If I have a regression result where the positive coefficient of the interaction term is not strong enough to reverse the effect of each of its components with negative coefficients, how shall I interpret the coefficients of the interaction term? For instance:

 y = constant -1.5a - 1.4b + 0.5ab

that even if I increase a by one unit while keeping b constant as one, the interaction effect ab is not strong enough to reverse the effect of variable a in this case.


One way to think about it would be the marginal effects:

Here you have estimated the expected value of y, given a and b

$E[y|a,b] = -1.5a + -1.4b + .5ab$

If you take the derivative of the above expression with respect to a:

$\frac{\partial E[y|a,b]}{\partial a} = -1.5 + .5b$

This expression is increasing in b. You could then set b to "interesting" values for interpretation. I put interesting in quotes because this will depend on the application. For some applications b=0 might be interesting. The sample mean may also be interesting.

A better way would be calculate what is called the Average Partial Effects. Suppose there were n data points in your sample the average Partial Effects would be as follows:

$$APE_a = -1.5 + .5*\frac{1}{n}\sum_{i=1}^{n} b_i$$ Where $b_i$ is the value for b for the ith observation. This would have the interpretation of what the average effect for an additional a would be for the actual sample that you are using.


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