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

$$[1] \quad  E(u|x)=0$$

However, I often see this condition written as: 

$$[2] \quad E(ux)=0$$

Are these two things equivalent? I see that if [1] and $E(u)=0$ then we get [2]; however I don't understand why [2] would imply [1].
[2] does not imply [1]. [2] and $E(u)=0$ imply $cov(u,x)=0$, which is about linear independence. [1] 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 &lt;- runif(100, min = -1, max = 1)
e &lt;- rnorm(100, mean = 0, sd = 0.1)
x &lt;- 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:



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