We all know that we can fit a linear regression model even in cases when our predictors are not linearly related to the dependent. Like the example below (polynomial regression) -
(y = a + b X^2) can be written as (y = a + b Z) where Z = X^2
My question is, can there be changes to the form of the Beta coefficients as well?
More over, I am thinking of a few peculiar cases like this -
y = a + e^b X (where e is the Euler number)
would this still qualify for a linear regression or these mutations cannot be applied on the coefficients?
Thank you.
Is e the Euler number? If so then let B=e^b and you can find b by taking the natural logarithm of whatever coefficient you got
The linear regression will attempt to fit a linear model to the data, so even if the relationship is non linear it will only give you a coefficient for $X$ and not $X^2$. If you wanted the coefficient for $X^2$ I guess you could square all your data points and it would give you the coefficient. This is never done though, you would specify a quadratic term instead.
I do not think it makes sense to express the coefficient as a function of some true coefficient. It may help in interpretability, as in logistic regression with a binary outcome variable, you must exponentiate your coefficients to determine the effect a one unit increase will have on the odds of the response variable being $1$. It increases the odds by a factor of that much.