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I have time series data and it has following ACF plot

enter image description here

I read The data should be stationary

"The data is non-stationary when there is a large spike at lag 1 that slowly decreases over several lags. If you see this pattern, you should difference the data before you attempt to identify a model. To difference the data, use differences. Once you difference the data, obtain another autocorrelation plot."

Adf test telling me the data is stationary as its p values is less than 0.05.

For stationary series , I read many places that "A stationary time series has a mean, variance, and autocorrelation function that are essentially constant through time."

do we really need to have constant autocorrelation for each lag for data to be stationary?

Based on Mauritis response here i am attaching graph highlighted with seasonal regionenter image description here

user172500
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  • As I explain in the comments to my answer, the highlighted regions in the data are *not* seasonal patterns. The underlying data generating process is a simple AR(1) model without any seasonality. – Maurits Evers Aug 17 '22 at 10:20

1 Answers1

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Is autocorrelation an indication of Non Stationary Series

The short answer is no.

To demonstrate, let's consider a stationary AR(1) process: I'm using R here to simulate data and plot the ACF.

set.seed(2020)
ts <- arima.sim(model = list(ar = 0.8), n = 100)
plot(ts)

enter image description here

acf(ts)

enter image description here

Notice how the sample autocorrelation tapers off; to be specific, the ACF decreases with phi^h where h is the lag and phi is the slope in the AR(1) model (here phi = 0.8).

Maurits Evers
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  • don't you see any seasonal pattern in your simulated data - i uploaded your graph in my original question – user172500 Aug 17 '22 at 07:17
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    @user172500 No. By design, these are *not* seasonal patterns. Time series data can be tricky to interpret and the eye can be deceiving. Even more, not all time series data with cyclic patterns are seasonal, see e.g. [Rob Hyndman's section on stationarity and differencing](https://otexts.com/fpp2/stationarity.html). – Maurits Evers Aug 17 '22 at 10:10
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    @user172500 The key answer (and explanation) to your question is that autocorrelation does not imply a non stationary process. As I show, an AR(1) process has a very well defined autocorrelation trend with increasing lag. – Maurits Evers Aug 17 '22 at 10:18
  • I agreed but these things are puzzling me https://stats.stackexchange.com/questions/372145/adf-test-showing-stationary-for-a-non-stationary-series – user172500 Aug 18 '22 at 04:37