# Backtransform a log-odds transformation of the dependent variable in a Fixed Effects panel regression

I'm modeling a fixed effects (within) panel regression for a fractional dependent variable (DV) bounded between 0 and 1. My aim is to model the relationship so that I can predict the frational DV on its original scale.

The relationship between a fractional DV and independent variables (IV) is most likely of a non-linear nature. A conventional within regression applied to a non-linear relationship is rarely feasible due to the incidental parameter problem.

If I understood correctly, there are two common modeling options for such a case.

1. Parametric approach: Correlated random effects (CRE) model with a non-linear link function (logit, probit)
2. Nonparametric approach: Make a log-odds transformation on the fractional DV, run a within regression and then retransform the fractional DV with the smearing estimate method described by Duan (Smearing Estimate: A Nonparametric Retransformation, 1983).

At the moment I'm going for option 2. The problem is, I'm confused with a small detail regarding the smearing estimate when applying it to a panel regression. Duan describes the smearing estimate as the following:

$$\hat{E}\space Y_0 = \frac{1}n\sum_i^n\space h(x_0\hat{\beta}\space+\space\hat{\epsilon}_i)$$,

where $$\hat{E}\space Y_0$$ is the expectation or prediction of the DV (Duan calls it the expecation of the "individual's response"), $$h()$$ is the inverse of the transformation function (in my case: the inverse of the log-odds transformation), $$x_0\hat{\beta}$$ is the prediction with the explanatory variables $$x_0$$ of the "individual" (again, as Duan calls it) and $$\hat{\epsilon}_i$$ are the residual errors.

Expressed in words, the smearing estimate estimates the unknown error distribution by the empirical cummulative distribution function of the estimated regression residuals $$\hat{\epsilon}_i$$; this is partly done by taking the sum of the residuals (coming from the log-odds transformed scale model estimate) and dividing it by the total number of residuals ($$\frac{1}n$$). And here comes my question:

Applied to a panel regression and given that I want to predict the outcome of the DV for a specific cross-section: Which are the residuals I need to sum up? Do I need to consider all residuals of the whole panel? Or do I only have to consider the residuals of the specific cross-section estimates? In other words, what are the $$i=1,...,n\space$$ for $$\space\sum_i^n$$ in the above equation?