Limited dependent variables arise when some minimum threshold value must be reached before the values of the dependent variable are observed and/or when some maximum threshold value restricts the observed values of the dependent variable.
A limited dependent variable causes the standard model to become
where restricted values don’t allow you to always observe Y*. Specifically, you observe
if the dependent variable is limited by a lower threshold and/or
if the dependent variable is limited by an upper threshold. Because the ordinary least squares (OLS) technique estimates the model without accounting for the missing data or the values that are at the threshold (rather than their actual values), the resulting estimated coefficients are biased.
Situations where the dependent variable is discrete (meaning it has a finite number of possible outcomes) or where measurement of the dependent variable takes place while the process is still ongoing (like the amount of time unemployed) are also problematic for OLS estimation.
A number of techniques (multinomial probit, multinomial logit, ordered probit, ordered logit, Poisson, negative binomial, and duration models) can be used for these scenarios, but treatment of these topics is usually reserved for advanced or graduate-level econometrics courses.
A limited dependent variable results in either a censored sample or a truncated sample. In other words, censored and truncated dependent variables are the two types of specific limited dependent variables you’ll encounter.