Simple Autocorrelation Function


The Simple Autocorrelation Function (FAS) is a statistical analysis tool that allows us to find the level of autocorrelation of the data and at what delays, k, it occurs.

In other words, the Simple Autocorrelation Function (FAS) or, from English, Autocorrelation Function (ACF), is a mathematical function that helps us to know what dependence the data of a given period have with the same data from k previous periods.

The importance of the FAS lies more in its representation than in its mathematical formula since it is the results that we represent and from which we will draw our conclusions.

Objective of the Simple Autocorrelation Function

The utility of the FAS is to measure the inertia or trend of a time series, that is, to see what degree of dependence the data now shows with the data from k previous periods.

Since the work methodology is time series, we establish the analysis on a single variable at different moments in time. A typical example would be the listing price of a financial asset between 1990 and 2020. Even if prices change, the study variable will be the same: listing price.


We recall the calculation to estimate the autocorrelation coefficient:

Autocorrelation Coefficient
  • The numerator is the covariance of xt with its past xt-k, with respect to the estimated population mean.
  • The denominator is the variance of xt with respect to the estimated population mean.
  • The time horizon is delimited by 0 and T. Where T is the maximum number of time periods available and 0 is the minimum for k but not for t, because t has to be greater than 0.
  • In the same way as the correlation coefficient, the autocorrelation coefficient is bounded between -1 and 1.

The key to understanding autocorrelation is to simply think about the correlation coefficient and change the "y" to "xt-k."

As we have said before, each lag, k, has its own autocorrelation coefficient. In other words, the trading price will not always follow the same trend with the same intensity, there will be periods of strong trend and there will be others that will trade in range and more randomly. Although it is not very common to calculate the FAS by hand because we use statistical programs, the formula is the following for stationary processes:

Autocorrelation function

We will always work with the estimation of the correlation coefficient (first formula) and not with the population values ​​(second formula). You can see that both result in the same quotient but the first has "^" and the second does not.


Depending on the type of data, the FAS or ACF, in English, will change since not all the data are the same or have the same level of correlation with the past.

  • "Lag" means lag in English.
  • The dashed lines represent the default 95% confidence bands.

Simple Autocorrelation Function Example

Some examples of graphics:

Example of any FAS Example of any FAS Example of any FAS

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