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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 <- 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:
Read more about this here.
[1] is stronger than [2]. 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 [1] implies [2] but [2] does not necessarily imply [1]
