# Can time series be used to predict series which has a changing trend?

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 $t$ ?

• 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 – AOGSTA Jul 24 '13 at 20:24