Is there such a thing as a combination of linear and non-linear regression in one form?

Question

Let's say I have a dataset D with many variables. I can get a multiple linear regression from that in the form Y=M1X1+M2X2+M3X3+...+MnXn+B. I can also get a multiple non-linear regression by taking ln(D) then performing a normal multiple linear regression and raising the final product up by e to get something in the form of: Y=X1^(W1)*X2^(W2)*X3^(W3)*...*Xn^(Wn)*C.

Now for the titular question: is there a mixed form that involves both linear and non-linear components? Where would I start to learn more?

Answer 1

Perhaps this is something similar to what you have in mind?

Let $Y=e^{X_1\beta_1}*X_2^{\beta_2}*X_3^{\beta_3} $

Taking logs of both sides yields:

$ ln(Y) = X_1\beta_1 + \beta_2ln(X_2) + \beta_3ln(X_3) $

which contains both a linear ($X_1\beta_1$) and non-linear ($\beta_2ln(X_2) + \beta_3ln(X_3) $) components.

One place we see these a lot is in Duration Models