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Wooldridge Introductory Econometrics: A Modern Approach (2018), pages 561 and 572, gives the following definitions:

Latent variable model (LVM): $$ y^*=\beta_0+\mathbf{x} \boldsymbol{\beta}+e, y=1\left[y^*>0\right] $$ where the indicator function takes the value 1 and if the event in the brackets is true and 0 otherwise. So $y$ is 1 if $y^* > 0$.

Tobit: $$ y^*=\beta_0+\mathbf{x} \boldsymbol{\beta}+u, u \mid \mathbf{x} \sim \operatorname{Normal}\left(0, \sigma^2\right) \\ y=\max \left(0, y^*\right) $$

where $y=\max \left(0, y^*\right)$ implies that the observed variable $y$ equals $y^*$ when $y^* \geq 0$, $0$ otherwise.

Questions:

So it seems to me that one difference is that $y^*$ in the LVM takes the value 0 or, while in the Tobit model $y^*$ takes the values 0 or any positive values otherwise. Do I have that right?

Are there any other key takeaways w.r.t to the differences between the models?

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1 Answer 1

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Your understanding is correct.

The latent-variable representation is convenient because you can express various models in this form.

The LVM in your question corresponds to a probit model. If the errors come from a standard logistic (instead of standard normal) distribution, you get a logistic regression model.

If instead of a single threshold at 0 you have multiple cutpoints at $\zeta_1<\zeta_2<\zeta_3<\ldots$, you get an ordinal probit model (or ordinal logistic if you have logistic errors).

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