I am not sure if this question satisfies the guidelines, but here goes.
I'm reading a paper (text is below) from a financial mathematics journal, and the authors make a calculation step that I don't get. They reach the conclusion that $$E_i(\theta)=\frac{\alpha y + \beta x_i}{\alpha +\beta}$$
Whereas I would guess that the result should be either $$E_i(\theta)=y$$ or $$E_i(\theta):=E_i(\theta|x_i)=E_i(x_i)-E_i(\epsilon_i)=x_i$$
In the second case I'm interpreting the expectation as conditional expectation, and this seems to me the most sensible interpretation.
However the author's result is completely different as you can see, and I have no idea how they get to that result. In fact, if this weren't published in a peer reviewed journal, I would assume they made a very weird mistake. But since it's in a peer reviewed journal, I must miss something.
Here is the entire text:
Here is some context:
Here is the part my question is about:
How do they get to this result? How should I interpret the setup of their problem so that the conclusion makes sense?
It's conditional expectation. The error in your calculation is assuming $\mathbb E_i(\epsilon_i)=0.$
One way to do the calculation is to notice that the variable $x'_i=\theta - (\beta/\alpha) \epsilon_i$ has zero covariance with $x_i=\theta+\epsilon_i,$ and jointly normal variables with zero covariance are independent. This gives $\mathbb E_i \theta=\mathbb E_i (\alpha x'_i+\beta x_i)/(\alpha+\beta)=\mathbb E_i (\alpha y+\beta x_i)/(\alpha+\beta).$