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I am trying to estimate the quantiles of the function $f(x_i)$ in the equation:

$$ y_{it} = \alpha_i + f(x1_{it}, x2_{it}) + \epsilon_{it} $$

My current, probably naive, approach is to run the linear regression $y_{it} = \alpha_i + X_i\beta+ \epsilon_{it}$, calculate the predicted value $X_i\beta$ and its empirical CDF. I am then bootstrapping to estimate the confidence intervals of the various quantiles.

I also considered using a hierarchical model and assuming that that $f(x1_it, x2_it)$ follows a specific distribution but this seemed like almost assuming away the problem.

Any suggestions appreciated. Also, as this is my first post, suggestions on how to better frame the question also appreciated.

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