I'm developing a model in R estimating vehicle miles traveled (VMT) using Tobit specification, since my data include a cluster of zero VMT values.

I understand that the coefficients given by a Tobit regression relate to an uncensored latent variable; however, since VMT actually cannot dip below zero, I'm interested in effects on the censored variable. I've found that typically the marginal effects are used to discuss Tobit results in this case vs. the regression coefficients.

For predicting values, I know one can make adjustments to generate expected values of the censored latent variable (as outlined here). In R, the VGAM package also allows you to specify type.fitted ="censored" for Tobit; when then fed into the predict function, it provides estimates for the censored latent variable as well.

My question is, is using these censored-variable estimates practical in generating real-world predictions, or is shifting the predicted values to reflect a censored latent variable more of an academic exercise? Hopefully that makes some sense -- I just haven't seen Tobit models used predictively anywhere, and I'm trying to generate how shifts in my independent variables will actually impact VMT from predicted values.

Thank you! This is my first Stack Exchange question, so forgive any mistakes here (and obviously I'd happily incorporate any changes you might suggest to future questions).


1 Answer 1


Yes, it is quite common to use tobit (and related) models for predictive modeling of non-negative variables with a point-mass at zero. Often probabilistic forecasts are used, e.g., the probability for a zero outcome or certain quantiles (median, 90% quantile, etc.).

Whether there really is an underlying latent variable that is actually censored is not so important. The "trick" of using a zero-censored Gaussian distribution for the model to accomodate the point mass at zero also works in many situations where an underlying uncensored variable is less plausible. For example, in our own work we often use tobit models for probabilistic forecasting of precipitation.

The more crucial question is whether the probability for a zero outcome is driven by the same effects as the mean outcome when it is positive. This is a fundamental assumption in the tobit model but it can be relaxed, e.g., by using a two-part model (also known as Cragg model in the econometrics literature). This uses a binary response model (typically probit) for $y=0$ vs. $y > 0$ in the first part and a zero-truncated Gaussian model for $y | y > 0$ in the second part.

A worked example using our crch package for "censored regression with conditional heteroscedasticity" is available in:

Messner JW, Mayr GJ, Zeileis A (2016). "Heteroscedastic Censored and Truncated Regression with crch." The R Journal, 8(1), 173–181. doi:10.32614/RJ-2016-012.


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