I'm learning time series analysis and I have a confusion about the concept of auto-correlation. Is the auto-correlation calculated for the time series or the so-called residuals? I don't know what they mean by residuals here. To me, residuals mean the different between predicted values and ground-truth values.

The same confusion arises when I learn the concept of Ljung-Box test. For example, in here, they define the test with residuals, but on wikipedia, they define it on the time series values.

  • $\begingroup$ Is your question about how to specifically compute the auto-correlation or the relation between auto-correlation and residuals? $\endgroup$ Commented Aug 18, 2020 at 12:56
  • $\begingroup$ My question is about the definition of autocorrelation, and now I realized that it is defined on time series. The residuals can be represented as time series, thus the autocorrelation can be computed for residuals. Still, definitions on the Ljung-Box are confusing as one is on time series and the other uses residuals. $\endgroup$
    – lenhhoxung
    Commented Aug 18, 2020 at 13:57

1 Answer 1


Autocorrelation can be calculated for any time series regardless of its interpretation. It can be raw values, residuals from a model or something else.

Residuals. In many models, errors are assumed to be free of autocorrelation. When it comes to model diagnostics, it is common practice to obtain residuals as estimates of model errors and test them for presence of autocorrelation. If found, presence of autocorrelation is interpreted as a problem with the model.

Raw data. Autocorrelation in raw data may be of interest, too. E.g. if a model without exogenous regressors such as ARIMA is considered, one would look for autocorrelation in the data. If present, one would try to select an appropriate model for it, facilitating a description of the pattern and making forecasts of future values based on past values of the same series.


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