White noise terms in moving average model

As many before, I lack the clear intuition behind the Moving Average model. Eventhough I read quite a few threads on CV.

The Moving Average $$MA(q)$$ model consists of a constant and White Noise terms.

$$X_{t}=μ+ε_{t}+θ_{1}ε_{t−1}+⋯+θ_{q}ε_{t−q}$$

By definition White noise terms are uncorrelated with zero mean and constant variance. Given that white noise is uncorrelated with respect to past realizations, makes it unpredictable. So why do we bother to include them in the model?

Assuming the DGP follows an $$MA(q)$$ process:

Why does it make sense to relate white noise terms with eachother ?

How can $$ε_{t}$$ be White noise if it is by definition related to its past value $$ε_{t-1}$$? Since:

$$X_{t}=ε_{t}+θ_{1}ε_{t−1}$$

$$ε_{t}=θ_{1}ε_{t−1}-X_{t}$$

• Let us assume I have two independent white noise processes $e_t$ and $\epsilon_t$. I can add them: $\eta_t = e_t + \epsilon_t$. Just because I can add them - the realizations are just numbers, after all - doesn't mean they are related to each other (i.e., not independent,) even though I can write $e_t = \eta_t - \epsilon_t$. Commented Oct 22, 2019 at 21:56
• The sum of the two independent white noise does not mean they are related. I understand that. However, why would you bother take some part of a previous White noise $\theta_1\epsilon_{t-1}$ for the estimation of the next time step, knowing that White noise terms are unrelated over time? Commented Oct 23, 2019 at 7:59
• If a hurricane hits Houston, there is going to be an impact on sales at, say, Target. From a high level, this is a big negative $e_t$ for that week. Next week, there will be another weekly innovation, $e_{t+1}$, but the effect of the hurricane may not have fully worn off at the week boundary, so there will be some carryover effect on sales from the previous week's innovation. Not a perfect example, but in the right general direction, I think. Commented Oct 23, 2019 at 13:29
• Thanks for the clarification. An (unpredictable) shock may have a longer impact than at one time instance. The $\mu$ is a constant (mean, presumably) and does not vary over time. So basically, an MA process models a single constant (flat horizonal line) with some unpredictable variations (white noise) around, right? Commented Oct 23, 2019 at 14:12
• Right, and the effect of the variations may cross the boundaries dividing continuous time into discrete chunks. So, if you want to forecast "next week" (to continue my example), having an estimate of this week's innovation is helpful. Commented Oct 23, 2019 at 14:28