Interaction terms in level-log regression

[Andrea Baldin]

I have the following regression: y=a + b*ln(x1) + c*(ln(x1)*ln(x2)).
How do I interpret the marginal effect of x1 on y? Should I take the derivative of y with respect to x1? I haven't find anything about the interaction term in a level-log regression. Thanks a lot

[Clyde Schechter]

First, it is unusual to have a model that includes an interaction term that does not also include the "main effects" of each component of the interaction. Why is there no term in the model for ln(x2)? (The usual reason is that you are doing a panel regression and x2 does not vary within panels--but if this isn't the case, you need some justification.)

In any model with interaction, there is no such thing as "the effect" of x1 on y. The expected difference in y associated with a unit increase in x1 will depend on the concurrent value of x2. In the case of your particular model, even without an interaction term, there wold be no such thing as "the effect" of x1 on y because of the non-linearity in the model: a unit increase in x1 will have different impacts on y depending on what the initial value of x1 is. So you need to decide at what base values of x1 and x2 you are interested in knowing the differences in y associated with a 1 unit difference in x.