Meaning of Moving Average Term in ARIMA

Question

I am working on a forecasting project and want to reaffirm my knowledge on different techniques before blindly hitting run in my Python code. I am testing several forecasting techniques such as exponential smoothing, Holt-Winters Methods, and Box-Jenkins/ARIMA modeling as there is not excellent exogenous data to perform other forms of forecasting. I am basing my research off of Bowerman, O'Connell, and Koehler's "Forecasting, Time Series, and Regression" Fourth Edition. I have a good grasp of what the autoregressive(AR) term represents according to the function $z_t=\phi_1z_{t-1}+\alpha_1$. This shows that an AR process is one in which future values are constant multiples of past values.

The book describes moving average (MA) processes, however, as "less intuitive but equally useful" and I seem to agree. Following the formula $z_t=\alpha_t-\theta_1\alpha_{t-1}$, the best I can interpret to mean is that future values are based on the errors of past estimates. Is this interpretation correct, and if so, what does that mean about the data if it relies on past errors? For AR, this means that the data is directly related to the past values, but I cannot connect the dots to what conclusion I should draw about the existance of a MA process in my data.

Answer 1

For time series, it is better to think about the $\alpha_t$ terms (supposed to be an independent mean zero series) as innovations, not errors. They are not modeling errors of measurements, but the really new information that arrives in the series, not predictable from the history of the series.

So a pure AR series depends on former values, while a pure MA series depends on former innovations. Natural examples can be found in this CV post. The following example

When trying to get an intuitive real world picture of MA or AR (or ARMA or ARIMA if you are extending it) I often find it useful to think of carry over effects, that is something happening in one period carries over into the next.

Here's an example: say you are modelling newspaper sales. The noise (random error) in such a model could sensibly incorporate the relatively short lived effect of newspaper headlines while the rest of the model deals with more stable things like trend and seasonality (now I'm assuming an ARIMA model but if you want a pure MA model imagine no trend or seasonality for the paper). Although the newspaper headline effect is modelled as error we might decide that this effect does indeed carry over into the next few days (a good story brings in readers who then fade away again). This would invite the inclusion of an MA term in the model - the carryover of the effect of the previous error term into the current time period.

You can think in the same way about the AR term only what is carried over here is part of the effect of the whole of the previous days sales.

is taken from an answer here.

Answer 2

Kjetil's answer is very good, and the distinction between errors and innovations are important to understand. But there are also applications where measurement issues do result in MA-type errors.

For example, statistical time series are often collected from a rotating sample panel. In something like an employment survey, households might be selected for the sample for eight months at a time; each month data on employment status is collected from those households, then one-eighth of the sample is "rotated out" and replaced by newly-selected households.

This approach gives both operational advantages (it's more trouble to recruit a new household than to re-interview one that you've already contacted) and statistical advantages (greatly reduces the effects of sample churn on month-to-month movements, improving ability to detect real changes).

It also results in MA(8) behaviour: if you select an unusual group of households in one month, then those households will affect the time series for eight months before suddenly disappearing from the time series. Many economic time series are subject to similar phenomena that can cause MA-type behaviour.