Maximum likelihood variance estimator of simple linear regression is biased

Question

Colleagues, I understand that bias is defined as $b(\theta)=\mathbb{E}(\hat{\theta})-\theta$. How can one show that the variance estimator for maximum likelihood estimators of simple simple linear regression is biased?

The estimator is given by $$ \sigma ^2 = \frac{1}{n} \sum (Y_i -\hat{\beta}_0 -\hat{\beta}_1 X_i)^2$$

Answer 1

If you assume normality of the error term, then you can use the $\chi^2$ distribution. Namely, $$ \frac{1}{n}\sum_{i=1}^n( y_i - \hat\beta_0 - \hat\beta_1x_i)^2 = \frac{\sigma^2}{n}\sum_{i=1}^n( y_i - \hat\beta_0 - \hat\beta_1x_i)^2/\sigma^2 = \frac{\sigma^2}{n}\chi^2(n-2) , $$ where the expected value of $\chi^2(n-2)$ is $n-2$.