Interaction term between 2 continuous variables

I'm running a multiple linear regression in RStudio of revenue ( rev ) on identity, ID (an index which measures a customers identity towards a company) and firm age, Age (years of corporation). These two variables are expected to be depentend 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

Coefficient of ID and the interaction term are statistically significant. I'm not sure about how to interprete the negative interaction term. Is it correct to interprete 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.

Thank you so much for your help!

I would interpret it using some hypothetical situations - also I don't know what range of values your ID and Age variables take, so I'll just assume some values. Let's first fix ID = 1. An additional year (I guess) of age will increase revenue by 0.1 - 0.003 = 0.097. However, if ID = 10, an additional year of age will increase revenue by 0.1 - 0.003*10 = 0.1 - 0.03 = 0.07. So, the higher the ID the less an additional year of Age contributes to revenue. I guess that might also make some economic sense in terms of diminishing marginal returns.

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yes exactly, thank you @valeri! And then vice versa, for an additional 10 units in ID, the increase in revenue is 0.7 - 0.00310 = 0.67, if additional 1 unit in ID -> 0.7 - 0.0031 = 0.697. So younger firms benefit more from identity den older firms - right?

Yes - you can of course make an argument in terms of increasing ID, but you need to fix Age - so I don't quite follow your example. I'd say that if we fix Age = 1 then an increase of ID by 1 has an effect of 0.7 - 0.003 = 0.697, while if Age = 10, an increase of ID by 1 leads to a change in revenue of 0.7 - 0.03 = 0.67.

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