Interpreting Coefficients in the Presence of Interaction Effects

Question

I have a model which estimates the average sqft price based on whether the estate needs renovation or not and whether it is downtown, suburbs or in the transition zone between those two ares. All variables are binary:

$price = 402.45 − 23.10 ∗ renovation + 56.10 ∗ downtown − 10.01 ∗ transition + 15.90 ∗ renovation × downtown − 0.15 ∗ renovation × transition$

The reference groups are suburb estates that do not need renovation which price is represented by the intercept. But how do I e.g. interpret the coefficient of $renovation$? Does it mean that the price of estates that need renovation is 23.1 less than those which do not need renovation or is it that the price of suburb estates that need renovation is 23.1 less than suburb estates that do not need renovation?

The same question goes to the coefficient of the interaction terms such as e.g. $renovation × downtown$.

Answer 1

The coefficients of the main effects are the effect when the other variable in the interaction is 0. The coefficients of the interactions are the extent of non-additivity in the whole equation. This is much easier to understand by example.

In your example, I'm guessing that all three variables are 0/1. Then you can easily make a table with the estimate of price for each of 8 combinations. E.g.

R      P      D     price
0      0      0     402.54
0      0      1     402.54 + 56.10
....
1      1      1    402.54 - 23.10 + 56.10 - 10.01 .... -0.15

I will leave the calculation to you. If the variables are not 0/1 you will have to add some products.

You can also make graphs. A lattice plot would be useful.

Answer 2

I think if the sample data only consists of suburb estates, then all $\beta$-coefficients determine the effect on that sample data, i.e. suburb estates. This means - given your $\beta$-coefficients for the specific variable are significantly different from zero - the price of suburb estates that need renovation is 23.1 less than suburb estates that do not need renovation (exactly as you already said). Same applies to the interaction term, which only has an effect if they are both $renovation_i = 1$ and $transition_i = 1$ for the $i$-th data point.