So, I have just started going through binary choice models, and while explaining probit models- they start with how by specifying alternate distributions of the error term- we can use different econometric models.

According to my textbook, if we assume the error term to be normally distributed, we use the probit model. However, given that we are dealing with binary models, isn't it impossible for the error term to be normally distributed? Since $Y_i$ can only take values 0 or 1, implying corresponding errors to take just two values?


In a probit model, the observed data is binary, and thus is certainly non-normal. However, the probit model can be described as having an underlying process that includes a normal error. Specifically, we let

$Z_i = X_i^T \beta + \epsilon_i$

However, $Z_i$ is not observed, but only an indicator of whether $Z_i > 0$. In otherwords, we observe

$Y_i = I\{ Z_i > 0 \}$

Note that this make $Y_i$ a binary response. So we consider $\epsilon_i$ to be the "error term" for the underlying process, but only observe the binary outcome $Y_i$.


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