Conditional mean independence assumption in linear regression

Question

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?

Answer 1

Thanks for your questions.

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.

2. 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.