Using the law of iterated expectations with multiple variables

Question

I'm given that:

$E[\epsilon_i | dist_i, range_i] = 0$

How can I apply the law of iterated expectations to find that

$E[\epsilon_i|dist_i] =0$

is also true?

Answer 1

Tower property

if $F_1 \subset F_2$ so $E(E(Z|F_2)|F_1)=E(Z|F_1)$

by taking $F_1=\sigma(X)$ and $F_2=\sigma(X,Y)$ so $E(E(Z|\sigma(X,Y))|\sigma(X))=E(Z|\sigma(X))$

on the other hand

$E(E(Z|X,Y)|X)=E(Z|X)$

choose $(Z=\epsilon_i,X=dist_i,Y=range_i)$

$E[Z|X]=E\bigg(E[Z | X, Y])|X \bigg) =E(0|X)=0$

(just for hint $E[Z=\epsilon_i|X=dist_i]=E(E[Z=\epsilon_i | X=dist_i, Y=range_i])|X=dist_i) =E(0)=0$ )