Defining additive and multiplicative regression models

 

Additive model

Y = a + b*x1 + c*x2 + e 

The coefficients describe the absolutes effects of unit changes in the independent variables on the dependent variable.

Multiplicative models

“Semi-logarithmic” models

Y = exp(a) * exp(b*x1) * exp(c*x2) * exp(e)
this can be written like follows:
Y = exp(a + b*x1 + c*x2 + e)
which in turn can be written like an additive model:
ln(Y) = a + b*x1 + c*x2 + e
where the coefficients are interpreted as percent changes in the response variable due to absolute unit changes in the independent variables.

“Logarithmic” models

Y = exp(a) * (x1 ^ b) * (x2 ^ c) * exp(e)
this is linearized with logarithmic transformations:
ln(Y) = a + b*ln(x1) + c*ln(x2) + e
where the coefficients describe the percent change in the dependent variable due to a 1% change in the independent variable; they are sometimes called “elasticities”.

​

Links:

https://www.sciencedirect.com/topics/mathematics/multiplicative-model

http://mail.sd-group.com.au/en/blog/additive-versus-multiplicative-marketing-mix-model