How to use the Autocorreation Function (ACF)?

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.

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).

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:

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:

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:

$Y_t = \epsilon_t$

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

21 thoughts on “How to use the Autocorreation Function (ACF)?”

1. amin ahsan says:

thank you very much…..

1. Bahaa says:

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?

2. Ankush Jamthikar says:

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

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5. Anonymous says:

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

1. 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!

6. Anonymous says:

Hi Abbas,

Just a non scientific comment to edit the post:
The word autocorrelation on the title is missspelled and needs a “L” 🙂

7. Felix Asare says:

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

8. Anonymous says:

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

9. 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.

10. Anonymous says:

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

11. Martin D says:

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?

12. inamul haq says:

awesome explanation… Thank you sir….

13. Sophie Feng says:

It helps a lot. Thanks!