Pooled cross-section or very highly unbalanced panel data? Suppose I observe high school graduates applying to a certain college.
I want to model the probability to get admitted on the basis of individual characteristics (cognitive ability, test score, GPA, exchange experiences, interviewer's scores, etc.) with a logistic or a probit regression.
Suppose I observe 10 years of applications, i.e., 10 cohorts of applicants. In year 2010 some students apply to college, and in year 2011 some other students apply to college. This looks like a pooled cross-section (different individuals observed at different points in time).
But, some persons, like 0.5% of the total observed population, applied to college multiple times. For example, Mr. Joe Sixpack applied to college in 2010, didn't get admitted, and re-applied in 2011 (and still didn't get admitted).
Does this make my dataset a (highly) unbalanced panel dataset, rather than a pooled cross-section dataset? How to take into account autocorrelation?
I have an analogous objection from a referee for a paper I'm writing.
Related (without an answer).

This is not panel data, as a longitudinal panel design implies to follow the same individuals over time. For example, in An Overview of Longitudinal Research Designs in Social Sciences, Jyoti Bala says in the definition she gives of longitudinal panel design:

The researcher repeatedly measures the same variables in the same way
for the same set of cases at each wave of data collection.

Similarly, in Panel Studies, Heather Laurie says:

The key feature of panel studies is that they collect repeated
measures from the same sample at different points in time.

It doesn't fit the situation you describe, as the purpose of the study design you describe was not even to follow the same individuals over multiple years of applications.
This is a pooled cross-sectional design, as in this kind of study you can sample by accident the same individuals. To quote Applied Panel Data Analysis for Economic and Social Surveys by Andreß et al. (page 5), in a pooled cross-sectional design, "some rare cases" can "incidentally have been sampled in both years". It fits better the situation you describe.
As for the autocorrelation issue you mention, unfortunately I don't have an answer to offer, and you should probably ask a separate question about it to get more feedback. About that, an interesting question you could ask yourself may be: is the unit of analysis the individual or the application? Also, you could consider using a variable "applied multiple times" in your model, to take into account those multiple applicants. It might be informative relative to your research question.
