I am trying to find the order of integration of a time series. I checked for stationarity using the ADF and KPSS tests. Both the tests indicated non-stationarity, so I differenced the series once and conducted the tests again. Now, the p-value for both tests is coming out to be less than 0.5: The ADF test seems to indicate that the series is stationary, while the KPSS test indicates that the series is non-stationary. This answer seems to imply that the differenced series has increasing variance but it is reverting to the mean - clearly the data is not covariance-stationary. I think taking subsequent differences of such a series will always give a series with increasing variance.
According to Wikipedia: In statistics, the order of integration, denoted I(d) of a time series is a summary statistic, which reports the minimum number of differences required to obtain a covariance-stationary series.
I also read here that the order of integration of a series is equal to the number of unit roots of that series.
Finally, this page indicates that the ADF test just indicates the presence of unit root and not stationarity.
How do I make conclusions about the order of integration of this series? If I go by definition 1, my data is not stationary even after multiple rounds of differencing and so the order of integration is infinity(?). If I go by the second definition, the order of integration is 1.
I'm sure there's a gap in my understanding, please guide me.