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When calculating ACF and PACF, what is the rule regarding differencing? For example, I have an increasing time series- looking at instrument drift. Differencing returns what looks like white noise. When I do ACF and PACF, do I use the raw values or the differenced values?

I've done both, just playing around. The raw values return positive ACF values for many lags. The differenced values return a negative value and the rest negative and insignificant.

What is the correct procedure? Since the mean increases over time. Do I difference before ACF? Or ACF on the raw data?

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  • $\begingroup$ Beware of overdifferendcing. Differencing is only appropriate when the original series are integrated, but not otherwise. Mean increasing over time does not warrant differencing, but rather modelling with a deterministic (e.g. linear) time trend. $\endgroup$
    – Richard Hardy
    Commented Sep 25, 2017 at 5:20

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You look at ACF and PACFs of the differenced series. This is because these are tools for looking at stationary processes. You mention that the undifferenced values have a mean increasing over time...right off the bat that process can't be stationary.

Looking at the ACF and PACF will help you distinguish between some sort of noise and an ARMA(p,q) process. If you're looking at financial time series, it would be unsurprising to see ACF and PACF values that look nonsignificant--white noise is a pretty common model. Also keep in mind that stationarity can be broken in more ways than the mean function depending on time.

Edit: Valerie, what Richard is referring to might happen if your true model is something like $y_t = \beta_0 + \beta_1 t + \epsilon_t$ where $\epsilon_t$ is iid or white noise. In this case, if you incorrectly difference your series, you will have $\bigtriangledown y_t = \beta_1 + \epsilon_t - \epsilon_{t-1}$. Then, looking at the ACF plot, it will look like you have an MA(2) model. This will help you determine whether the series is trend-stationary or difference-stationary, or in other words, if it has a deterministic trend or a stationary trend.

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    $\begingroup$ Thanks, new to this method, so I appreciate the non judgement. Also gives me more to base my next searches on. $\endgroup$
    – Valerie S
    Commented Sep 25, 2017 at 2:56
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    $\begingroup$ In general, I would advise someone new to time series like this: Beware of overdifferendcing. Differencing is only appropriate when the original series are integrated, but not otherwise. Mean increasing over time does not warrant differencing, but rather modelling with a deterministic (e.g. linear) time trend. $\endgroup$
    – Richard Hardy
    Commented Sep 25, 2017 at 5:21
  • $\begingroup$ @RichardHardy How do you tell if it is "over differencing". If as OP says, the original data has an upward trend and looks non-stationary. After differencing it one time, it pass the stationary test (e.g.ADF test). Does it mean the differencing is the right thing to do or does it not? $\endgroup$
    – an offer can't refuse
    Commented Feb 17, 2020 at 15:32
  • $\begingroup$ @Taylor If the model is that linear model, after differencing and then use the ARIMA(0, 1, 2) model, does it have any problem? $\endgroup$
    – an offer can't refuse
    Commented Feb 17, 2020 at 15:33
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    $\begingroup$ @anoffercan'trefuse, you should check the existing questions tagged with augmented-dickey-fuller, your questions have most likely been answered there. $\endgroup$
    – Richard Hardy
    Commented Feb 19, 2020 at 6:58

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