<p>Essentially, <span class="math-container">$Cov(u,W) = 0$</span> is implied by <span class="math-container">$E(u|W) = E(u)$</span> by the <a href="https://stats.stackexchange.com/questions/306620/conditional-expectation-and-covariance">Law of iterated expectations</a></p>

CC BY-SA 4.0
2018-12-05T10:40:12.210Z
In general, uncorrelated random variables need not be mean independent. Unless you are talking about an asymptotic case when they are normal......?
Student
CC BY-SA 4.0
2018-12-06T21:04:30.427Z
I think $\mathrm{Cov}(u,W)=0$ is *implied by* $E(u|W)=0$, not the other way around.
Herr K.
CC BY-SA 4.0
2019-01-05T17:20:34.407Z
you are right, I edited this
E. Sommer
CC BY-SA 4.0
2018-12-05T09:13:46.527Z
2019-01-05T17:21:01.833Z
2019-01-05T17:21:01.833Z
E. Sommer
<p>My professor states the validity condition in the following way:
<span class="math-container">$W$</span> (the matrix of instrumental variables) must be such that 
<span class="math-container">$ \text{plim} \, \frac{W'u}{N}=0$</span>, where <span class="math-container">$u$</span> is the error term.
Intuitively, I can see that in some way, this relates to the covariance between <span class="math-container">$W$</span> and <span class="math-container">$u$</span>, but am not entirely convinced.
Moreover, why do we look at this condition in an asymptotic setting? Why not impose mean independence, i.e., <span class="math-container">$E(u|W)=E(u)$</span>?</p>

<econometrics><instrumental-variable>
Validity condition for Instrumental Variables
Student
# Validity condition for Instrumental Variables

(Asked by: username_0 on 2018-12-05)

My professor states the validity condition in the following way:
$W$ (the matrix of instrumental variables) must be such that
$ \text{plim} \, \frac{W'u}{N}=0$, where $u$ is the error term.
Intuitively, I can see that in some way, this relates to the covariance between $W$ and $u$, but am not entirely convinced.
Moreover, why do we look at this condition in an asymptotic setting? Why not impose mean independence, i.e., $E(u|W)=E(u)$?

**Tags:** econometrics, instrumental-variable


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## 1 answers

### Answer by username_1 on 2018-12-05. Score: 2 | Accepted answer

Essentially, $Cov(u,W) = 0$ is implied by $E(u|W) = E(u)$ by the [Law of iterated expectations](https://stats.stackexchange.com/questions/306620/conditional-expectation-and-covariance)

> **Comments:**
> **username_0**: In general, uncorrelated random variables need not be mean independent. Unless you are talking about an asymptotic case when they are normal......?  
> **username_2**: I think $\mathrm{Cov}(u,W)=0$ is *implied by* $E(u|W)=0$, not the other way around.  
> **username_1**: you are right, I edited this  

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