Can we still use OLS on truncated a $Y$ if its conditional distribution is normal? I was recently reading about Heckman selection models, and got sidetracked by how little I knew about truncated data. I was reading these slides, and on page 78 Baum mentions that if part of a sample has a truncated $Y$ variable, we can’t even use regression analysis on that $Y$ to make inferences about the sub-sample (whose $Y$ variable is truncated).
I know that having a truncated $Y$ means the distribution will be skewed, but if the conditional distribution of $Y$ is normal, can we still use basic OLS? Also, does a tobit model work on the assumption that the conditional distribution is truncated?
 A: It's not about the distribution being skewed (in the statistical sense of having a non-zero third moment). It is about the conditional mean (given $X$) being shifted by truncation. So no, OLS is not making sense. You can still use it, but the estimates will be biased. No statistical software would spit out a diagnostic message that your sample was truncated, and that's something like metadata that comes along with the data themselves that you are bringing into data analysis.
A tobit model is one way to take truncation into account. It tends to be overly sensitive to the assumption of underlying normality, and breaks for whatever mild violations that linear regression can withstand, like heteroskedasticity. So it is a step in the right direction, but it rarely is a step that's good enough.
Motivation for Heckman comes from a counterfactual experiment of what a person's wage would be should they participate in the labor market -- what would a market pay if a housewife did not have that independently wealthy husband and have to work instead. From a finite population perspective, the distribution of observed wages is what it is -- skewed or not, lognormal, uniform, whatever -- a set of numbers. It is only truncated in the modeling sense that you don't observe the wage for everybody who is of working age and of good health (unless you are in a Communist country where everybody has to work or would go to jail if they don't... but the wage does not make an economic sense in such countries, anyway). So for Heckman model, the truncation problem is rather a missing data problem. It is mentioned as such in some missing data books written by pure statisticians who otherwise don't look very often into econometrics.
