Any recommendations for models (or applications in papers) to test theoretical predictions about non-continuous effects, e.g., based on a threshold, rather than marginal effects? Some examples of the sort of threshold theoretical predictions I'm interested in testing:

The simplest example would be where the threshold is known. e.g.,

\begin{equation}
Y\_i=1 \ \text{if} \ X\_i>0.5 \ \text{and}\ Y\_i=0 \ \text{otherwise} \ \ \ \ \ \ \ \text{(1)}
\end{equation}

A more useful example would be where we theorize a threshold, but we don't know its value, or even assume that it's constant across individuals.

\begin{equation}
Y\_i=1 \ \text{if} \ X\_i>T\_i \ \text{and}\ Y\_i=0 \ \text{otherwise} \ \ \ \ \ \ \ \text{(2)}
\end{equation}
Where $T\_i$ is a threshold that is allowed to differ across units indexed in $i$.

I don't just want to test whether $X$ has a positive effect on $Y$ (e.g., $Y$~$X$), but rather I want to test for evidence for the threshold prediction.