# Interaction effects with three dummy variables - interpretation

I am doing an OLS regression with interaction effects and dummy variables, where the interaction effect is given by the dummy variables. Remaining variables are continuous. I am quite new to statistics so i would need some help for interpreting results. I have the following simplified model:

return = 0.44 + 0.5size + 0.3cost + 1.2D1 + 1.3D2 + 0.4D1*size -0.55D2*size + 0.32D1*cost - 0.8D2*cost

The model includes three dummy variable that are defined as employed (D1), self-employed (D2) and unemployed (base condition).

Am i correct in interpreting results in the following way: - First beta coefficient (0.5size) indicates the effect of size on return when a person is unemployed - 0.4D1*Size indicates the additional effect of size compared to the base condition (unemployed) when the person is employed -> the total effect is 0.5+0.4 - (-0.55D2*size) indicates the additional effect of size compared to the base condition when the person is self-employed -> the total effect is (0.5 -0.55) for self-employed individuals

Also: say instead of size i have a male/female dummy variable with male =0. Simplified model: return = a +0.3Dmale +0.4D1 + 0.5D2 +0.6Dfemale+ 1.8Dmale*D1 +0.8D2*Dmale +... 0.3 is the effect that being male and unemployed has on returns, 0.4 the effect of being employed, 0.5 of being self-employed. 1.8 is the effect that being male and employed has on returns compared to being male and unemployed (so total effect 0.3+1.8), same for 0.8 only with different dummy definition.

Am i correct? Thank you!!

The equation for the fitted model that you stated is really a collection of 3 equations - one for each type of employment.

1) D1 = 0, D2 = 0 (unemployed)

return = 0.44 + 0.5size + 0.3cost


2) D1 = 1, D2 = 0 (employed)

return = 0.44 + 0.5size + 0.3cost + 1.2 + 0.4*size + 0.32*cost


or, equivalently,

return = (0.44 + 1.2) + (0.5 + 0.4)*size + (0.3 + 0.32)*cost


3) D1 = 0, D2 = 1 (self-employed)

return = 0.44 + 0.5size + 0.3cost + 1.3 - 0.55*size - 0.8*cost


or, equivalently,

return = (0.44 + 1.3) + (0.5 - 0.55)*size + (0.3 - 0.8)*cost


So the coefficient of size, 0.5, in the first equation represents the effect of size on return for unemployed people, after controlling for the effect of cost.

The coefficient 0.5 + 0.4 of size in the second equation represents the effect of size on return for employed people, after controlling for the effect of cost.

The coefficient 0.5 - 0.55 of size in the third equation represents the effect of size on return for self-employed people, after controlling for the effect of cost.

If you compare the magnitude of the coefficients of size in equation 2) (employed) and 1) (unemployed), you will conclude that the difference between (0.5 + 0.4) and 0.5 - namely, 0.4 - represents the difference in the effect of size between employed and unemployed people, after controlling for the effect of cost.