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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?

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  • $\begingroup$ This is an expression of the distributive law (of multiplication over addition). $\endgroup$
    – whuber
    Apr 24, 2023 at 13:10

1 Answer 1

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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 $\sum_i \left(\frac{(x_i-\bar x)}{s}\right)u_i.$

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