# How is ARMA/ARIMA related to mixed effects modeling?

In time series analyses, I have used multi-level or random/mixed effects to deal with auto-correlation issues (i.e., observations are clustered within individuals over time) and added controls are added for some specification of time and for shocks of interest. ARMA/ARIMA seem designed to address similar issues.

The resources I've found online discuss either ARMA/ARIMA or mixed effect models but I don't understand the relationship between the two. Might one want to use ARMA/ARIMA from within a multilevel model? Is there a sense in which the two are equivalent or redundant?

Answers or pointers to resources that discuss this would be great.

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I think the simplest way to look at it is to note that ARMA and similar models are designed to do different things than multi-level models, and use different data.

Time series analysis usually has long time series (possibly of hundreds or even thousands of time points) and the primary goal is to look at how a single variable changes over time. There are sophisticated methods to deal with many problems - not just autocorrelation, but seasonality and other periodic changes and so on.

Multilevel models are extensions from regression. They usually have relatively few time points (although they can have many) and the primary goal is to examine the relationship between a dependent variable and several independent variables. These models are not as good at dealing with complex relationships between a variable and time, partly because they usually have fewer time points (it's hard to look at seasonality if you don't have multiple data for each season).

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:Peter Very nice summary. I would only add that time series data is not usually "long" when dealing with weekly/monthly/annual data BUT can get long when dealing with daily/hourly/second data. –  IrishStat Oct 26 '11 at 12:53
Your explanation is quite good, in practice, though I would add a slight caveat. ARIMA models can be implemented as State Space models (R's arima does this, under the hood), also known as Dynamic Linear models (DLMs). DLMs are also extensions from regression (in a different way than Mixed Effects), so I'd guess that there is a deep-down relationship between ARIMA and Mixed-effect models. That doesn't change the differences in practice, which you summarize well. –  Wayne Oct 26 '11 at 23:16
This is very useful. I will point that out that adding a moving average to a multi-level model is certainly possible (and, in the simplest form, is done all the time by adding lagged variables (e.g., the dependent variable at $t-1$). –  Benjamin Mako Hill Oct 27 '11 at 19:26
Benjamin: The whole idea of statistics is to IDENTIFY STRUCTURE not assume it. –  IrishStat Oct 27 '11 at 22:51