In class we discussed two options for removing the linearity assumption in regression: $$ Option $1$: fit linear model, but interpret it as a best linear approximation.

$$ Option $2$: use nonparametric methods $($partitioning$/$kernel methods, adaptively chosen splines$/$polynomials$)$ to actually estimate non$-$linearities.

What are the costs and benefits of each option? Name at least one of each for both.