Measure of central tendency in ARIMA

I know very little about the Auto-Regressive Integrated Moving Average model (ARIMA) but I am interested in what type of central tendency is represented by "average". Is it simply the mean? If so, is it possible to use other measures such as the mode, median or even another type?

• The title does not quite match the body. Also, did you mean central tendency? – Richard Hardy Apr 7 '16 at 14:47
• Whilst creating this question clicked enter outside of the text box and it saved down. Going to flesh this out! – josh Apr 7 '16 at 14:48
• @RichardHardy Yes. I assume Central Tenancy would be renting an inner building of a set of terraced houses. – josh Apr 7 '16 at 14:54

[W]hat type of central tendency is represented by "average"[?]

Let us forget about the letter "I" in ARIMA and consider stationary ARMA models in the beginning. Let $y_t$ be a time series satisfying

$$y_t = \alpha + \varphi_1 y_{t-1} + \dotsc + \varphi_p y_{t-p} + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \dotsc + \theta_q \varepsilon_{t-q}$$

where $\varepsilon_t$ are zero-mean $i.i.d$ random variables. Hence, $y_t$ is a linear combination of a constant, own lagged values $y_{t-1}$ through $y_{t-p}$, random shocks/innovations/errors $\varepsilon_t$ and their lagged values $\varepsilon_{t-1}$ through $\varepsilon_{t-q}$. This constitutes an ARMA($p$,$q$) model.

An ARIMA model (with "I" in the middle) is obtained when cumulatively summing up a process that itself is an ARMA process. E.g. $z_t:=\sum_{\tau=1}^t y_{\tau}$ will be an ARIMA($p$,1,$q$) process, $w_t:=\sum_{\tau=1}^t z_{\tau}$ will be an ARIMA($p$,2,$q$) process, etc.

As you can see, there is no simple average here even though there is the word "average" in the acronym. There is a weighted sum, though, which by nature is somewhat similar to a simple average, but it differs from it since the weights are generally unequal and do not sum to one.

See this thread for some more details and references, including where the name might have originated from (Slutsky "The Summation of Random Causes as a Source of Cyclical Processes", 1927).

• Thanks for the SLUTSKY reference . He and Yule are in my pantheon of revered elders ..... – IrishStat Apr 7 '16 at 21:29
• The reference is due to @GraemeWalsh (in the other thread). – Richard Hardy Apr 8 '16 at 5:00