Bar 03 · Dependence

Stationarity and autocorrelation

A series is stationary when its probabilistic character does not move as the clock advances. This page separates weak from strict stationarity, defines the autocovariance and the autocorrelation function, follows their decay, and explains how differencing can rescue a wandering series.

Weak stationarity

Weak stationarity asks three things of a series. The mean must be the same at every time. The variance must be the same at every time. And the covariance between two times must depend only on the gap between them and not on where they sit. A series with these three properties is stationary in the wide sense, and its behaviour may be described by a single covariance function of one argument, the lag.

Weak stationarity is a modest requirement and a very useful one. It permits a record to be summarised by its mean, its variance and a curve of covariances, and it makes the spectral account given on the next page possible. Many series that look irregular at first glance are stationary once a trend has been removed.

Strict stationarity

Strict stationarity is stronger. It requires the entire joint law of any collection of values to be unchanged when all the times are shifted by the same amount. Strict stationarity need not give a finite variance, so it does not by itself imply weak stationarity; the two conditions coincide for the Gaussian family, where the law is fixed by its first two moments alone.

The autocovariance function

The autocovariance at lag k is the covariance between a value and the value k steps later. At lag zero it reduces to the variance. As a function of the lag it is even, since the covariance between two times does not care which is named first, and it can never exceed the variance in magnitude. A sequence of numbers can serve as an autocovariance function only if it is even and if every finite block built from it is positive semi-definite; this is the same requirement met earlier for a covariance matrix, now written in the time dimension.

The autocorrelation function

Dividing the autocovariance at each lag by the variance standardises the curve so that it begins at one for lag zero and lies within minus one and one elsewhere. The result is the autocorrelation function, and it is the customary summary of linear memory in a series. A curve that falls quickly signals a short memory; a curve that decays slowly signals a long one.

Figure 1. An autocorrelation function at successive lags. The stems fall toward zero, showing memory that fades as the lag grows.

Decay and memory

The speed of decay is the practical content of stationarity. Rapid geometric decay, as in a stable autoregressive series, means that values far apart are nearly unrelated. Slow decay, as in a long-memory series, means that dependence lingers across many lags and the sample mean converges unusually slowly. The shape of the curve, not merely its height at one lag, is what carries the information.

Differencing

A series with a drifting level is not stationary, but its successive differences often are. Differencing removes a linear trend and turns many wandering walks into stationary series to which the usual summaries apply. Over-differencing, by contrast, injects artificial memory, so the step is taken once and then checked rather than repeated by habit.

Once a series is stationary, its memory may be viewed in a second way, through the frequencies that compose it. That is the subject of the spectral page.