Very stuck with this problem. Not meant to make any more assumptions than laid outin the question.
I have a regression model: $\ln(_)$ = $_0 + _1\cdot \ln (_i) + _i$ (where sav is annual savings and inc is annual income) and an error term: $_i = _i^{0.2}\cdot e_i$ It is also assumed that $_i|\sim \mathcal N \left(0, _{}^2 \right)$ so $_i$ and $_i$ are independent. The question is to find $\mathbb E (_|)$ and $(_i|)$.
I understand that the error term $_i$ is normally distributed, with mean 0 and variance $_{}^2$ . I also understand that, since e and inc are independent, their covariance will be 0. I just can't figure out how to get further than this and what values need plugged in to which equations at this point.
Any help would be amazing, even just to point me in the right direction - I'm getting very frustrated, and have spent hours trying to figure out what I'm missing. Thanks!
Given $e$ and $inc$ are independent, we know $e$ is normally distributed with (0,$\sigma_e^2$), and Inc is known. $$E(u_i|inc_i)= E(inc_i^{0.2}\cdot e|inc_i)=inc_i^{0.2}E(e|inc_i)$$ $$Var(u_i|inc_i)= Var((inc_i^{0.2}\cdot e |inc_i)=inc_i^{0.2(2)}Var(e|inc_i) = inc_i^{0.4}\sigma_e^2$$