Conditional mean independence assumption in linear regression

In Linear Regression, we have the Conditional mean independence assumption:

E(u|x) = 0, where u is the error in the linear relationship.

May I clarify: my understanding of this is that implications of this assumption are that:

1. u is normally distributed around 0, therefore expectation = 0 (I understand that it may not be a normal distribution now after seeing an answer here, but at the very least the average value is at 0)

2. x doesn't influence anything about u (i.e. gives no information about u, and by extension its expectation) since x and u are independent (so this shows the independence)

Is my understanding correct?

In words, the assumption $$E(u|x_1, ..., x_k)=E(u)=0$$ states that the error term $$u$$ has an expected value of zero given any value of the independent variables.

1. Therefore, the zero conditional mean assumption itself does not make a statement about which distribution $$u$$ has, only a statement about its expected value/mean.

For example, if you check the textbook "Introductory Econometrics" by Wooldridge you can compare assumptions MLR.4 and MLR.6. Only in assumption MLR.6, it is assumed that the error term follows a normal distribution. However, the more important assumption is MLR.4 which is needed for the OLS estimator to be unbiased.

1. If the assumption $$E(u|x_1, ..., x_k)=0$$ holds $$u$$ and $$x$$ are said to be mean independent (technically, they must not be fully independent). An implication of this is that $$u$$ and $$x$$ are not correlated.

I'm not sure what you mean by the statement

x doesnt influence anything

But assume that the true model is $$y=b_0 + b_1x + u$$. Here, by definition, $$x$$ has an effect on $$y$$ of $$b_1$$, irrespective of whether the zero conditional mean assumption holds. If the assumption holds the OLS-estimator $$\hat{b1}$$ is an unbiased estimator of $$b_1$$. If the zero conditional mean assumption does not hold, this is not the case.

• I think I meant to say in the second point that another implication of the zero mean conditional assumption is that "x gives no information of u" or like that "having a given value of x will not tell us anything of u and therefore not it's expectation as well" Oct 13, 2020 at 14:23
• thank you so much for your answer nevertheless! I dont have the textbook but i'll see if i can get a hold of it Oct 13, 2020 at 14:24
• Yes. Interpretations such as "x gives no information about u" are useful in thinking about the zero conditional mean assumption. You can see an overview of the assumptions in Wooldridge here . Compare MLR.4 and MLR.6 to see the difference. If I answered your question, please consider checking my answer as right. Thanks! Oct 13, 2020 at 14:31