Compute the inverse of a conditional quantile regression output

Question

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.

EDIT : Simplfying the question : How does 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 $$ a\left(\tau_{a}\right)=s-\bar{Q}\left( \tau_{a} \mid X, z \right) $$

[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 $Q_{w}\left(\tau_{u} \mid s, X, z\right)$ and $Q_{s}\left(\tau_{a} \mid X, z\right)$ 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 $$ \begin{array}{c} Q_{w}\left[\tau_{u} \mid Q_s\left(\tau_{a} \mid X, z\right), X, z\right]=Q_s\left(\tau_{a} \mid X, z\right) \Pi\left(\tau_{a}, \tau_{u}\right)+\gamma_{w} X+G_{a}^{-1}\left(\tau_{a}\right)+G_{u}^{-1}\left(\tau_{u}\right)& (8) \\ Q_{s}\left(\tau_{a} \mid X, z\right)=\gamma_{s} X+\pi z+\xi G_{a}^{-1}\left(\tau_{a}\right) & (9)\end{array} $$ Given the restrictions imposed by (6) and $(7),$ the key parameter of interest $\Pi\left(\tau_{a} \tau_{u}\right)$ is a matrix with the following structure $$ \begin{array}{c} \Pi\left(\tau_{a}, \tau_{u}\right)=\beta+\lambda G_{a}^{-1}\left(\tau_{a}\right)+\phi G_{u}^{-1}\left(\tau_{u}\right) \quad \end{array} $$

EDIT : Added some clarification to the equation numbering, fixed the equation in the first qoute and added more information on variable ycomp/z. Fixed the main quantile regression equation (8) from subscript v to w (wages). Clarified the importance-sampling tag. Edited the subscripts of the distribution functions $\tau_a$ and $\tau_u$

Answer 1

I think I found an answear to this on my own. Following Ma and Koenker one simply does the following :

For tau = 0.1 :

$G_{a}^{-1}\left(\tau_{a}\right)$ =

   ability_male_tau_01*qnorm(0.1,0,var(ability_male_t01))

$G_{u}^{-1}\left(\tau_{u}\right)$ =

   labour_luck_male_tau_01*qnorm(0.1,0,var(labour_luck_male_t01))

Maybe someone can verify this theoretically?