Multivariational regression

Question

I have been given following model

$\ln(y)=\beta_0+\beta_1a+\beta_2a^2+\beta_3a^3+\beta_4a^4+\beta_5b$

and a set of observations that describe relation between $y$ and $\{a, b\}$. The goal is to estimate $\beta$-coefficients such that the model best explains the observations. How can I estimate them?

Also, why would one want to put the logarithm on $y$?

I have tried least squares (with $\ln(y)$), but the model tends to overestimate $y$. This might be due to outliers (which are present in the observational data), or maybe least squares is not a fit for this in a first place?

Answer 1

For the model $$z=\ln(y)=\beta_0+\beta_1a+\beta_2a^2+\beta_3a^3+\beta_4a^4+\beta_5b$$ all the $\beta$'s will be obtained using multilinear regression just as you apparently did.

The problem is that this is just a first step since what is measured is $y$ and not $\log(y)$. In such a case, you need to start a nonlinear regression of the same data for the model $$y=\exp(\beta_0+\beta_1a+\beta_2a^2+\beta_3a^3+\beta_4a^4+\beta_5b)$$ for which you have good starting values from the first step.

Edit

If you do not want (or cannot) use nonlinear regression, you can stay with weighted multilinear regression. This is a good compromise (see here).