# Maximum likelihood variance estimator of simple linear regression is biased

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

• Start by writing down the variance estimator and take the expectation? edit your post to include your working. Commented Jun 17, 2021 at 8:18
• @SiongThyeGoh, I have done the did Commented Jun 17, 2021 at 9:21
• perhaps consider what do you know about $\hat{\beta}$. Commented Jun 17, 2021 at 9:34

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 $$n-2$$.