What you are talking about are basic inteaction effets. You could think forexample of simple model:
y = a + b1*x1 + b2*x2 + e
If we add and interaction-term to the model depicted, the following term consequences.
y = a + b1*x1 + b2*x2 + b3*x1*x2 + e
By simply reordering, we find
y = (a + b1*x1) + x2*(b2+b3*x1) + e
The first parenthesis depicts the intercept and the second the regression’s slope. However, both elements are subject to the level of x1. Despite term appearing rather econometrically tame, Braumöller (2004) shows that interpreting models with multiplicative terms often goes askew especially when turning to the lower-order coefficients
The regression’s slope will equal
b1 only if
x2 equals zero. Thus
denotes the simple effect of
x2 denotes the simple effect on
y = a + b2*x2 + e ; for x1=0. Similarly,
a is the intercept when both
x2 are zero.
b3 finally depicts the interactions between
x2. And here lies the crux:
First, you can only interpret
x3if you incorporate both
x2in your regression. For a detailed analysis of this matter see Whisman, Mark A., and Gary H. McClelland. "Designing, testing, and interpreting interactions and moderator effects in family research." Journal of Family Psychology 19.1 (2005): 111. Second, you need "meaniful" values for
x2=0. One idea might be a centering at meaningful values or at means. For a more detailed discussion of this matter check here: When conducting multiple regression, when should you center your predictor variables & when should you standardize them?
Statistic packages like Stata and R usually do everything for you. Practically, you can incorporate x1, x2 and a third variable x1*x2. But usually it is better to merely incorporate x1 and x2 and "tell" R/Stata to gauge an interactio effect too ( x1##x2 or x1#x2 in Stata). R is a little different. Check here: Different ways to write interaction terms in lm?