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.


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


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

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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 🙂


  3. Pingback: How to Use Autocorreation Function (ACF) to Determine Seasonality | CoolStatsBlog

  4. Hi Abbas,

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

  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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