When I ran ADF test with my data set, I got following results. I am confused about why "alternative hypothesis" is always (even for real non-stationary series) showing as "stationary"?

     Augmented Dickey-Fuller Test

data:  stlts
Dickey-Fuller = -10.717, Lag order = 0, p-value = 0.01
alternative hypothesis: stationary

Warning message:
In adf.test(stlts, k = 0) : p-value smaller than printed p-value

When I looked at R code, it seems alternative is by default "stationary", I am really confused on why "alternative hypothesis", if it is always considered as "stationary" (or) I might misunderstood r implementation with other literature available online?

I understood that "unit root" doesn't guarantee "stationary" and I am not worried about it at this point. I want to make sure we are interpreting ADF results properly.

Thanks for your time and help!


The test regression under the ADF test is

$$ \Delta y_t = \alpha + \beta t + \gamma y_{t-1} + \delta_1 \Delta y_{t-1} + ... + \delta_{p-1} \Delta y_{t-(p-1)} + \varepsilon_t$$

($\alpha$ and $\beta t$ may or may not be included).
The null hypothesis is that $\gamma = 0$, which implies that $y_t$ is an integrated series.
The alternative hypothesis is that $\gamma < 0$, which implies that $y_t$ is a stationary series.

The null hypothesis and the alternative hypothesis do not depend on the data you are applying the test on. In your case, the test result is evident from the test statistic -10.717 and the corresponding p-value 0.01 (but actually smaller than that as noted below in the Warning message). The p-value suggests that the data is very unlikely given the null hypothesis (of integrated $y_t$), so you would rather "believe in" the alternative hypothesis (stationary $y_t$).

By the way, Lag order seems to suggest that you have not included any lagged $\Delta y_t$'s. You should consider including some (the lag order could be selected by BIC, for example) to remove potential autocorrelation from the residuals $\varepsilon_t$ to make the test valid.

  • $\begingroup$ Thanks for your answer! Fixed lag (added 4). I am comparing R results with excel implementation here, specially that non-stationary one with (r=1) which confused me with understanding this concept.xlstat.com/en/learning-center/tutorials/… $\endgroup$
    – kosa
    Mar 16 '15 at 21:08
  • $\begingroup$ More I read your answer and compare with excel implementation, it seems null hypothesis is "integrated series" only in case of alternative hypothesis is "stationary". If alternative hypothesis is "explosive" (which is an option R provides), in that case I think "null hypothesis" is series is "stationary", am I correct? $\endgroup$
    – kosa
    Mar 16 '15 at 21:19
  • 1
    $\begingroup$ I am not sure, but I would guess that the null is still "integrated"; that would correspond to $\gamma=0$. It is most common to have a null hypothesis where the parameter of interest equals something (in this case, zero), while the alternative can be either bidirectional "not equals" or unidirectional "greater" or "lesser". But try to look up R documentation to be sure. $\endgroup$ Mar 16 '15 at 21:28
  • $\begingroup$ Thanks! I looked at r documentation, no clear explanation about explosive option. Will explore more.inside-r.org/packages/cran/tseries/docs/adf.test $\endgroup$
    – kosa
    Mar 16 '15 at 21:37

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