Stationary Process

Basic Concepts

A time series is stationary if the properties of the time series (i.e. the mean, variance, etc.) are the same when measured from any two starting points in time. Time series which exhibit a trend or seasonality are clearly not stationary.

We can make this definition more precise by first laying down a statistical framework for further discussion.

Definitions

Definition 1: A stochastic process (aka a random process) is a collection of random variables ordered by time.

In economics, GDP and corporate profits (by year) can be modeled as stochastic processes. In biology the number of elephants in the wild, in meteorology the average temperature of the planet, in medicine the number of Ebola cases, etc. can all be modeled as stochastic processes.

Thus, if we are interested in GDP from 2001 until 2015, we can define the random variables y_i = the GDP in 2000 + i, and so the series y₁, y₂, …, y₁₅ is a stochastic process.

Corresponding to the individual populations of the random variables in a stochastic process are the samples for each random variable. Any such realization of samples is called a time series. Note that the sample of each random variable in a time series contains just one element.

Definition 2: A stochastic process is stationary if the mean, variance and autocovariance are all constant; i.e. there are constants μ, σ and γ_k so that for all i, E[y_i] = μ, var(y_i) = E[(y_i–μ)²] = σ² and for any lag k, cov(y_i, y_i+k) = E[(y_i–μ)(y_i+k–μ)] = γ_k.

A time series is stationary if the above properties hold for the time series (in the same way as we extend properties of a population to its samples). We will make this more precise shortly.

Observation: The above definition of stationary is what is usually called weakly stationary, but fortunately it is sufficient for our purposes. A stochastic process is truly stationary if not only are the mean, variance, and autocovariances constant, but all the properties (i.e. moments) of its distribution are time-invariant.

Example

Example 1: Determine whether the Dow Jones closing averages for the month of October 2015, as shown in columns A and B of Figure 1 is a stationary time series.

Figure 1 – Dow Jones Time Series

As you can see from Figure 1, there is an upward trend to the data. This is an indication that the time series is not stationary.

We now take the first differences of the Dow Jones closing averages on consecutive days, as shown in Figure 2.

Figure 2 – Differences of Lag 1

Here cell N5 contains the formula =M5-M4 and similarly for the other cells in column N. This time the chart shows what looks like a random pattern. This is indicative of a stationary time series.