where $\epsilon_1$,$\epsilon_2$, and $\epsilon_3$ are uncorrelated random variables with $E(\epsilon_i)=0$ and $Var(\epsilon_i)=\sigma^2$ for $i=1,2,3$

I need help finding $Var(\hat{\beta}_1)$ and $Var(\hat{\beta}_2)$


Assuming $\hat \beta_1$ and $\hat \beta_2$ means least square estimates and using there values from your earlier question Find the Least Squares Estimates, we have$$\hat \beta_1=\frac{y_1+2y_2+y_3}{6}, \hat \beta_2= \frac{-y_2+2y_3}{5}$$
Now as $\epsilon_i $ are uncorrelated we have $y_i$'s as uncorrelated and then by using the fact that for uncorrelated variables $X_i$'s, $Var(\sum X_i)=\sum Var(X_i)$ and that $Var(aX)=a^2Var(X)$, we have $$Var (\hat \beta_1)=\frac{1}{36}(Var(y_1)+4Var(y_2)+Var(y_3))=\sigma^2/6$$ Using $Var(a+X)=Var(X)$, in the last equality. Similarly $Var(\hat \beta_2)=\sigma^2/5$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.