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

${\hat {\beta }}_{1}={\frac {\sum {x_{i}y_{i}}}{\sum {x_{i}^{2}}}}={\frac {\sum {x_{i}(Y_{i}-{\bar {Y}})}}{\sum {x_{i}^{2}}}}={\frac {\sum {x_{i}Y_{i}}}{\sum {x_{i}^{2}}}}-{\frac {\sum {x_{i}{\bar {Y}}}}{\sum {x_{i}^{2}}}}$

$={\frac {\sum {x_{i}Y_{i}}}{\sum {x_{i}^{2}}}}$, 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.