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Suppose we have some financial time series. When we calculate the standard ACF, $\mu$ is considered as the average of all series' values. However, if we have a volatile series, the average can be biased towards high prices that were long time ago and, intuitively, "local" average (for example, the average of prices during current week) seems to be more relevant to calculate autocorrelations.

Can you please explain me, am I right given that I try to predict next day return? Why is calculating ACF on the whole series is considered more standard (at least in textbooks) than the rolling one?

Thank you!

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  • $\begingroup$ because it is not only easy but incorrect as you are pointing out. $\endgroup$
    – IrishStat
    Commented Apr 19, 2020 at 19:11

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In basic time series modelling, no matter you use AR(Auto Regressive) or MA(Moving Average) models, they all inciur some sort of autocorrelations. So ACF is not a model, it's just a way to show the autocorrelation properties of different models.

No model is more standard than the other, as for what model to pick, it depends on your model selection criteria, such as BIC, or expected prediction error. Most text books start with AR models is because AR is simple.

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  • $\begingroup$ Let me clarify myself a little bit. I understand that ACF is not a model, what I mean is that when we calculate ACF for the whole series, we, for example, obtain insignificant autocorrelation at the first lag. But suppose when we calculate rolling autocorrelations we observe that many 1st lags are significant. Moreover, in many cases autocorrelation of the whole series is about 0 but rolling, for example, in my sample shows more-or-less 0.5. At the end of the day, I want to know which series show significant dependencies. Which of two cases leads to that? $\endgroup$
    – Nik
    Commented Apr 19, 2020 at 12:06
  • $\begingroup$ This is a GENERAL problem with all time series modelling that most textbooks neglect to talk about: The instability of parameters of ANY KIND, be it the autocorrelations, coefficient estimates, etc. The best thing to do is make a decusion on your own about what you think will be most representative of the future. One month, one week, 6 months etc. This decision depends on what horizon you're trying to forecast. Are you trying to forecast monthly, daily, hourly etc. Basically, there is no answer to your good question. $\endgroup$
    – mlofton
    Commented Apr 19, 2020 at 14:23
  • $\begingroup$ there is an answer estimate the model for sub-groups and test for homogeneity of parameters using the 1960 CHOW TEST $\endgroup$
    – IrishStat
    Commented Apr 19, 2020 at 19:13
  • $\begingroup$ @mlofton, thank you for your response! Still working on this problem so will try such strategy! $\endgroup$
    – Nik
    Commented May 29, 2020 at 14:32
  • $\begingroup$ @IrishStat thanks! Will read about this test $\endgroup$
    – Nik
    Commented May 29, 2020 at 14:33
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You should have a look at locally stationary processes. They are defined as typical time series models (e.g. ARMA) but where the parameters of model change smoothly over (unit interval) time.

In particular for white noise deviations for locally stationary alternatives with non zero auto correlations (locally) see "Testing for white noise against locally stationary alternatives", (Goerg 2012).

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