Models for binary response that appear like logit/probit regression but never approach 1 I'm reading the textbook "Introduction to Econometrics" by Stock and Watson. Chapter 11 discusses regression with a binary response and it teaches the logit/probit models mostly. These seem to make sense when the plots show an S shaped relationship between X and Y, but what if I'm working with a dataset where the model calculates a probability in the shape of a "hill".
For example what if my continuous predictor X ranges from 0-100 and my response Y is only equal to 1 for X's at values of 30-70 but gets more concentrated at X's around 50. So lets say around X=50 we could have a 40% probability of Y=1, and then at X=40, X=60 we'd have 25% probability of Y=1 until the probability of Y=1 dissipates around 30-70. I cant find any models that model such a relationship through google - Are there any models that are more appropriate than logit/probit for this type of relationship?

What you are a describing suggests that there is not a complete relationship between X & Y. If only 40% of all X's equal to 50 associate with Y=1, then there is some unexplained variance in Y not explained by X.
That means this simple model is not enough, there is some other variable (or variables) needed to properly predict Y.
