Suppose we have the relation $Y_{i}=\beta_{1}+\beta_{2}X_{i}+u_{i}$ with $X_{i}$ and $Y_{i}$ observed values, $\beta_{1}$ and $\beta_{2}$ the parameters and $u_{i}$ the disturbance term. We want to construct estimators of $\beta_{2}$.
The method of least ordinary squares will minimize $\sum u_{i}^2$. This means minimizing $(\sum Y_{1}-\beta_{1}-\beta_{2}X_{i})^2$, in other words minimizing at the same time the parameters $\beta_{1}$ and $\beta_{2}$.
Now we take another method of estimation only for $\beta_{2}$, which is summing the slope of the lines formed after each consecutive observation (slope between first and second points, second and third points, etc...) and averaging the result.
How can we compare these two methods of estimation? Which estimator for $\beta_{2}$ will be more accurate? Which one will be bigger than the other?