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?



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