I am trying to compute $Var(e_i)$.

So far I have

$Var(e_i)=Var(y_i-\hat y_i)=Var(y_i)+Var(\hat y_i)-2cov(y_i,\hat y_i)$

Now, I know that

$Cov(y_i,\hat y_i)=var(\hat y_i)$

but how do I prove this? (without using matrices)

But anyway, from there I have $Var(e_i)=var(y_i)-var(\hat y_i)= \sigma^2 -var(\overline y+\hat \beta_1 (x_i-\overline x))$

$=\sigma^2-var(\overline y)-(x_i-\overline x)^2var(\hat\beta_1) -2cov(\overline y,\hat \beta_1 )(x_i-\overline x)$

$\sigma^2-var(\sum y_i /n) - (x_i-\overline x)^2\sigma^2\sum(x_i-\overline x)^2$

$var(\sum y_i /n)=\sum(var(y_i))n^2 = \sigma ^2 / n$

So I end up with $Var(e_i)=\sigma^2(1-(1/n)-(x_i-\bar x)^2\sum (x_i-\overline x)^2)$

Is this correct?


The (Estimated) Variance of residuals in an OLS regression is simply: $$ Var(e)=\frac{e'e}{n-(k+1)} $$ where $k+1$ is the number of regressors (plus a constant).

  • $\begingroup$ Note the numerator is simply the squared sum of the residuals. $\endgroup$ – ChinG Nov 4 '15 at 16:17

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.