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In probit models the latent variable is assumed to be of the form: $y=\alpha+\beta x+\epsilon$ with $\epsilon \sim N(0,1)$.

What if instead $\epsilon \sim N(0,\sigma^2)$?

Is there a way to estimate $\sigma^2$? Does it make a big difference to the model?

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Assuming that $\epsilon \sim \mathcal{N}(0,\sigma^2)$ is the same with $Y*:=y=\alpha+\beta x+\sigma\epsilon$ where $\epsilon \sim \mathcal{N}(0,1)$

Then $\mathbb{P}[Y=1|X] = \mathbb{P}[Y^*>0] = \mathbb{P}[\epsilon<\tfrac{1}{\sigma}(\alpha+\beta x)] = \Phi\left(\frac{\alpha+\beta x}{\sigma}\right) = \Phi\left(\tfrac{\alpha}{\sigma}+\tfrac{\beta}{\sigma}x\right) = \Phi\left(\tilde{\alpha}+\tilde{\beta}x\right)$.

The only thing that change is the parameters. Then if you suppose that the innovations follow a non standard normal distribution, only your parameters will change.

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A model with $\sigma^2$ as a parameter to be estimated is not identified. We sent $\sigma^2=1$ simply to identify the model. In reality, we are estimating $\frac{\beta_k}{\sigma}$ for each $k$. See Breen, Karlson, & Holm (2018) for more details.

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