construct three random variables X Y Z so that
E(X|Y=y) , E(X|Z=z) are both constant whilst E(X|Z=x,Y=y) isnt a constant??
not really sure where to go with this??
I assume you typoed and you meant to sayconstruct three random variables X Y Z so that
E(X|Y=y) , E(X|Z=z) are both constant whilst E(X|Z=x,Y=y) isnt a constant
(X|Y=y) , E(X|Z=z) are both constant whilst E(X|Z=z,Y=y) isnt a constant
Let Y,Z be independant bernoulli variables. with p=0.5 for both.
Define:
$\displaystyle X=-1^{Y+Z}$
$\displaystyle E(X|y=0) = E(-1^Z) = 0.5*1 + 0.5*-1 = 0$
$\displaystyle E(X|y=1) = E(-1^{1+Z}) = 0.5*-1 + 0.5*1 = 0$
So $\displaystyle E(X|Y) = 0$
$\displaystyle E(X|z=0) = E(X|z=1) =0~~$ (by analogy)
But consider $\displaystyle E(X|Y,Z)$
$\displaystyle E(X|y=1,z=0) = -1 $
$\displaystyle E(X|y=1,z=1) = 1$
...etc. Obviously not constant so i wont do the other two cases.
That establishes the required properties. Not bad for an economist, eh?