How to show $\operatorname{Cov}(b_0,b_1)=-\frac{\sigma\bar{x}}{S_{xx}}$

Question

Consider the equation $y_i=\beta_0+\beta_1x_i+\epsilon_i$ for $i=1, \dotsc, n$.

We have unbiased estimators $b_0$ and $b_1$ for $\beta_0$ and $\beta_1$ respectively, where $b_0=\bar{y}-b_1\bar{x}$ and $b_1= S_{xy} / S_{xx}$.

How does one show that $\operatorname{Cov}(b_0,b_1)=-\frac{\sigma\bar{x}}{S_{xx}}$

I tried using $\operatorname{Cov}(b_0,b_1)=E(b_0b_1)-E(b_0)E(b_1)$ to no avail as it just equals $0$ when I try and do that. Thanks!

Answer 1

You have not defined your symbols, so presumably $$S_{xx}=\sum (x_i-\bar x)^2 \quad\text{ and }\quad S_{xy}=\sum (x_i-\bar x)(y_i-\bar y)$$

Use the bilinearity of covariance.

We have

\begin{align} \operatorname{Cov}(b_0,b_1)&=\operatorname{Cov}(\bar y-b_1\bar x,b_1) \\&=\operatorname{Cov}(\bar y,b_1)-\bar x\operatorname{Var}(b_1) \end{align}

Recall that $\bar y$ and $b_1$ are independently distributed, so their covariance vanishes.

Moreover, the exact distribution of $b_1$ is $$b_1\sim\mathcal N\left(\beta_1,\frac{\sigma^2}{S_{xx}}\right)$$

Hence you finally get as the covariance between the least square estimates $$\operatorname{Cov}(b_0,b_1)=-\frac{\bar x\sigma^2}{S_{xx}}$$