How many two-way interaction terms can be included in a regression model and are there any necessary considerations which apply in interpretation?

Question

A model which looks like this Y=b0+X1+X2+X3+X4+(X1*X2)+(X1*X3)+(X1*X4) is X1 was a dummy variable = female.

Answer 1

Hi Ben.

First, I am going to rewrite your regression equation so it has coefficients:

Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₄ + β₅(X₁ * X₂) + β₆(X₁ * X₃) + β₇(X₁ * X₄) ( Equation #1)

When you include a two-way interaction term (such as you described), you are assuming that the marginal effect is statistically different for different values of the interaction term. For concreteness, assume that Y is wages and X₂ is years of experience. By having an interaction term between female and years of experience (X₁ * X₂), you are assuming that the change in wages due to years of experience is different for men and women. This is an implicit assumption when you specify this regression equation.

You asked how many two-way interaction terms can there be in a model. There can be as many two-way interaction terms as you specified in your stated regression equation, However, there can be even more interaction terms, say X₃ * X₄ or X₂ * X₃. What you need to make sure of, however, is that there are no colinearities in your specification.

For interpretation, I assume that X₁ = 1 when the observation is female. For males, they will have the following regression:

Y = β₀ + β₂X₂ + β₃X₃ + β₄X₄

For females, they will have the following regression

Y = β₀ + β₁ + (β₂ + β₅) X₂ + (β₃+ β₆)X₃ + (β₄+ β₇)X₄

Let's focus on the interaction term on X₂. In order for the two-way interaction term X₁ * X₂ to be justified, the estimated coefficients for male and female for X₂ should be different. In other words:

β₂ != β₂ + β₅

Subtracting β₂ from both sides, we get:

β₅ != 0

Therefore when you run the full regression equation specified in Equation #1, in order for this specification to be justified, the estimated coefficient of β₅ should be significantly different than zero. To do see this, look at the reported p-value and t-statistics.

You can do this procedure for the rest of the interaction term coefficients (β₆, β_7, β₈)