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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.

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  • $\begingroup$ I'm confused looking at the process $x$, this process is not adapted to the natural filtration generated by $u$. $\endgroup$
    – Q9y5
    Feb 11, 2022 at 11:22
  • $\begingroup$ Is this a Macroeconomics problem? I would guess rational expectations based? $\endgroup$
    – Rumi
    Feb 12, 2022 at 3:44

2 Answers 2

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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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$u_t$ 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.

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  • $\begingroup$ OP did say it's IID, which I interpret as serially uncorrelated. $\endgroup$ Feb 11, 2022 at 21:18
  • $\begingroup$ I should've noticed, thank you. IID is even stronger than serially uncorrelated. $\endgroup$ Feb 13, 2022 at 8:02

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