I need some clarification on where I can expect a closed-form solution. My research so far suggests the following:
- Logit model for binary dependent variables: no closed-form solution
- Probit model for binary dependent variables: no closed-form solution
Source: Ben Lamberts video on the topic https://www.youtube.com/watch?v=WflqTUOvdik. (Though he says in the video that the solution is typically "analytical", I guess from the context, that he actually meant "numerically")
- PDF of logistic distribution as likelihood function for continuous dependent variables: no closed-form solution
- PDF of the normal distribution as likelihood function for continuous dependent variables: $\hat{\beta} = (X'X)^{-1} X'Y$ (thus provides us with the same estimator as OLS, if the error term is normally distributed)
Can you verify that? In particular, I am interested in whether it is correct that the logistic model has no solution for the continuous dependent variables. Could that change if I assume some more specific assumptions, like in the case of the normally distributed model (error term normally distributed)?
