Okay. So, your time series is loaded into your favourite statistical package. What now? Probably the first thing you need to do is produce a plot of your time series. The plot will give you an idea of the overall levels and variability of the series. The plot will give you an idea of any trends or seasonality in the series. This kind of evaluation is part of an initial data analysis and an excellent description can be found in Chapter 2 of Chatfield, listed in the references. After trend and seasonality are assessed they are often removed and the residuals are then further analyzed for stochastic structure.
Often, the next step commonly advocated is to compute autocorrelations or autocovariance (again, see the brief introduction to stationary series for more details on this). However, these statistics rely on the assumption that the series is stationary, i.e. has statistical properties that do not change with time.
We were a little vague in the introduction about the definition of a stationary series. In fact, there are different, related (and more technical) definitions of stationarity. We will attempt to describe them here in informal terms.
Strict stationarity is the strongest form of stationarity. It means that the joint statistical distribution of any collection of the time series variates never depends on time. So, the mean, variance and any moment of any variate is the same whichever variate you choose. The formal mathematical definition of strictly stationary series can be found on the Wiki page. However, for day to day use strict stationarity is too strict. Hence, the following weaker definition is often used instead.
For everyday use we often consider time series that have:
Such time series are known as second-order stationary or stationary of order 2.
From now on, whenever we mention stationarity, we mean second-order stationarity.
Note: it is possible to consider a weaker form of stationarity still: a series that is first-order stationary which means that the mean is a constant function of time. Economists are very keen on this kind of stationarity, particularly in how to combine time series with time-varying means to obtain one which is second-order stationary (for example). This latter concept is known as cointegration which is important and famous.
So far we have explained that stationarity (second order or strict) is about imposing constancy of certain time series quantities. Why is this a useful concept? Certainly, much data seem to obey this rule in that future statistical behaviour is identical to past behaviour.
On the other hand, much data is not stationary or at least only approximately stationary. A real problem is that although there are tests for stationarity we submit that they are not used much in practice. Why is this?
Why do analysts persist with stationary models that are not appropriate and potentially risky? We offer four reasons. 1. Fear of diversity. There is a single mathematical model (the Fourier-Cramer model) for stationary time series. For nonstationary series the situation can be complex and the diversity of potential models can be daunting. 2. Education. Many undergraduate time series courses only have time or the ambition to consider stationary models, and 3. Mathematical expediency. Stationary models are mathematically easier to study and develop asymptotic theories for (that is, mathematically we understand how our modelling works for larger and larger samples). 4. Maturity. Stationary theory is mature, widely known and widely applicable.
However, it is the case that many real time series are just not stationary. Series often display trends (invalidating first-order stationarity), or seasonalities, or changes in variance (invalidating second-order stationarity). Hence, statistics and related fields have a second armoury of techniques that can manipulate time series to become stationary (differencing, variable transformations such as taking logs or square roots). After manipulation the series can be treated as stationary and standard methods used.