<p><span class="math-container">$u_t$</span> being mean 0 does not imply convergence of <span class="math-container">$\sum_{i=0}^\infty u_{t+i}$</span> because the variance will probably not converge to 0.</p>
<p>I feel like there should be more details given in this question. If they did not give you any details, then you would say that <span class="math-container">$Var\left(\sum_{i=0}^\infty u_{t+i}\right) = \sum_{i=0}^\infty Var(u_{t+i})$</span> which does not converge to 0 (I am assuming that <span class="math-container">$u_t$</span> is serially uncorrelated, otherwise you would have covariance terms).</p>
<p>EDIT: The <span class="math-container">$u$</span> are definitely serially uncorrelated because they are IID.</p>

CC BY-SA 4.0
2022-02-11T21:18:24.280Z
OP did say it's IID, which I interpret as serially uncorrelated.
Acccumulation
CC BY-SA 4.0
2022-02-13T08:02:27.033Z
I should've noticed, thank you. IID is even stronger than serially uncorrelated.
Michael Gmeiner
CC BY-SA 4.0
2022-02-11T12:14:08.653Z
2022-02-14T07:44:33.460Z
2022-02-14T07:44:33.460Z
Michael Gmeiner
<p>In the context of rational expectations, i.i.d zero mean error means that the on expectation you expect <span class="math-container">$u_{t+i}$</span> to be zero for any time in the future. I suspect that when deriving the general pattern you forgot to include the <span class="math-container">$E_t$</span> somewhere. In that case, you will get the convergence as desired.</p>

CC BY-SA 4.0
2022-02-12T03:55:54.063Z
2022-02-12T03:55:54.063Z
Rumi
<p>I am working on a question where I have derived a general pattern for my variable of interest <span class="math-container">$x$</span> in terms of the error term <span class="math-container">$u$</span> across time:</p>
<p><span class="math-container">$x_t = \sum_{i=0}^{\infty} u_{t+i}$</span>. The only information I have is that <span class="math-container">$u_t$</span> is i.i.d random error of zero mean. What implications does that have in terms of convergence for the series? I arrive at the correct solution if <span class="math-container">$\sum_{i=0}^{\infty} u_{t+i} = u_t$</span>.</p>
<p>What does zero mean error exactly mean here? I thought you would have the same error across all times but this seems to suggest that error for any time besides <span class="math-container">$t$</span> is zero.</p>

CC BY-SA 4.0
2022-02-11T11:22:18.057Z
I'm confused looking at the process $x$, this process is not adapted to the natural filtration generated by $u$.
Q9y5
CC BY-SA 4.0
2022-02-12T03:44:29.077Z
Is this a Macroeconomics problem? I would guess rational expectations based?
Rumi
<econometrics>
What does Zero Mean Error mean?
Kinno
# What does Zero Mean Error mean?

(Asked by: username_0 on 2022-02-11)

I am working on a question where I have derived a general pattern for my variable of interest $x$ in terms of the error term $u$ across time:

$x\_t = \sum\_{i=0}^{\infty} u\_{t+i}$. The only information I have is that $u\_t$ is i.i.d random error of zero mean. What implications does that have in terms of convergence for the series? I arrive at the correct solution if $\sum\_{i=0}^{\infty} u\_{t+i} = u\_t$.

What does zero mean error exactly mean here? I thought you would have the same error across all times but this seems to suggest that error for any time besides $t$ is zero.

**Tags:** econometrics

> **Comments:**
> **username_1**: I'm confused looking at the process $x$, this process is not adapted to the natural filtration generated by $u$.  
> **username_2**: Is this a Macroeconomics problem? I would guess rational expectations based?  

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## 2 answers

### Answer by username_2 on 2022-02-12. Score: 2 | Accepted answer

In the context of rational expectations, i.i.d zero mean error means that the on expectation you expect $u\_{t+i}$ to be zero for any time in the future. I suspect that when deriving the general pattern you forgot to include the $E\_t$ somewhere. In that case, you will get the convergence as desired.


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### Answer by username_3 on 2022-02-11. Score: 4

$u\_t$ being mean 0 does not imply convergence of $\sum\_{i=0}^\infty u\_{t+i}$ because the variance will probably not converge to 0.

I feel like there should be more details given in this question. If they did not give you any details, then you would say that $Var\left(\sum\_{i=0}^\infty u\_{t+i}\right) = \sum\_{i=0}^\infty Var(u\_{t+i})$ which does not converge to 0 (I am assuming that $u\_t$ is serially uncorrelated, otherwise you would have covariance terms).

EDIT: The $u$ are definitely serially uncorrelated because they are IID.

> **Comments:**
> **username_4**: OP did say it's IID, which I interpret as serially uncorrelated.  
> **username_3**: I should've noticed, thank you. IID is even stronger than serially uncorrelated.  

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