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I have a time series of a stock and use its log differenced daily returns. I have conducted an ADF test for a presence of unit roots, a KPSS test as well and both confirms stationarity in the time series.

However, when I use auto.arima or conduct a matrix letting BIC select number of terms, I get a ARIMA(0,0,0) model (when I let AIC select terms I get an ARIMA(1,0,2)). From what I have read an ARIMA(0,0,0) model can be interpreted as white noise and random walk.

But can a random walk be stationary on variance? Enders example. I have attached a picture from a book, where it says that it can't be stationary.

What can be a possible reason for the fact that my series seems to be stationary, but that the function still suggests ARIMA(0,0,0)?

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ARIMA(0,0,0) is white noise and therefore stationary. It is not a random walk, and random walk is not ARIMA(0,0,0). (Random walk is ARIMA(0,1,0), i.e. a cumulative sum of a white noise process.)

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