Order of integration for a time series with constant mean and increasing variance

Question

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.

1. 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.

2. I also read here that the order of integration of a series is equal to the number of unit roots of that series.

3. 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.

Answer 1

Point 1. implicitly deals only with processes that can be rendered stationary by differencing. There are other processes that do not belong to this category. With that in mind, any contradiction between 1. and 2. disappears.

Regarding 3., the ADF test deals with a certain set of processes that may have a unit root. Again, this does not encompass each and every stochastic process. This matches the fact that while presence of a unit root implies nonstationarity, absence of a unit root does not imply stationarity.

Without seeing your time series and detailed test results it is a bit difficult to comment on its stationarity and unit roots. However, note that many of the processes discussed in textbooks are simple, idealized versions of what is encountered in reality. The actual data generating process may be much more complex and it also may be changing over time, either gradually or abruptly. Perhaps your time series is plagued by that. This does not mean you cannot approximate it by a simple model to a decent extent, but diagnostics may well show that the approximation is not ideal.

Regarding your title question,

Order of integration for a time series with constant mean and increasing variance

such a series is not integrated (it does not contain a unit root), yet it is not stationary (because of nonconstant variance).