In our analysis of regression models thus far, we have employed models that implicitly assume that the dependent variable is continuous, rather than discrete, and complete, rather than restricted. But many variables that historians are interested in studying do not share these characteristics. If the historian wishes to analyse why a state voted Republican or Democrat in a presidential election, or why some households decided to invest in the stock market in 1929 and others did not, or which country a family decided to migrate to, she is dealing in every case with responses that can take on only a small number of possible values. Similarly, other variables are limited in the range of values they can take – a labour historian investigating individual variation in the length of the working year in weeks will find many more values in her data set, but none above 52 or less than 0.

Such variables are known as limited dependent variables. Dependent variables that can take on only a small number of discrete values (such as 0, 1, 2) are known as qualitative dependent variables; those that are continuous, but restricted in their range (e.g. never becoming negative), are known as censored variables. Because of the restriction on the value of the dependent variable, these and other such questions involve different estimating strategies and require different regression techniques from the standard OLS approach.