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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.

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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.

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    $\begingroup$ 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? $\endgroup$
    – weez13
    Dec 4 '12 at 21:25
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    $\begingroup$ 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. $\endgroup$ Dec 4 '12 at 22:08
  • $\begingroup$ 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 $q$. $\endgroup$
    – weez13
    Dec 5 '12 at 14:17
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    $\begingroup$ 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...) $\endgroup$
    – Basj
    Jan 28 '16 at 21:12
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    $\begingroup$ @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. $\endgroup$ Jan 28 '16 at 21:51
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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).

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It is true that MA processes are more difficult to explain to users than AR processes. However they are very ubiquitous. The most common MA type of the process that you didn't know about is a low pass filter. enter image description here

The active versions would be a "TREBLE" knob on your car stereo, or a tone control knob on your guitar.

enter image description here

Here's how the most primitive passive RC series circuit works. At high frequencies it integrates: $$V_C \approx \frac{1}{RC}\int_{0}^{t}V_\mathrm{in}\,dt\,,$$ You should recognize the continuous time version of the MA process in this equation. The reason why this happens is because capacitor's impedance changes with frequency of input.

The filter is called a low pass because at low frequencies it doesn't integrate, and lets them pass as is: $$V_\mathrm{in} \approx V_C$$

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Suppose you are producing some good, stockpiling some of it and selling the rest. Your production in time period $t$ is $x_t=m+\varepsilon_t$ with $\mathbb{E}(\varepsilon_t)=0$ and your stock is $y_t$. The sequence of $\varepsilon$s is i.i.d. A $1-\theta$ fraction of the period's production is sold during the next period, and the remaining $\theta$ during the one after that. Then your stockpile is \begin{aligned} y_t&=x_t+\theta_1x_{t-1} \\ &=\mu+\varepsilon_t+\theta_1\varepsilon_{t-1}, \end{aligned} where $\mu=(1+\theta_1)m$. Thus, $y_t$ follows and MA(1) process.

If it took a longer time ($q+1$ periods instead of $2$ periods) to sell a period's production, you would have an MA(q) process.

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Consecutive multiple-step-ahead forecast errors from optimal forecasts will be MA processes.

For example, suppose the data generating process is a random walk: $X_t=X_{t-1}+\varepsilon_t$ where $\varepsilon_t\sim\text{i.i.d.}(0,\sigma_\varepsilon^2)$. If you are at time $t$ predicting the value of the process at time $t+3$, the optimal forecast is $X_t$. The forecast error is therefore $e_{t+3|t}=X_{t+3}-X_t=\varepsilon_{t+3}+\varepsilon_{t+2}+\varepsilon_{t+1}$. If you repeat the forecasting exercise at time $t+1$, you have the optimal prediction $X_{t+1}$ and the forecast error $e_{t+4|t+1}=X_{t+4}-X_{t+1}=\varepsilon_{t+4}+\varepsilon_{t+3}+\varepsilon_{t+2}$.

Now $e_{t+3|t}$ and $e_{t+4|t+1}$ will be (positively) correlated because they share two elements, $\varepsilon_3$ and $\varepsilon_2$. Similarly, $e_{t+3|t}$ and $e_{t+5|t+2}$ will be (positively) correlated because they share an element $\varepsilon_3$. $e_{t+3|t}$ and $e_{t+6|t+3}$ will however not be correlated because there is no shared element and $\varepsilon_t$ is an i.i.d. sequence.

The fact that autocorrelations cuts off abruplty after several periods is characteristic of MA processes. Indeed, it is not difficult to show that the sequence of consecutive 3-step-ahead forecast errors $(e_{t+3|t},e_{t+4|t+1},e_{t+5|t+2},\dots)$ is an MA(2) process. More generally, when predicting $h$ steps ahead, consecutive errors from an optimal forecast form an MA($q$) process with $q\leq h-1$. (The precise value $q$ depends on the memory of the process being forecast. For a random walk, $q=h-1$; for some processes with shorter memory, $q<h-1$. For a process with no memory, $q=0$.)

Processes of consecutive multiple-step-ahead forecast errors are common. You see them in macroeconomics (long-term forecasts of GDP, inflation, unemployment, etc.), finance (forecasts of asset returns, currency exchange rates, etc.) and beyond. While hardly any of the forecasts are optimal, some are close to that, and their forecast errors will resemble MA processes quite closely. An example could be the random-walk based multiple-step-ahead forecast of daily stock prices as detailed above.

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    $\begingroup$ A downvote on a textbook example of an MA process? I would appreciate some constructive comments. $\endgroup$ May 7 at 11:50
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Increments of cumulative processes measured over overlapping periods of time are MA processes when increments are i.i.d. If $$ x_t=\sum_{\tau=0}^t\varepsilon_\tau $$ where $\varepsilon_\tau\sim i.i.d.$, then $$ (x_t-x_{t-s},x_{t+1}-x_{t+1-s},\dots)=(\sum_{\tau=s}^t\varepsilon_\tau,\sum_{\tau=s+1}^{t+1}\varepsilon_\tau,\dots) $$ is an MA($s-1$) process.

A prime example of an approximate* MA process is multi-period asset returns (concretely, price changes). E.g. a daily series of yearly returns on a stock has a one-year-minus-one-day overlap between consecutive observations.

  • The yearly return on January 2 is the cumulative return from January 3 the previous year through January 2 the current year.
  • The yearly return on January 3 is the cumulative return from January 4 the previous year through January 3 the current year.
  • Etc., etc.

If there are $252$ trading days a year, we have an MA($252-1$) process. (This can be seen e.g. from inspecting the theoretical autocorrelation function of the series; the autocorrelation cuts off at lag $251$.) The same holds for logarithmic returns, as they are additive just as price changes are. If we look at percentage returns on the other hand, these may be approximated by MA processes as long as the price level is not too close to zero and does not vary too much over the course of the sample. The latter conditions ensure that percentage returns are approximately proportionate to price changes (which are MA processes themselves, as pointed out above).

Similar examples could be macroeconomic processes such as year-on-year quarterly GDP growth that could be considered approximately MA($3$).

*Asset returns are not strictly i.i.d. but usually not very far from that, so MA may be a good approximation for overlapping cumulative returns.

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    $\begingroup$ As is always the case with downvotes, I would appreciate some constructive comments. $\endgroup$ May 7 at 11:50

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