Basic Concepts in Time Series Analysis (Part-4)

This post is the fourth part of a series of tutorials regarding time series analysis. Here is the first part discussing about the definition of times series, and here is the second part discussing on time series components and here is the third part.

Stationarity

A stationary time series is one whose properties do not depend on the time at which the series is observed. More precisely, if y_t is a stationary time series, then for all s the distribution of y_t, y_t+1,…,y_t+s does not depend on t. The statistical properties such as the mean and variance for the time series do not depend upon time. Therefore, time series with seasonal and/or trend components are not stationary. White noise, on the other hand, is stationary since there is no particular pattern in the data that depends on time. In the time series shown in the figure above, only the bottom right graph shows a stationary time series. There is also the concept of weak stationarity in which only the mean and autocovariance have to be independent of time.

In order to make a time series stationary we can compute the differences between consecutive observations. This methodology is known as differencing. By taking the n-th difference of a time series we can stabilize the mean of observations eliminating trend and seasonality and by using power transformations, we can limit the variance of the data. Usually, a first order difference is sufficient to make a time series stationary but sometimes, a second order differencing might be needed. Also, in order to check if a time series is stationary, we can use a unit root test such as the Dickey-Fuller test.