Econometric Theory/Proofs of properties of β1

To be linear, ${\displaystyle {\hat {\beta }}_{1}}$ must be a linear function of ${\displaystyle Y_{i}}$, as shown below

${\displaystyle {\hat {\beta }}_{1}=\sum {k_{i}Y_{i}}}$

where ${\displaystyle k_{i}}$ is a constant, at any given observation 'i'.

From the deviation-from-means form of the solution of the OLS Normal Equation for ${\displaystyle {\hat {\beta }}_{1}}$, we have

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

${\displaystyle ={\frac {\sum {x_{i}Y_{i}}}{\sum {x_{i}^{2}}}}}$, since ${\displaystyle {\sum {x_{i}}}=0}$.

${\displaystyle =\sum {k_{i}Y_{i}}}$, where ${\displaystyle k_{i}={\frac {x_{i}}{\sum {x_{i}}}}}$, which is a constant for any given 'i'-value.