simplfying the question : How do one calculate $G_{a}^{-1}left(tau_{a}right)$ and $G_{u}^{-1}left(tau_{u}right)$?

(2):First, we estimate the conditional quantile functions of schooling $s$
and compute the control variate $$
aleft(tau_{a}right)=s-bar{Q}left( tau_{a} mid X, z right) $$

(3): Omitting subscripts for simplicity, the earnings-cum-education model
presented above can be written in the format of an exactly identified
triangular model, as in Chesher’s approach $$ begin{array}{c} ln(w)=beta s+s(lambda a+phi u)+gamma_{w} X+a+u &(6) s=gamma_{s} X+pi
z+xi a &(7)end{array} $$

(4): Define $tau_{a}=G_{a}left(a_{tau_{a}}right) text { and } tau_{u}=G_{u}left(u_{tau_{u}}right)$, where $a_{tau_{a}}$ and
$u_{tau_{u}}$ are the $tau-$ quantiles of the distributions of $a$
and $u,$ respectively. Furthermore define $Q_{w}left(tau_{u} mid s,
X, zright)$ and $Q_{s}left(tau_{a} mid X, zright)$ as the
conditional quantile functions corresponding to log wages and years of
education. Ma and Koenker (2006) show that recursive conditioning
yields the following model $$ begin{array}{c} Q_{w}left[tau_{u}
mid Q_sleft(tau_{a} mid X, zright), X, zright]=Q_sleft(tau_{a}
mid X, zright) Pileft(tau_{a}, tau_{u}right)+gamma_{w}
X+G_{a}^{-1}left(tau_{a}right)+G_{u}^{-1}left(tau_{u}right)& (8)
Q_{s}left(tau_{a} mid X, zright)=gamma_{s} X+pi z+xi
G_{a}^{-1}left(tau_{a}right) & (9)end{array} $$ Given the restrictions
imposed by (6) and $(7),$ the key parameter of interest
$Pileft(tau_{a} tau_{u}right)$ is a matrix with the following
structure $$ begin{array}{c} Pileft(tau_{a},
tau_{u}right)=beta+lambda G_{a}^{-1}left(tau_{a}right)+phi
G_{u}^{-1}left(tau_{u}right) quad end{array} $$

Metatopic : I really think that this topic would benefit if there would be a specific quantile-regression tag. Since there is none and my reputation is too low to create a new one, I will go with just the regression tag.