# Linear regression estimates

For the simple linear regression model

$$y_i=\beta_0 +\beta_1 x_i$$

I have derived the estimates of $\beta_0=\bar y -\beta_1\bar x$ and $\beta_1=\frac{\Sigma (y_i-\bar y)(x_i-\bar x)}{\Sigma (x_i-\bar x)^2}$ but my book says that it is possible to express $\beta_0=\Sigma a_i Y_i$ and $\beta_1 =\Sigma c_i Y_i$.

How ever I can't seem to factor out $Y_i$ from either expression. For $\beta_0$ all I can get is $\beta_0=\Sigma \frac{y_i}{n}-\beta_1\bar x$ but this has constant term $\beta_1 \bar x$.

For $\beta_1$ I get $\beta_1=\Sigma y_i\frac{x_i-\bar x}{\Sigma (x_i-\bar x)^2}-\frac{\bar y(x_i-\bar x)}{\Sigma(x_i- \bar x)^2}$ again with constant term that I can't get rid of. Does anyone know how to solve it, why these constants should go or how to get rid of them?

• What is $\bar{x}_0$ in the expression of $\beta_1$? Commented May 20, 2018 at 13:01
• Should be written $\hat{\beta}_0$ and $\hat{\beta}_1$ instead of just $\beta_0$ and $\beta_1$ while writing the estimates. Commented May 20, 2018 at 13:03