Interaction term between a dummy variable and a quadratic term

[Barbara Zarate]

Hi,

I am using stata 12 and running a multilevel model. I want to test three hypotheses: 1) Individuals' income has a curvilinear (inverted U-shaped) relationship with protest participation1) Individuals with economic motives (dummy variable) have a higher probability to participate in protest and 2) the effect of economic motives is only significant for middle class individuals (in terms of income). Thus, the central variables in my model are economic motives, income, and income^2.

First, I run the model without the interaction effect between income and motives, then I add the interaction effect. However, since income is hypothesized to show a curvilinear relationship with protest participation, I include the squared term of income but I am not sure how to include the interaction term. Do I have to interact income*motives? or income^2*economic motives?

This is the code I use for the regression without interaction terms:

xtmelogit economic_protest income_PPP_adjusted2011 i.motives income_PPP_adjusted2011_sq i.insurance ///
i.maritalst i.female age i.urban || ctrycode:, variance

Many thanks for your help,

Barbara

[Clyde Schechter]

When you include a quadratic term, the effect of income is represented by both the linear and quadratic terms. Neither one alone fully represents it. So if your hypothesis is that the effect of motives is modified by income, and in particular that this peaks in the middle range, you need to interact motives with both income and income^2. Interaction with only the linear term would not model a peaking of the interaction in the middle of the income range, and interaction with the quadratic term alone would force the peak effect to be seen at income = 0. So you must include both the linear and quadratic terms in the interaction to model what you are seeking.

By the way, you should not compute your own quadratic terms or interactions. You should use factor variable notation and let Stata do it for you. Apart from that minor convenience, using factor variable notation will make it possible for you to then use the -margins- command to get (and, if you wish, graph) predicted probabilities and marginal effects calculated correctly with simple one-line commands . So

xtmelogit economic_protest i.motives##c.income##c.income i.insusrance i.maritalst ///
     i.female age i.urban || ctrycode:, variance

Note: I have shortened income_PPP_adjusted2011 to just income to make the typing quicker and easier. That may not be possible for you if there are other income* variables in your data set. In that case, you need to use the full name.

The interpretation of this model will be complicated, and using -margins- will make it a lot simpler and clearer.

[Barbara Zarate]

Dear Clyde,

Many thanks for your answer, It is of great help. I will also follow your advice on the notation for the interactions and the use of margins to interpret the interactions.

Very best wishes,

Barbara