# Conditional independence and no correlation

I have a question regarding basic econometrics. Consider the model $$y_i=\alpha +\beta x_i +u_i$$

I understand that assumption 4 of the linear regression model states

$$ \quad E(u|x)=0$$

However, I often see this condition written as:

$$ \quad E(ux)=0$$

Are these two things equivalent? I see that if  and $E(u)=0$ then we get ; however I don't understand why  would imply .

• Note that  implies , even without the condition $E(u)=0$. In fact this condition is implied by . Applying the law of iterated expectations to the LHS of , we get $E_x[E_u(u\vert x)]=E(u)$. But since $E_u(u\vert x)=0$ by , we have $E_x[E_u(u\vert x)]=E_u(u)=E_x(0)=0$ Oct 2, 2017 at 19:25
• I believe this is off topic as it belongs on Cross Validated. Oct 3, 2017 at 16:36

 does not imply .  and $E(u)=0$ imply $cov(u,x)=0$, which is about linear independence.  is stronger, as it refers to any type of dependence.

The classic counterexample to show this is $x=u^2$ over a symmetric domain. These are dependent yet linearly independent.

The R code below shows this:

set.seed(1)
u <- runif(100, min = -1, max = 1)
e <- rnorm(100, mean = 0, sd = 0.1)
x <- u^2 + e

plot(u,x)
abline(lm(x ~ u)) # Yields an R^2 of 0.006539

cov(u,x) # Yields 0.01206663


The plot is, where the black line represents the regression line: is stronger than . By applying the law of total probability expectation and the properties of conditional expectation, we have: $$E(ux)= E(E(ux|x)) =E(xE(u|x))$$ Thus  implies  but  does not necessarily imply