# Real-life examples of moving average processes

Can you give some real-life examples of time series for which a moving average process of order $q$, i.e. $$y_t = \sum_{i=1}^q \theta_i \varepsilon_{t-i} + \varepsilon_t, \text{ where } \varepsilon_t \sim \mathcal{N}(0, \sigma^2)$$ has some a priori reason for being a good model? At least for me, autoregressive processes seem to be quite easy to understand intuitively, while MA processes do not seem as natural at first glance. Note that I am not interested in theoretical results here (such as Wold's Theorem or invertibility).

As an example of what I am looking for, suppose that you have daily stock returns $r_t \sim \text{IID}(0, \sigma^2)$. Then, average weekly stock returns will have an MA(4) structure as a purely statistical artifact.

• I really like this question! I have not read of any example in the literature. I will answer something that might be of interest. – Ric Dec 3 '12 at 21:30
• – Stephan Kolassa Mar 15 '18 at 7:55

One very common cause is mis-specification. For example, let $y$ be grocery sales and $\varepsilon$ be an unobserved (to the analyst) coupon campaign that varies in intensity over time. At any point in time, there may be several "vintages" of coupons circulating as people use them, throw them away, and receive new ones. Shocks can also have persistent (but gradually weakening) effects. Take natural disasters or simply bad weather. Battery sales go up before the storm, then fall during, and then jump again as people people realize that disaster kits may be a good idea for the future.

Similarly, data manipulation (like smoothing or interpolation) can induce this effect.

I also have "inherently smooth behavior of time series data (inertia) can cause $MA(1)$" in my notes, but that one no longer makes sense to me.

• If I am not mistaken, what you say seems to apply to any kind of dynamic misspecification. Of course, that can be dealt with by using some ARMA model for the error terms. From what you wrote above, I do not see any particular reason to believe that weather shocks or coupon campaigns have an MA(q) structure. Am I missing something? – weez13 Dec 4 '12 at 21:25
• Tell me if this makes sense. At time 1, we have 100 unobserved coupons and assume the take-up rate is always 50% ($\theta_1$). So 50 incremental sales will take place at that time. At time 2, we have 80 new coupons and 50 remaining ones from last period. This gives you $40 + 25=0.5 \cdot 80 + 0.5^2 \cdot 100$ bonus sales. Combine that with an assumption about coupon expiration, and you get a finite $MA(q)$ process. – Dimitriy V. Masterov Dec 4 '12 at 22:08
• Thanks, I think I see it now! I suppose the key point I failed to see before is that there exists an "expiration date" for the coupons, which kills serial correlation after some lag $q$. – weez13 Dec 5 '12 at 14:17
• From the point of view of a learner, I really don't understand this example : grocery sales, coupons (what kind of coupon?), "vintages" (?), shocks, disasters, battery sales, disaster kits? I don't get the big picture of this example. (Maybe it's because I'm not native english...) – Basj Jan 28 '16 at 21:12
• @Basj In the US, stores and manufacturers frequently issue coupons that can be redeemed for a financial discount or rebate when purchasing a product. They are often widely distributed through mail, magazines, newspapers, the internet, directly from the retailer, and mobile devices such as cell phones. Most coupons have an expiration date after which they will not be honored by the store, and this is what produces "vintages". Coupons possibly boost sales, but how many there are out there or how big the rebate is not always known to the data analyst. You can think of them a positive errors. – Dimitriy V. Masterov Jan 28 '16 at 21:51

in our article Scaling portfolio volatility and calculating risk contributions in the presence of serial cross-correlations we analyze a multivariate model of asset returns. Due to different closing times of the stock exchanges a dependence structure (by the covariance) appears. This dependence only holds for one period. Thus we model this as a vector moving average process of order $1$ (see pages 4 and 5).

The resulting portfolio process is a linear transformation of a $VMA(1)$ process which in general is an $MA(q)$ process with $q\ge1$ (see details on pages 15 and 16).