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

${\hat{\beta }}_1=\frac{\sum x_i{y}_i}{\sum x_i^2}=\frac{\sum x_i(Y_i-\overline{Y})}{\sum x_i^2}=\frac{\sum x_i{Y}_i}{\sum x_i^2}-\frac{\sum x_i\overline{Y}}{\sum x_i^2}$

$=\frac{\sum x_i{Y}_i}{\sum x_i^2}$, 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.