My time series is on Life expectancy at birth from 1960 to 2018. Obviously, it has an increasing trend and ACF also supports this because the ACF values dampen so slowly. However, ADF test p-value is less than 0.01 (stationary) and kpss test p-value is also less than 0.01(non-stationary). This is contradicting. Even after applying several differencing, ADF p-values get larger(>0.01) which makes them non-stationary while KPSS p-values still are less than 0.01 which means the differenced data still is non-stationary. How do I go about it?


1 Answer 1


Both ADF and KPSS rejecting their null hypoheses does not have to be a contradiction. ADF test indicates your series does not have a unit root. This is not the same as stationarity. E.g. your series may have variance that is growing with time, a sinusoidal time trend or yet something else making it nonstationary. Something like that might be what the KPSS test is picking up.

  • $\begingroup$ I am about to forecast using ARIMA and hence, I need to make the TS stationary by differencing. However, even after 3rd difference, KPSS test reveals a p-value of less than 0.01 (non-stationary). I noticed also the plot of the differenced series, it's like a smooth sinusoidal plot. Is there something wring with the data? $\endgroup$
    – Francis
    Jun 6, 2021 at 12:55
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    $\begingroup$ @Francis, the common mistake* with differencing is not differencing too few times, it is differencing too many times. This has been discussed on Cross Validated before; search for overdifferencing. If I were you, I would first double check whether any differencing is needed. Perhaps your data has a deterministic trend, in which case no differencing is warranted. *Which appears to have become more widespread recently. $\endgroup$ Jun 6, 2021 at 14:04
  • $\begingroup$ @Francis, how is it going? If my answer is helpful and clear, you may accept it by clicking on the tick mark to the left. Otherwise, you may ask for further clarification. This is how Cross Validated works. $\endgroup$ Jun 18, 2021 at 13:22

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