Econometric Theory/Proofs of properties of β1

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Linearity[edit]

To be linear, \hat{\beta}_1 must be a linear function of Y_i, as shown below

\hat{\beta}_1 = \sum{k_i Y_i}

where k_i is a constant, at any given observation 'i'.

Proof[edit]

From the deviation-from-means form of the solution of the OLS Normal Equation for \hat{\beta}_1, we have

\hat{\beta}_1 = \frac{\sum{x_i y_i}}{\sum{x^{2}_i}} = \frac{\sum{x_i (Y_i - \bar{Y})}}{\sum{x_{i}^2}} = \frac{\sum{x_i Y_i}}{\sum{x_{i}^2}} - \frac{\sum{x_i \bar{Y}}}{\sum{x_{i}^2}}

 = \frac{\sum{x_i Y_i}}{\sum{x_{i}^2}} , since {\sum{x_i}} = 0.

 = \sum{k_i Y_i}, where k_i = \frac{x_i}{\sum{x_i}}, which is a constant for any given 'i'-value.