Models for testing non-linear dynamics (e.g., threshold effects), where threshold is not necessarily known 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.

You could model this with probit or logistic regression. Then you model the probability
$$\mathbb{P}(Y_i = y) = \begin{cases} p & \qquad \text{if $y=1$}\\
1-p & \qquad \text{if $y=0$}
\end{cases}$$
This $p = \mathbb{P}(Y_i = 1)$ is equivalent to the probability $\mathbb{P}(T_i < X_i) = F_{T_i}(X_i)$. That is, the probability of observing $Y_i = 1$ is equal to the cumulative distribution function of $T_i$ in the point $X_i$.
When $T_i$ is normal distributed then you have a probit model. When $T_i$ follows a logistic distribution then you have a logistic model. (Personally I feel that a probit model is more natural here, and logistic regression is more appropriate in classification problems)
