A.   Regress the response variable on the explanatory variable. What is the residual at the last point (5,0)?

B.   Regress the first four response variables on the first four explanatory variables. What is the studentized residual at the last point (5,0)?

C.   Explain intuitively why the residual is correlated with the response variable.

D.   Is the studentized residual correlated with the response variable? Why or why not?

 

 

Can anyone give me some guidance on answering letters C and D? Thanks!

[NEAS: If the response variable is large, it is likely to be greater than its expected value, so the residual is positive . . .]

I got (A) E5 = -1.4 and (B) E5* = -1.917. Anybody get something different?

For Part A I also got -1.4. For Part B I got -1.4857.

To get that I calculated hi= (1/n) + (Xi-Xbar)^2/sum(Xi-Xbar)^2 and got hi = .6

Then I calculated the Se(-i) and got 1.48997

So Ei* = Ei / (Se(-i)(1-hi)^.5) = -1.4 / (1.48997*.4^.5) = -1.4857

I am not sure if I am right but that is how I got my answer.

Horshack, I also got h5 = 0.6. However, I got SE(-5) = 0.866.

I also got SE(-5)=.866, which gives E5*=-2.556

[NEAS: Correct]

Oooh, thanks for posting that. I had the correct equation and everything was right until the last step. I had apparently made a mistake plugging it into my calculator and got a different answer (but close enough that I didn't notice it was wrong).

I'm not sure how to calculate SE(-i).  I'm coming up with what horshack calculated, but apparently that's not correct.

To calculate SE(-i), you delete the i-th observation and run a regression using the remaining n-1 observations. The standard error of this regression is SE(-i). For each studentized residual that you want to calculate, you would have to run a separate leave-one-out regression. Since the assignment only asks you to compute ONE studentized residual, you will only need to run ONE extra regression.

If the residuals, Ei = Yi - Yi^ , then how could Ei and Yi not be correlated? Is this related to why we plot residual against the Yi^ values in order to detect nonconstant spread?

[NEAS: Correct. The residual is always correlated with the observed response values. We examine the correlation with the fitted values to test for heteroscedasticity.]