# What does Zero Mean Error mean?

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.

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

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.
$$u_t$$ 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.