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I have a time series of Micex Index volatility and cannot define whether it is stationary or not. On the one hand, Dickey–Fuller test, performed in Python (statsmodels adfuller) gives me a p-value of 0.000003 which is really small. On the other hand, the autocorrelation plot is not random and variance seems to change over time. I would like to use this series to forecast another stationary process.

Should I differentiate this series to make it more stationary, or not?

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Presence of autocorrelation does not imply presence of a unit root. ACF and PACF consider the former, the ADF test considers the latter. Since the ADF test suggests absence of a unit root, you should not difference. (You can also see from the plot that the variance does not behave like a random walk and tends to come back to lower levels after each spike, suggesting some form of mean reversion.) Note that differencing when there is no need for it is called overdifferencing and can lead to problems (namely, introduce an integrated MA(1) component in the differenced series).

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