If you want to model this in stan, then y is a random variable on equal footing with your other variables and you should define a probability distribution over it (the likelihood), usually conditioned on the remaining variables. I imagine your full joint distribution looks like

p(y,\beta_0,\beta_1|x)=p(y|\beta_0,\beta_1,x)p(\beta_0)p(\beta_1)

so the above transformation was more correctly for the conditional distribution p(y|\beta_0,\beta_1,x) and so does not require a Jacobian term adjustment for the betas.

I am not quite familiar with the model you are pursuing, but I think I get the gist.

In short, your original model specification is sufficient, but you might want to add priors to the betas. I think you can safely use priors designed for general regression problems.

```
beta0 ~ normal(0,1);
beta1 ~ normal(0,1);
erro_y~normal(0,sigy);
```

p.s. you can use $âŚ$ for latex typesetting and `âŚ` for code. You can also post blocks of code using ```âŚ```