# Conditional heteroscedasticity for the tobit model in Layman's terms

I have been using the crch package for modelling censored data with the tobit model. I noticed early on that the errors (by far) are not normally distributed by means of QQ-plot. Furthermore, I know from earlier posts on cross-validated that the Breusch-Pagan test was not option to test for heteroscedasticity. So I chose to employ two models (one with conditional heteroscedasticity, and the other without). After that I looked to see if there is a big difference in AIC to see if conditional heteroscedasticity is statistically present.

Now, I would like to fully understand what a conditional heteroscedasticity tobit model really is? Does it assume that if X (explonatory variables) increases, the scale (sigma) of heteroscedasticity increases as well? Does the crch model impose a linear scaling between X and sigma?

• Have you read the vignette for the package you are using? cran.r-project.org/web/packages/crch/vignettes/crch.pdf. The model is defined in that vignette answering all your questions. You can always return if there is something in the vignette you do not understand. Oct 15, 2019 at 11:26
• I have read the vignette, and it showed me perfectly well how to use the crch model and how the model is constructed. However, I could not grasp the clear intuition of what is meant with 'conditional heteroscedasicity' tobit modelling. Oct 15, 2019 at 11:38
• The way I read it - not being an expert - it simply refers to the model assumption that $g(\sigma) = \mathbf z^\top \gamma$ where $g$ is a strictly increasing and twice differentiable function; It follows that $g$ is invertible and the inverse function is also increasing so $\sigma = g^{-1}( \mathbf z^\top \gamma)$ so the variance is allowed to vary - hence heteroscedasticity - but only conditional on data $\mathbf z$ and some apriori specified link function $g^{-1}(;\gamma)$ known up to parameters $\gamma$, that I assume is estimated. Oct 15, 2019 at 15:15