Econometric Theory/Proofs of properties of β1

Linearity
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
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.