Why You Should Test

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.

The plot left shows a series recorded on an explosion (details below). Is it stationary? It certainly looks not, but can we be more objective about this statement?

Precisely, what do we mean by "stationarity"?

Strict Stationarity

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.

Stationarity of order 2

For everyday use we often consider time series that have:

• a constant mean

• a constant variance

• an autocovariance that does not depend on time.

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 keen on this kind of stationarity, particularly in how to combine time series with time-varying means to obtain one which is first-order stationary (for example). This latter concept is known as cointegration. Tests, continued>