Solved – Tobit for corner solution models: normality and homoskedasticity tests

hypothesis testingrtobit-regression

I would like to regress several explanatory variables (called X) on a dependent variable called Y where Y is strictly positive and continuous. It is left-censored at 0. Y represents how much a household is willing to pay to protect the environment in a specific area. It is cross-section data.

I asked R to run a Tobit I model using censReg package or AER package:

tobit <- tobit(Y ~ X, left=0, data = mydata)


tobit2 <- censReg(Y ~ X, left=0, data = mydata)

and then

plot(fitted(tobit), residuals(tobit))

When observing the residuals' plot, I see the distribution is not normal whereas it's necessary that the errors from this regression be normally distributed and homoskedastic.

Does anyone know how I can test these errors for normality and heteroskedastcity in R with the specific censored data issue? Using bptest() from the lmtest package – the Breuch Pagan test- does not work. I know that in some programs, like Stata, it's pretty straightforward to test residuals, but I have no idea how in R.

Thank you!


Best Answer

The residuals from such tobit models will often look non-normal (typically right-skewed) due to the censoring. This makes it difficult to apply standard classic techiques such as graphics or diagnostic tests for linear regression models (like the Breusch-Pagan test).

One thing you can do, though, is to fit a heteroskedastic tobit model where the dependence of the variance on some covariates is included. And instead of using a latent normal distribution you could also employ a distribution with heavier tails such as the logistc or t distribution. All these approaches are available in the R package crch (for censored regression with conditional heteroskedasticity). So you can fit models, say with constant variance and with heteroskedasiticy, and then compare these models by means of information criteria (AIC, BIC, ...) or likelihood ratio tests etc.

Of course, there are further sources of misspecification problems, e.g., omitted regressors, misspecified functional form etc.