How many parameters should I report in an AIC table for a tobit model?

Question

I am considering using a tobit model in R to estimate velocity of a number of animals moving north across a landscape. I am working with simulated data, so when an animal goes "south" or "sideways" on the simulated landscape, we can't measure it, and the velocity is given as zero (I cannot alter this behavior at the current time). Because my data are continuous, I am choosing to use a tobit model for left-censored data. The animals are in two types of landscapes, forested and agricultural.

I am testing several models and want to include an AIC model selection table in my results, including k, the number of model parameters.

For a linear model, e.g.,

lm(velocity ~ landscape)

there would be 3 parameters: the intercept, the agricultural landscape (vs forested), and the error.

However, consider the tobit model, e.g.,

library(crch)
crch(velocity ~ landscape, left = 0)

In a tobit model, how many parameters would there be? I am not sure whether the latent variable counts as "another parameter" here.

Answer 1

The tobit model you are estimating also has three parameters: intercept, slope (for landscape), and the latent error (log-)variance (of the underlying uncensored normal distribution). The crch package by default estimates the log-standard-error rather than the variance to assure positivity. But other link functions are also available, namely the identity (i.e., standard error) and the quadratic link (i.e., variance). Moreover, you may add regressors to the scale equation to obtain heteroscedastic tobit models.

In all cases the summary() and logLik() methods report the number of estimated parameters as the degrees of freedom. And consequently these are used by the AIC() and BIC() methods.