# Parameter estimation in linear regression

This is a regression problem. Let $$r_i=f_iu_i+e_i,i=1,...,n$$, where $$u_i\sim^{i.i.d}N(0,1)$$, and $$e_i=O_p(\alpha_n),\alpha_n\rightarrow 0$$ as $$n\rightarrow 0$$. When I take the log transformation of $$r_i$$ and get \begin{align} \log|r_i|&=\log|f_iu_i+e_i|\\ &=\log|f_i|+\log|u_i|+\log|1+\frac{e_i}{f_iu_i}|. \end{align} The third term is $$O_p(\alpha_n)$$ since $$\log(1+x)\sim x$$ as $$x\rightarrow0$$. My question is that, if I want to model $$\log|f_i|=x_i'\beta$$, and denote by $$\epsilon_i=\log|u_i|+\log|1+\frac{e_i}{f_iu_i}|$$ the remainder. In regression problem, we need basic assumption that $$E\epsilon_i=0$$, but how can I achieve this? We can calculate the mean of $$\log|u_i|$$, denote by $$c$$, and hence \begin{align} \log|r_i|-c&=x_i'\beta+(\log|u_i|-c)+\log|1+\frac{e_i}{f_iu_i}|, \end{align} but what is the mean of $$\log|1+\frac{e_i}{f_iu_i}|$$? In this case, how to derive the convergence rate of $$\hat{\beta}$$?

• Maybe I can derive the result directly as usual, since the mean of $\epsilon_i^\star=(\log|u_i|-c)$ is 0 and the variance is finite. Hence $\epsilon_i^\star=O_p(1)$. The term $\log|1+\frac{e_i}{f_iu_i}|=O_p(\alpha_n)$ can be ignored since it does not affect the asymptoic result. Sep 5, 2021 at 2:13