Stata tip 135: Leaps and bounds

A simple way of adding a variable nonlinearly to a model is to transform that variable. Common transformations are adding a quadratic term or taking a logarithm, but other transformations are also possible, such as taking the cube root (Cox 2011) or adding splines (see [R] mkspline). The purpose of this tip is to discuss yet another underused alternative transformation: the combination of continuous variables and indicator (dummy) variables.

Sometimes, a continuous variable consists of qualitatively different segments. A good example of such a variable is the number of hours a respondent usually works per week. In many countries, numbers less than 40 on such a variable represent respondents who work part-time, the number 40 represents respondents who work full-time, and numbers above 40 represent respondents who routinely work overtime. Using nlsw88.dta, which comes with Stata, we could analyze how one’s number of hours worked per week influences one’s average hourly wage, that is, the total earnings in a week divided by the hours worked that week. If we just add hours linearly, then we would conclude that an extra hour working is related to a four-cent increase in average hourly wage.

Figure 1.

Linear effect of hours worked per week

However, we might hypothesize that working “normal” hours makes it easier for companies to standardize the allocation of tasks to the workers. As a consequence, companies might be willing to pay a premium for working full-time. This means that working more hours may increase average hourly wage, but there is an extra “jump” at 40. To test that, we can add both the variable hours and an additional indicator variable for full-time workers to our model. Cox and Schechter (2019) wrote a useful tutorial on how to effectively create indicator variables. In this model, an extra hour working is still associated with a 4-cent increase in average hourly wage, but those working full-time get a 35-cent “bonus”. In this case, the indicator variable introduced a single spike at 40 hours worked per week.

Figure 2.

Linear effect of hours worked per week with a jump at working full-time

Sometimes, overtime is paid at a higher rate. So we might expect that working more hours generally increases the average hourly wage, but after 40 hours there is an extra jump that does not immediately disappear like before but persists. To test that, we can introduce the variable hours and an indicator variable for those respondents that routinely work overtime to the model. However, the results show that working overtime leads to a persistent (nonsignificant) 11-cent decrease in average hourly wage.

Figure 3.

Linear effect of hours worked per week with a persistent jump for overtime

We forgot that not everybody gets his or her overtime paid. For those who get paid for working overtime, overtime will increase their average hourly wage. However, for those who are not paid for overtime, overtime will decrease their average hourly wage. We might expect that unpaid overtime happens in professions where people are intrinsically motivated (for example, academics), so they may work long hours. Whereas paid overtime happens in occupations where people are less intrinsically motivated, in which case both the workers and the employers have an incentive to keep the amount of overtime within bounds. So we hypothesize that the group of respondents working small amounts of overtime mainly consists of people getting paid overtime, while the group of respondents working large amounts of overtime consists mainly of people who do not get (completely) paid for overtime. In that case, we would expect a sharp increase in average hourly wage at 41 hours per week but a decrease after that. This is implemented by including an interaction between the overtime indicator variable and the hours variable. In this case, it makes sense to center the hours variable at 41; that way, the effect of the overtime indicator variable can be interpreted as the jump that occurs when one starts to work overtime. In this model, working an extra hour increases the average hourly wage by six cents if one works part time. If one starts working overtime, there is an immediate bonus of 1 dollar and 9 cents, but every extra hour decreases the average hourly wage by 11 cents (6 − 17 = −11). This type of regression is sometimes called segmented, broken-stick, or piecewise regression. This type of model is also closely related to a regression discontinuity design (Calonico, Cattaneo, and Titiunik 2014; Calonico et al. 2017).

Figure 4.

Different linear effects of hours worked per week for respondents working overtime or not with a jump

By combining continuous and indicator variables, one can allow for nonlinearity by adding spikes, persistent jumps, or complete breaks to the regression line. This flexibility allows one to tailor the kind of nonlinearity in the model to the research question and what one knows about the variables involved with only a few parameters. Moreover, those parameters are easy to interpret.