A.     Choose a sample for the explanatory variables. The exhibits below use 1 to 100.

B.     Simulate response variables whose observed values are heteroscedastic. The exhibits below use Y_j = X_j + X_j × Ö(0,1): the variance of the error term is proportional to the value of the X_j.

C.    Form the regression equation and derive the fitted values at each point.

D.    Plot the residuals against the observed values (left pane below).

E.     Pot the residuals against the fitted values (right pane below).

F.     Explain why the two plots are so different.

 

We show sample exhibits below. Your exhibits will look different, since you simulate points.

 

If your variance is too small, you won’t see the effect that the textbook discusses.

We should use Y_j = X_j + ŷ_j × Ö(0,1): the variance of the error term is proportional to the fitted value of Y_j. This is harder to simulate, so we use the simpler formula above.

 

Form the exhibits in Excel, SAS, R, or any other software. Sample exhibits are below.

 

I must be doing something wrong...

I am using Excel to run my regression. Using X from 1-100 and Yi = Xi + Xi*RAND():

X Y

1 1.944

2 2.205

. .

. .

. .

100 169.911

When I run my regression my Resid vs Observed plot looks identical to my Resid vs Fitted plot (both look like the Resid vs Fitted plot illustrated in the HW). I obviously made a mistake somewhere along the way (maybe in the X's and Y's I am using in the regression?).

I think what you are doing is correct because I got the similar results. I think the graph they provided for residuals vs observed values is incorrect. What the NEAS plotted in the first graph looks more like fitted values vs observed values.

NEAS: RAND() gives a random number from a uniform distribution on (0,1). To get a random number from a standard normal distribution, use NORMSINV(RAND()), which will give you the plots in the homework posting. RAND() has a smaller variance and doesn’t show the difference between the two plots. You can also use R to generate the plots:

xvv <- 1:100

yvv <- xvv + xvv * rnorm(100,0,1)

lm.hsce <- lm(yvv ~ xvv)

rsds <- residuals(lm.hsce)

fits <- fitted(lm.hsce)

par(mfcol=c(1,2))

plot(yvv, rsds, xlab="observed values", ylab = "residuals", main="residuals vs observed values")

plot(fits, rsds, xlab="fitted values", ylab = "residuals", main="residuals vs fitted values")

Using random draws from a uniform distribution gives the plots which you formed, as with

yvv <- xvv + xvv * runif(100,0,1)