When dealing with response variables that have lots and lots of zeros, is there a clear argument for when hurdle models are preferred and when left censored or tobit models are preferred?
My understanding is that the best option is to try to reproduce the data-generating model as closely as possible. The data-generating model for a left censored tobit is the following: a random value is drawn from a Gaussian distribution, and if that value is less than zero, it is given a value of zero. This implies it's theoretically possible to get a score below zero, but some mechanism recodes such scores to zero (e.g., the measuring instrument doesn't go lower than zero). The data-generating model for a hurdle is the following: 0/1 value is draw from a Bernoulli distribution, and if that value is 0, the individual is given a value of 0. If the Bernoulli value is 1, a value is drawn from a zero-truncated (typically exponential or Poisson-like) distribution.
In most applications, I think a hurdle makes more sense. It's unlikely that a theoretical variable could take values less than 0 but is censored at 0, which is required for a tobit. Though a possible example might be some health score, where 0 is death (i.e., had a person not died, they could have been even worse off than before they died). Another example might be GPA, where (in the US) a zero means failing all classes, but it's possible to fail with a 20% grade in all classes or a 40% grade, and clearly the 20% grade demonstrates less proficiency, but the measurement scale simply stops at zero and gives both grades the same value of zero. When discussing more natural scales likes counts or amounts, though, typically a hurdle makes more sense.