Introduction to Econometrics 3rd Edition by James Stock, Mark Watson

Exercise 20

Suppose that Y 1 , Y 2 ,..., Y n are random variables with a common mean y , a common variance
and the same correlation (so that the correlation between Y i and Y j is equal to for all pairs i and j, where i j ).
a. Show that

b. Suppose that n = 2. Show that
and

c. For n 2, show that
and

d. When n is very large, show that