I have a doubt about the posterior distribution of the variance parameter for the Tobit model as provided by Koop, Poierier, Tobias (2007) in "Bayesian Econometrics Methods" page 221.
Posteriors for parameters in Tobit model are derived by using the auxiliary variable: $$y_z=Dz+(1-D)y $$ where $D=1$ when observations $y$ are censored and $z$ are latent data: $z_i=x_i\beta+\epsilon_i$ with $\epsilon_i\sim N(0,\sigma^2)$. Given the prior on variance $ IG(a,b)$, the posterior is directly provided as: $$ \sigma^{2}|\beta,z,y\sim IG\left[\frac{n}{2}+a,\:\left(\frac{\sum_{i}(y_{z_{i}}-x_{i}\beta)^{2}}{2}+b^{-1}\right)^{-1}\right] $$
When I try to derive it, I use the normality assumption on the latent variable and, because of the conjugacy with the inverse gamma prior for variance, its posterior turns out to be: $$ \sigma^{2}|\beta,z,y\sim IG\left[\frac{n}{2}+a,\:\frac{\sum_{i}(y_{z_{i}}-x_{i}\beta)^{2}}{2}+b\right] $$
Why should it instead be like the first one? What is the density for $y_z$ that they use?
I guess it is because I am neglecting somehow the censoring and considering variable $y_z$ as it were an uncensored one, so that my posterior for $\sigma^2$ has the same form of that for a regression with continuous dependent variable. My doubt is feeded by the fact that the posterior for $\beta$ they provide is actually the same as that for a regression with uncensored dependent variable, except for the use of $y_z$ instead of $y$.
Thanks