The residual of a seasonal_decompose (from python's statsmodels) yields the following ACF and PACF plots



The sampling frequency of the time series is hourly, so the ACF plot hints at daily (24 hour) seasonality, which is leaking from the seasonal_decomposition (I haven't been able to find a decomposition that doesn't leak some of this seasonality).

Anyway, I need to move forward and model this residual. However the SARIMAs I have so far used with orders (3,0,0)(0,0,4,24) haven't been doing very well. I see as a rule of thumb people take the significant ACF peaks to stand for MA orders, and significant PACF peaks to stand for AR orders, but I don't know the theoretical motivation behind that rule. Do you have any suggestions for how to capture this signal? It is normally distributed.

  • $\begingroup$ you didn't tell us how many observations are in the series or what is the sigma of the ACF values. that's why it's impossible to say anything. for econometric series, which are usually short, anything below 0.2 is often not significant. so this could tell me that there are no lags apparent in the series $\endgroup$
    – Aksakal
    Jan 16, 2021 at 23:14

1 Answer 1


interpreting the observed ACF and PACF an trying to match them to a candidate ARIMA model requires that the data under analysis

has no

1) step/level shifts 2) deterministic time trends 3) pulses 4) seasonal pulses


that the resultant AIMA model has constant parameters and constant error variance over time

thus one needs to simultaneously consider factors 1-4 along with an ACF and PACF to form a useful model.

You probably want to include 23 hourly indicators and closely examine the residuals for your arma model. Hourly data offers the possibility of using deterministic structure. You may find the need for level/step shift indicators and/or time trends to capture additional deterministic structure. How many observations do you have ? Perhaps you should post your data.

  • $\begingroup$ If you are happy with my answer , please accept it and close the question $\endgroup$
    – IrishStat
    Aug 15, 2019 at 8:00
  • $\begingroup$ best answer ever, but how to do it anyway with 1-4? i cant really know when trends change, to switch arimas, what model(s) could work well with step shifts, pulses , seasonal pulses .., seems data should be bucketed maybe is there stretch invariant encoding... $\endgroup$ Apr 20, 2022 at 3:02
  • $\begingroup$ thanks for the kudo. One way to answer your question is to actually examine the process by using an actual data example. If you wish we can pursue this offline using a commercially avAILable package called AUTOBOX which I have helped to develop,. $\endgroup$
    – IrishStat
    Apr 20, 2022 at 12:35

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