Expected value and the standard simple regression model

Question

Given the standard simple regression model: $y_i = β_0 + β_1 x_i + u_i$

What is the expected value of the estimator $\hat\beta_1$in terms of $x_i, \beta_0$ and $\beta_1$ when $\hat\beta_1=\sum x_i y_i/\sum x_i^2$?

Answer 1

\begin{align} \text{E} \left ( \frac{\sum_{i=1}^{n} x_i Y_i}{\sum_{j=1}^{n} x_j^2} \right ) &= \left ( \sum_{j=1}^{n} x_j^2 \right )^{-1} \sum_{i=1}^{n} x_i \text{E}(Y_i) \\ &= \left ( \sum_{j=1}^{n} x_j^2 \right )^{-1} \sum_{i=1}^{n} x_i (\beta_0 + \beta_1 x_i) \\ &= \beta_0 n\bar{x} \left ( \sum_{j=1}^{n} x_j^2 \right )^{-1} + \beta_1 . \end{align}

Answer 2

Denote $x=(x_1,\dots,x_n)$. Assuming that $\mathbb{E}[u_i|x]=0$

$$\mathbb{E}[\hat\beta_1|x]=\mathbb{E}\left[\frac{\sum x_i y_i}{\sum x_i^2}\mid x\right]=\mathbb{E}\left[\frac{\sum x_i (\beta_0 + \beta_1 x_i + u_i)}{\sum x_i^2}\mid x\right]=\frac{\beta_0\sum x_i+\beta_1\sum x_i^2+\sum x_i\mathbb{E}[u_i|x]}{\sum x_i^2}=\beta_1+\beta_0\frac{\sum x_i}{\sum x_i^2}$$

and

$$\mathbb{E}[\hat\beta_1]=\beta_1+\beta_0\mathbb{E}\left[\frac{\sum x_i}{\sum x_i^2}\right]$$