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.

Time series plot of non-stationary series
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
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
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 nonstationary 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:

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

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

    1. 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?

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  2. 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

  3. Hi Abbas,

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

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

  5. 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?

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