Predicting when a proportion will reach a certain value Suppose I am trying to model women entering the field of computer science and would like to make a prediction about when the proportion of women in studying computer science in college will reach $.5$.
I have data spanning back several years about the number of women and men entering college for each particular year, and also have a list on the proportion of women in college for some particular year. In other words, 1st list is number of people entering school, and the 2nd list is the total proportion of women across all grade levels. From this yearly data, I would then like to make a prediction about how many years it will take for parity to occur.
What statistical model would work well for this? So far I've only been able to rule out tests that are clearly flawed

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*Linear regression obviously wouldn't be very helpful as we are dealing with proportions and won't want unbounded predictions as time progresses

*Logistic regression seems valid, but having it reach $1$ as time progresses is not desired.

*Beta Regression seemed feasible, but I want to make a forecast rather than a guess on the mean true proportion.

This answer seemed pretty feasible, but a bit hacky and I wanted to know if there was a well established standard way of doing this. It seems like it would be a pretty standard problem (similar with estimating vaccination rates) so I was hoping to gain some insight

The typical approach is to fit your model, but your problem is that you don't even have a model. So the question is more like a modelling problem than a statistics problem. This requires domain knowledge.



*Logistic regression seems valid, but having it reach 1 as time progresses is not desired.


Fitting a logistic growth curve doesn't need to end up at a fixed final level of 1. A problem is that the motivation for logistic growth may not be not very strong. The model relates to a differential equation $$f^\prime \, = \, \frac{k}{a} f(a-f) \, = \, kf - \frac{k}{a} f^2 \, \underbrace{\approx \,  k f}_{\text{if $f \ll a$}}$$
It can be seen as an exponential growth model where the growth rate is a second order polynomial instead of first order. For small values or large values the model will be approximately following an exponential curve.
If there is no direct motivation then the models might still fit and predict well for short time scales because growth is more or less linear on a small time scale anyway.
But does it matter in that case (a small order polynomial model fit) that we use a model for exponential growth?

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*On the one hand no: Either the extrapolation to 0.5 will only be a short time and a linear model will be fine as well, or the extrapolation will be a long time away and the logistic model may not be accurate. The latter situation will be the case when the growth rate stagnates close to a 0.5 value.


*On the other hand yes: The potential use of the logistic curve can be to fit a model where the value stagnates around 0.5 or something close, and the question revolves around the question when the random fluctuations will first create a situation where there is a year that more women than men enter.
