Statistical Properties of OLS Estimators (Unbiasedness)

Question

In deriving the unbiasedness of OLS Estimators,

$$\hat{\beta_1} = \beta_1 + \frac {\sum_{i=1}^{n} (x_i - \bar{x})u_i}{\sum_{i=1}^n (x_i - \bar{x})^2}$$

My professor changes the above to:

$$\hat{\beta_1} = \beta_1 + \sum_{i=1}^{n} (\frac {(x_i - \bar{x})}{\sum_{i=1}^n (x_i - \bar{x})^2})u_i$$

How are these two equal?

Answer 1

Note: The denominator $ \sum_{i=1}^n (x_i - \bar{x})^2 := s$ is constant; it doesn't change when you are summing over $i,$ i.e. the summation would be $\sum_i \left(\frac{(x_i-\bar x)}{s}\right)u_i.$