Simple linear regression model with intercept parameter

Question

Consider a Simple Linear Regression model with intercept parameter $\beta_1$ included. Show that the following holds:

$\sum\limits_{i=1}^n (y_i-\hat\mu_i)^2 = \sum\limits_{i=1}^n (y_i-\bar{y})^2 - \hat\beta_1^2 \sum\limits_{i=1}^n (x_i-\bar{x})^2$

Can someone help me how to proceed with this problem? I have no clue how to start..

Thank you in advance for your help.

Answer 1

You should write down what you already know about the model. This means that $$ \hat{\mu}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i $$ where $$ \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}, \quad \hat{\beta}_1= \frac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^n (x_i-\bar{x})^2}. $$ The best way to approach this is to start from the LHS, square the expression and plug in the $\hat{\mu}_i$ and $\hat{\beta}_0$. That way, you are left with an expression which depends only on $x$, $y$ and $\hat{\beta}_1$, which you need to get in the end. And now it just takes a few lines to multiply everything out and simplify.