Let's analyse the following binary choice model: $$Y^*_i = X_i^T \beta + \varepsilon_i\\ \varepsilon_i \sim F \\ Y_i = \mathbb{I} \{X_i^T \beta + \epsilon_i > 0\}$$ where $F$ is determined by a finite number of parameters $\gamma$. Define $\theta = (\beta^T, \gamma^T)^T$.

$$L(\theta) = \sum_{i = 1}^{N} Y_iF(X_i^T \beta) - (1 - Y_i)(1 - F(X_1^T \beta)$$ For instance, assume that I have to show that $Y_iF(X_i^T \beta) - (1 - Y_i)(1 - F(X_1^T \beta)$ is Lipschitz in $\theta$. Should I take into account that $Y_i$ is a function of $\beta$? When we do take into account and when we don't? For instance, when we differentiate w.r.t $\theta$ in order to calculate the score, when we don't take into account or do we? I would be very grateful is someone could tell me a general rule.


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