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

تمرين 8

This exercise provides an example of a pair of random variables X and Y for which the conditional mean of Y given X depends on X but corr( X , Y ) = 0. Let X and Z be two independently distributed standard normal random variables, and let Y = X₂ + Z.
a. Show that E ( Y | X ) = X₂.
b. Show that Y = 1.
c. Show that E ( XY ) = 0.
d. Show that cov( X , Y ) = 0 and thus corr( X , Y ) = 0.