Problem 6. (Residual correlation) Suppose that you build a regression model given data Y and

design matrix X. We assume throughout this problem that there is an intercept (i.e., that the first

column of X consists of all 1's).

You apply ordinary least squares and find coefficients beta^

and fitted values ^Y== X(beta^)

. Recall the

residual vector is ^r = Y- ^Y

, and that (since there is an intercept)

the sume of residuals = 0.

a) Carry out the following steps in R. First, load the GaltonFamilies dataset:

install.packages("HistData") # if you don't have the package yet

library(HistData)

data(GaltonFamilies)

Next, for the following model:

lm(data = GaltonFamilies, childHeight 1 + father)

plot the residuals against the fitted values. Compare this to a plot of the residuals against the

observed values of childHeight, and describe what patterns you see.

Some helpful R code: the residuals of a fitted model fm are obtained by using residuals(fm);

the fitted values are obtained by using fitted(fm).

b) Show that in general, ^r ^Y

= 0, so that ^r x ^Y = ^r x Y||^r||2. Using these facts, explain the patterns

seen in the plots you created in (a). (Hint: What is the sign of the sample correlation between

the residuals and the observed values?)