# How can I derive OLS predicted error term ^ei as a function of ei?

First of all, I'd like to say that any kind of help would be really helpful, whether it's a hint or a good grad/undergrad book. Right now I'm working with Econometric Analysis of Cross Section and Panel Data by Jeffrey M. Wooldridge (MIT Press), but it isn't helping much.

I'm struggling to understand one particular case of the OLS procedure in econometrics. Let's suppose I want to estimate the correlation between $$Y_i$$ and $$F_i$$ by:

$$Y_i= \beta F_i + \epsilon_i$$

And let's assume I can't observe $$F_i$$ , but I observe

$$\tilde{F_i} = F_i + u_i$$

with $$E(u) = 0$$

$$\text{plim}_{n\to\infty} \frac{1}{n}(Y'u) = 0$$

$$\text{plim}_{n \to \infty} \frac{1}{n}(F'u) = 0$$

$$\text{plim} \frac{1}{n}(\epsilon'u) = 0$$.

How can I derive the estimator $$\hat{\beta}$$ for the following OLS regression? And the predicted error term $$\hat{\epsilon}_i$$ as a function of $$\epsilon_i$$?

$$Y_i = \beta\tilde{F_i} + \epsilon_i$$

My thinking is that $$\beta$$ is the ratio between the covariance of $$Y_i$$ and the regressor and the variance of $$F_i$$, so after a two-stage OLS procedure $$\hat{\beta} = \dfrac{\text{Cov}(Y_i,F_i)}{\text{Cov}(\tilde{F}_i,F_i)}$$.

Have I done it correctly or do I miss something? How can I proceed with the error? Any suggestion?

Thank you very much!