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?

  • $\begingroup$ 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. $\endgroup$ Oct 15, 2019 at 11:26
  • $\begingroup$ 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. $\endgroup$ Oct 15, 2019 at 11:38
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    $\begingroup$ 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. $\endgroup$ Oct 15, 2019 at 15:15


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