Time Series: Importance of time stationarity

Howdy Mates!

Today we’re explaining yet another interesting topic to all the enthusiasts out there.

First, let’s see the importance of time stationarity and why it is required to be stationary in the first place.

To understand the stationarity we need to understand how time series data is different from other kinds of data.

As we know, the assumption of Normality applies to all kinds of data (except time series) in inferential statistics e.g., t-tests, ANOVA, and simple regression. In other words, these statistical procedures are based on the assumption that the value of interest (which is calculated from the sample) will exhibit a bell curve distribution function if oodles of random samples are taken and the distribution of the calculated value (across samples) is plotted. This is why these statistical procedures are called parametric.

Now we have understood the basics. Let’s move on to the next part.

The most basic assumption of Normality is that the data is independent and identically distributed. Another important assumption of Normality is that the random errors follow a normal distribution. This assumption can break down when there are multiple sources of errors and they are correlated. If the error distribution is not normal and the assumption of normality is made, then there could lead to incorrect statistical analysis and thus erroneous conclusions. 

This property of independence is a nice property, since using it we can derive a lot of useful results. The problem is that in time series, this property of independence does not hold because the time series data is correlated with its own lagged values i.e. yt is correlated with yt-1 or yt-2 etc. Further, sometimes time series data also exhibits seasonality. All of this leads to non-constant mean and variance of the data, resulting in incorrect statistical analysis.

Stationarity is one way of modeling the dependence structure to bring that independence and is very important because, in its absence, a model describing the data will vary in accuracy at different time points. 

Examples of non-stationary time series:

SP500 & NASDAQ data from 1990 till 2020.

Auto correlation factor (ACF) and Partial Auto Correlation Factor of the SP500 is as per image below. We can clearly see high autocorrelation of stock price with its lagged variables.

After taking logarithm of first difference i.e. log(ytyt-1), we can convert non-stationary

time series to stationary time series as per image below. The data below now has constant means (around zero) and constant variance.

Further, there is no Auto correlation factor (ACF) and Partial Auto Correlation Factor of the stationary time series, can be seen in images below. Notice that both ACF and PACF have become insignificant.

To summarize, stationarity of time-series data is required for sample statistics such as means, variances, and correlations to accurately describe the data at all-time points of interest. 

That’s all folks for today!

See you next time, Lots of Love, take care of yourself 🙂 <3