I have to solve the following problem: I've been given that $E[y_i|x_i] $ is non-linear in $ x_i = (1, x_i1, . . . , x_ik) $ (I have not been told "how" is non linear) and $ Var[y_i|x_i] = σ^2 $. I've to prove that via linear model approximation of the Conditional expectation function, the difference between the regression line and the Conditional expectation function depends only on $x_i$. Substantially, I need to show that the variance $ u_i $ is heteroskedastic by construction. Also, as a hint I've been told to start from $ E[(y_i − x_iβ)^2|x_i]$ and rewrite is as a function of $ E[y_i^2|x_i]$ plus something else.
I noticed that the term $ (y_i − x_iβ)^2$ is the square of the residual, which I further decomposed in $E[y_i^2|x_i]$ plus $E[x_iβ(x_iβ-2y_i)|x_i]$. Then I basically got stuck, because I don't know how to further proceed. I've noticed, and for sure I'm wrong, that $ Var[y_i|x_i] = σ^2 $ which means that the conditional variance takes a constant value, whereas $y_i$, in order to be heteroscedastic, should have a variance which change with i. It quite reminds me the homoskedasticity assumption, rather that the heteroskedastic one. But what I need to prove is that the error is heteroskedastic. How can I proceed in order to prove it?