I am trying to model the logged returns of 5 different markets. When running the the Augmented Dickey-Fuller (ADF) tests on the logged market returns, the p-value is 0.000 for all markets regardless if the ADF includes 1) constant, 2) drift and constant or 3) neither constant or drift. Thus, it is assumed that none of these time series has no unit root, i.e. Y(t) is not dependent on Y(t-1), i.e. Y(t) is not autoregressive.

However, when running the ARMA(1,0) for the respective markets:
e.g. LN.Returns.GreenBonds = LN.Returns.GreenBonds(t-1) + e
the AR(1) parameter, LN.Returns.GreenBonds(t-1), has a p-value close to 0.000. How can this be the case when the ADF suggest that we do not have autoregressive time series?


1 Answer 1


Low p-values in an ADF test mean you reject the null hypothesis of presence of a unit root.

The ADF test does not tell whether the series are autoregressions or not. Instead, the ADF tests for presence of a unit root.

In sum, your findings are not contradictory.

  • $\begingroup$ Thank you for the quick reply Richard! So my ADF-tests just tell me that I do not have a unit root, i.e. the time series is stationary. Based on this information, I do not have to transform the time series. Whereas I would try to difference or detrend the data if the time series were non-stationary? I have proceeded by using the PACF of the time series as an indicator of the autoregressiveness of the respective marekts. Would you recommend that I compliment my autoregressiveness analysis with any more tests or is this sufficient? $\endgroup$
    – Isac
    Apr 14, 2021 at 8:46
  • $\begingroup$ @Isac, I think all you wrote in the comment makes sense. Regarding recommendations regarding your last question, looking at ACF and PACF and fitting an autoregressive model should be sufficient. You have fit an autoregression AR(1) (or equivalently, ARMA(1,0)); you could try AR(2) or AR(3). But realistically, approximating the conditional mean of financial returns by an AR(1) or even a constant is usually good enough. $\endgroup$ Apr 14, 2021 at 9:22
  • $\begingroup$ Hello again Richard, thank you for the quick reply! I am actually trying to model the comovement of the GB market and other markets over time using DCC-GARCH. This question has helped me in my preliminary testing. I am stuck a bit in my methodology of the DCC-GARCH and would greatly appriciate if you had an opportunity to look at my newly posted thread: stats.stackexchange.com/questions/520798/… $\endgroup$
    – Isac
    Apr 21, 2021 at 14:13
  • $\begingroup$ @Isac, I will take a look. FYI, if the answer above 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. (No pressure, though!) $\endgroup$ Apr 21, 2021 at 14:40

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