In a recent paper, Edelman et al. examine (amongst other things) how discrimination on AirBnB varies with the characteristics of hosts. First, they conduct a field experiment which involves sending a large number of fictitious applications, some of which are associated with African-American names and some of which are not. Then, they regress whether an applicant is accepted on the guest race, a host characteristic and the interaction. For example, they regress acceptance on guest race, host race and the interaction of the two. The idea is that if African-American hosts (for instance) are less likely to discriminate, then the coefficient on the interaction term should be non-zero since the effect of receiving an application from an African-American should depend on whether the host is African-American.
So far, so good. However, given that host characteristics are not randomly assigned, we should control for them. Otherwise, we might find that African-Americans are less likely to discriminate even though the real explanation is that they tend to offer expensive properties, and people who offer expensive properties are less likely to discriminate (made up example to illustrate the point). They seem aware of this, since they throw in various host and location characteristics as controls. But wouldn't it have been better to use the interactions of these covariates with applicant ethnicity as the controls? After all, we want to control for the fact that people with certain characteristics might respond to African-American applicants differently, and this means controlling for interaction effects.