Measure of central tendency in ARIMA

Question

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?

Answer 1

[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).