Supposing I have a stock return series from 2000-2013.

Looking at the data, it has pattern of long trending bull market: 2001-2008, and 2010-2013, while it also has great reversal in 2008 financial crisis. So if we denote the daily return series as $y_{t}$, there are two properties:

(1) $y_{t}$ has a non-zero mean part in the return series, denoted as $r_{t}$, which is something like a "macro factor". (2) $r_{t}$ can be correlated with $r_{t-1}, r_{t-2},... $, during the good years, or negative correlated during bad years.

Typically what people would do using ARMA model, is firstly estimate $r_{t}$ as $r_{t} = (y_{t-1} + ... + y_{t-k})/N $, and feed the series $y_t - r_t$ into ARMA identification toolbox to get the coefficients.

However, it really turns out that picking $k$ can be quite critical : the $k$ value tells how much we believe the past trend, which is really hard to say. For example, at 2008 when the financial crisis happen, you have no reason to believe the market would continue as the previous good years; while at the good years, you would assume the global economy works well and you can use a long $k$ window. Or maybe it is best not to make the model depend on choosing $k$.

So back to the question regarding time series modeling: What people usually do if the time series has a non-zero mean, which is quite different at different time point $t$ ?

  • $\begingroup$ You can try taking the first difference, run an ARIMA, and inspect the ACF afterward to see if it worked. That should work better than demeaning the series using rt $\endgroup$
    – AOGSTA
    Jul 24, 2013 at 20:24

1 Answer 1


Yes. Box-Jenkins ignored deterministic variables like trend when they laid out their ARIMA approach. Here is a blog post that speaks to your issue of identifying trends and searching for them as opposed to just leaning on ARIMA.


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    $\begingroup$ I am surprised to see such a dataset (age of kings at death) analyzed using methods intended for time series with regular intervals, because it differs fundamentally from them: it consists of interdependent events rather than observations at a given sequence of times. Not only is there then no reason to suppose that standard time series methods will be much good (except by accident or luck), one would expect that methods of data analysis more appropriate for such data would perform better. $\endgroup$
    – whuber
    Jul 24, 2013 at 20:38
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    $\begingroup$ Agreed. It comes from Rob Hyndman's time series data library found here robjhyndman.com/tsdldata/misc/kings.dat Yes, I point that in the blog. So, imagine it is just pizza sales and that pain all goes away. $\endgroup$
    – Tom Reilly
    Jul 25, 2013 at 11:27

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