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?


1 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.

  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.

  • $\begingroup$ 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" $\endgroup$
    – jojorabbit
    Commented Oct 13, 2020 at 14:23
  • $\begingroup$ 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 $\endgroup$
    – jojorabbit
    Commented Oct 13, 2020 at 14:24
  • 1
    $\begingroup$ 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! $\endgroup$
    – curious
    Commented Oct 13, 2020 at 14:31

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