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Assume the following linear relationship: $Y_i = \beta_0 + \beta_1 X_i + u_i$, where $Y_i$ is the dependent variable, $X_i$ a single independent variable and $u_i$ the error term.

According to Stock & Watson (Introduction to Econometrics; [Chapter 4][1]), the first least squares assumption is $E[u_i|X_i]=0$. Does this imply (law of iterated expectation): $E[u_i]=0$?

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    $\begingroup$ You appear to answer your question while you ask it! Are there perhaps some undisclosed conditions that cause you to doubt the applicability of your reasoning? $\endgroup$ – whuber Nov 4 '16 at 17:48
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    $\begingroup$ You said it: by law of iterated expectations, this holds. $\endgroup$ – Richard Hardy Nov 4 '16 at 19:39
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Yes, it does imply that, but it is a stronger assumption: Setting the conditional mean of the error terms to zero is a stronger assumption than just setting their marginal mean to zero. If we have $\mathbb{E}(U | X = x) = 0$ for all allowable $x$ then using the law-of-iterated-expectations we get:

$$\mathbb{E}(U) = \mathbb{E}(\mathbb{E}(U | X)) = \mathbb{E}(0) = 0.$$

In fact, we can also derive another useful property from this assumption:

$$\begin{equation} \begin{aligned} \mathbb{Cov}(U,X) &= \mathbb{E}(UX) - \mathbb{E}(U)\mathbb{E}(X) \\[6pt] &= \mathbb{E}(X \cdot\mathbb{E}(U | X)) - \mathbb{E}(U)\mathbb{E}(X) \\[6pt] &= \mathbb{E}(X \cdot 0) - 0 \cdot\mathbb{E}(X) = 0. \end{aligned} \end{equation}$$

This tells use that the conditional mean assumption implies that the marginal mean of the errors is zero, and the error terms are also uncorrelated with the explanatory variables.

Note that this is a stronger assumption than if we just assume that $\mathbb{E}(U) = 0$. That is, you can have situations where the error terms have a marginal mean of zero, even though they are correlated with the explanatory variable $X$.

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