# Probit and logit Model

Can you provide me with an intuition about what it means to derive a probit or logit model from an underlying latent variable model?

Logit and probit models can be derived from an underlying latent variable model. To see this, let $$y^{\ast}$$ be an unobserved (i.e. latent) variable, and suppose that $$$$y^{\ast} = \beta_0 + x\beta + \epsilon, ~y=1[y^{\ast}>\gamma] \label{latent}$$$$ where $$1[\cdot]$$ is an indicator function taking value $$1$$ if $$y^{\ast}>\gamma$$ and zero otherwise. Here, $$\gamma$$ is a chosen threshold. If the latent unobserved utility $$y^\ast$$ is above the threshold, then $$y$$ takes the value one; otherwise, it takes the value zero.
The idea is to see the binary outcome $$y=\{0,1 \}$$ as a dichotomization of a latent continuous variable $$y^{\ast}$$ modeled with a canonical multiple linear regression model. For example, a bank can issue a loan ($$y=1$$) or choose not to ($$y=0$$). Think of it as having an underlying continuous latent outcome, $$y^{\ast}$$ which can be regarded as a measure of the choice utility.
In the model above, we assume that $$\epsilon$$ is independent of $$x$$ and that $$\epsilon$$ either has the standard logistic distribution or the standard normal distribution.
Recall, the standard logistic and standard normal distribution are symmetric about zero. Thus, $$1-G(-z)=G(z)$$.
From the model above, we can derive the response probability for $$y$$. First, we assumed that $$y_i=1$$ if and only if $$y^{\ast}_i > \gamma$$, with $$\gamma$$ set arbitrarily to zero. Thus, \begin{align*} \mathbb{P}(y=1|x) =& \mathbb{P}(y^{\ast} >0|x) = \mathbb{P}(\beta_0 + x\beta + \epsilon >0|x) = \\ =&\mathbb{P}[ \epsilon > -(\beta_0 + x\beta ) |x] =1 - \mathbb{P}[ \epsilon \le -(\beta_0 + x\beta ) |x] \\ =& 1 - G[ - (\beta_0 + x\beta)] = G(\beta_0 + x \beta) \end{align*} where $$\mathbb{P}[ \epsilon \le -(\beta_0 + x\beta ) |x]$$ is the cdf of the error term of the latent outcome.