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Suppose I have a time series: $$ z_t = 0.01 t + 0.9 z_{t-1} + e_t $$ where $e_t$ is $N(0,1)$.

Now, this series is non-stationary as can be easily be checked with an ADF test (using statsmodels for example). My question is, if ADF tells me the that series is non-stationary, can I find the reason it is non-stationary? Is there a way I can find out if the series is $I(1)$ or $I(2)$ or has a drift with trend if it is not easily visible from a plot of the time series (unlike the example here) or do I need to run ADF tests on differences and so on?

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  • $\begingroup$ What does I(1) and I(2) mean? $\endgroup$
    – Michael R. Chernick
    Jan 23, 2017 at 18:58
  • $\begingroup$ I(1) and I(2) are integrated time series with respective orders. I meant it as an example. Given a non-stationary time series, if I dont know the process which generated it, is there a way to find out the reason it is non-stationary apart from running ADF tests on residuals a number of times. $\endgroup$
    – nimbus3000
    Jan 23, 2017 at 19:02
  • $\begingroup$ This question is clear enough to have been answered. I see no reason it should be closed. I'm voting to leave open. $\endgroup$
    – gung - Reinstate Monica
    Jan 24, 2017 at 1:53

2 Answers 2

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If you know your model, none of these tests are necessary.

If it was I(1), it would be stationary after differencing. Supposing your model is true, define the differenced series $\tilde{z_t} = z_t - z_{t-1}$. Then

\begin{align*} \tilde{z_t} &= .01[t- (t-1)] + .9\tilde{z}_{t-1} + [e_t - e_{t-1}] \\ &= .01 + .9\tilde{z}_{t-1} + \tilde{e}_t, \end{align*} which is stationary.

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Your best bet would be to try multiple tests...etc

Try first differencing the data and run the ADF test, and try second

differencing the data and then run the ADF test .... etc

It would seem, under my intuition from those coefficients, the series is likely I(1)

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  • $\begingroup$ Thanks a lot for the answer Kevvy, but my question is slightly more generic. I understand that the given time series is non-stationary because i know the model. My question is if i dont know the process, how can I find the reason it is non-stationary. It seems to me from your answer that one has to run multiple tests $\endgroup$
    – nimbus3000
    Jan 24, 2017 at 5:59

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