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I have a time series, AW1, with 365 daily observation, but I ignore if I should build my model taking the time series as given or taking the first difference. (I know many questions approach this problem, but I'm still confused as the time series presents different aspects. )

I rejected the null of a unit root, and hence it should be a stationary series, but looking tat the graph, I don't know if I should rely on the ADF test because of the presence of volatility etc,...

So here are the graphs of the series , the first one without first difference while the second one is with first difference. My ADF test gave me a value of -17,69. Any advice on that? AW1 with no first difference

AW1 with first difference

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There does not seem to be a unit root in your original series. The series does not resemble a random walk, and its values tend to return close to zero after any larger spikes. They seem to be mean reverting (the opposite of unit-root behaviour).

Differencing is a measure against a unit root -- but not other types of nonstationarity. Thus I do not see a reason for you to difference the series.

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