Inverse Gamma Posterior variance derivation in Tobit model

Question

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

Answer 1

I found the explanation. It's simply a matter of parameterization of the inverse gamma distribution. Once aknowledged the parameterization, my posterior for $\sigma^2$ and the one provided by the authors are the same.

The distribution of $y_z$ has to be normal, following the normality on $z$. As a matter of fact, the introduction of the latent variable enables us to "augment" the data and therefore to "ignore" the censoring of the dependent variable. As consequence, both the posteriors of $\beta$ and $\sigma^2$ need to be derived using such assumption.

The parameterization of IG that leads to my posterior is the one in terms of shape $\alpha$ and scale $\beta$ with pdf proportional to: $$x^{-\alpha-1}\exp\left\{ -\frac{\beta}{x}\right\}$$ The parameterization of IG used by the authors leads to a pdf proportional to the following: $$x^{-\alpha-1}\exp\left\{ -\frac{1}{x\beta}\right\}$$ See also the appendix of the textbook I quoted in the question. A given parameterization of the prior leads to a posterior expressed under the same parameterization.