Interpretation of interaction term between 2 continuous variables

Question

I'm running a multiple linear regression of revenue (rev) on identity (ID), an index which measures a customers identity towards a company, and firm age (Age). These two input variables are expected to depend on each other, so that the effect of identity on revenue depends on how many years a company is already established in the market. To test this, I've included an interaction term of ID*Age.

rev = b1*ID + b2*Age + b3(ID*Age) + ui

This gives me following estimated coefficients:

rev = 0.7*ID + 0.1*Age + -0.003(ID*Age) + ui

The coefficient of ID and the interaction term are statistically significant. I'm not sure about how to interpret the negative interaction term. Is it correct to interpret it as follow:

The effect of ID on revenue depends negatively on age. So that the effect on an increase in identity will be weaker (lesser) if the company is older (Age is bigger).

Is that equal to the following interpretation: Identity is weaker for older firms.

Therefore, I could conclude that identity does play a (slightly) more important role for younger companies.

Answer 1

1. You can't interpret the effect of any coefficient solely based on it's sign. While the effect is negative, we don't have the result of any significance test to support the hypothesis of interaction as you stated.

2. "Depend" is purely a mathematical (or probabilistic) construct, which is not useful for actually describing findings. Consider also that in probability theory, dependence is a commutative property. For instance, if lung cancer depends on smoking, smoking depends on lung cancer. Though we don't think developing lung cancer caused smoking, I know a lung cancer patient is way more likely to be a smoker than a lung cancer-free patient.

3. You have to report the units to make any sense of the results. Without knowing the univariate properties of ID and of Age, I can only observe that the interaction term is reported out to the third decimal place. If these were standardized (mean 0, SD 1) regressors, I would expect -0.003 is a spurious finding.

4. Interactions are easily interpreted as difference-in-differences. When in doubt, always plot it out. But in words: The difference in difference for expected revenue was -0.003 for groups differing in 1 year of incorporation for a unit difference in the ID scale.