The Autocorrelation function is one of the widest used tools in timeseries analysis. It is used to determine **stationarity** and **seasonality**.

**Stationarity:**

This refers to whether the series is “going anywhere” over time. Stationary series have a constant value *over time. *

Below is what a *non*-stationary series looks like. Note the changing mean.

Time series plot of non-stationary series

And below is what a stationary series looks like. This is the first difference of the above series, FYI. Note the constant mean (long term).

Stationary series: First difference of VWAP

The above time series provide strong indications of (non) stationary, but the ACF helps us ascertain this indication.

If a series is non-stationary (moving), its ACF may look a little like this:

ACF of non-stationary series

The above ACF is “decaying”, or decreasing, very slowly, and remains well above the significance range (dotted blue lines). This is indicative of a non-stationary series.

On the other hand, observe the ACF of a stationary (not going anywhere) series:

ACF of stationary series

Note that the ACF shows exponential decay. This is indicative of a stationary series.

Consider the case of a simple stationary series, like the process shown below:

We do not expect the ACF to be above the significance range for lags 1, 2, … This is intuitively satisfactory, because the above process is purely random, and therefore whether you are looking at a lag of 1 or a lag of 20, the correlation should be theoretically zero, or at least insignificant.

Next: ACF for **Seasonality**

Abbas Keshvani

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thank you very much…..

Thank you Abbas for simple and well explained topic.

My question is in non-stationary data how can we find auto correlation? is partial auto correlation is a good alternative?

Hi Bahaa, thanks for the kind words. An autocorrelation for a non-stationary series would look funny, kinda of like here: https://coolstatsblog.files.wordpress.com/2013/08/berlin2.jpeg. Are you trying to prove that the realizations/values are correlated?

You’re welcome Amin 🙂

Hi Can you explain relation between Auto correlation and Confidence Interval with same intuitive explanation

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Hi Good simple explanation – I’ve always believed if you can explain simply – the person has understood it thoroughly 🙂 Came across the term an hour or so ago (ACF term) and was looking for a simpler explanation

And after a few hits – here it is 🙂

Rajesh

I agree, Rajesh. I think the best part about understanding something fully is that you can take control of the language around it, and therefore simplify it. Thanks for visiting!

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Hi Abbas,

Just a non scientific comment to edit the post:

The word autocorrelation on the title is missspelled and needs a “L” 🙂

Thanks!

Can the acf be used to provide at least five comments about a series? If it is possible pls give me five of them

I am interested in knowing how do we assign the blue line in stationary series data

Hello,

a slight correction needed: MA(1) process is Y(t)=u(t)+b*u(t-1).

What you gave an example of above is a MA(0) process.

Good spot, Pranjal! It has been corrected. Thanks.

thanx sir, how can i get a pdf paper for this subject.

Thanks for this clarifying post!

The criteria for a stationary time series are (1) constant mean, (2) constant variance, (3) the covariance between today’s independent variable and tomorrow’s independent variable is not a function of time. In exactly what way does autocorrelation (correlation in the error terms) violate these three criteria?