Although being a mathematician, I am fairly new to time series and R. On an assignment I was being asked to check a time series for stationarity in R, only using the $\texttt{adf.test}$ function that is provided by the $\texttt{tseries}$ package. So in my case I coded
$$
\texttt{adf.test(nasdaq\\\$nasdaq_adj)}
$$
and received
$$
\texttt{Dickey-Fuller = -1.3079, Lag order = 21, p-value = 0.8717
}
$$
as output. My conclusion was that I can not reject the Null-Hypothesis because the $p$-value far exceeds $0.05$. Therefore the time series is non-stationary.
Is this correct? Furthermore: How can I use the $\texttt{Dickey-Fuller}$ test statistic in this case to interpret stationarity?
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1 Answer
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Yes, the null hypothesis of ADF test is that the series contains unit root (e.g. see Verbeek, A guide to modern econometrics pp 273).
So the results you present above indicate that you cannot reject the null of an unit root and consequently you should treat your series as non-stationary.
How can I use the Dickey-Fuller test statistic in this case to interpret stationarity?
Stationarity/non-stationarity is property of a time series. Non-stationarity just indicates that series has some non-deterministic trend. ADF test just tests for this property it does not tell you why the series is non-stationary. For that you need to draw upon some theoretical model.
For example, Hall's random walk hypothesis (see Romer Advanced Macroeconomics pp 373) implies that consumption should follow random walk and thus be non-stationarity so that would be an explanation of non-stationarity observed in consumption data. But there is no way of getting this sort of information from a statistical test you have to build some implicit/explicit theoretical model for that.
answered Jul 29, 2021 at 15:20
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$\begingroup$ Thank you! What exactly does the test statistic tell me? Does it give me information on if to reject $H_0$? $\endgroup$ Commented Jul 29, 2021 at 15:22
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$\begingroup$ @Meowdog the test basically test whether $(1-\alpha)$ is statistically different from zero (where $\alpha$ is coefficient from auxiliary time series regression where $\alpha<0$ is case of stationarity and $\alpha=1$ or $\alpha>1$ case of non-stationarity, the test is one-sided so it basically just tests whether $\alpha<0$ by checking if $1-\alpha < 0 $. Also yes of course, it gives you information to reject $H_0$ every statistical test must give you info on that otherwise it would cease to be test, but the test does not tell you why $H_0$ is rejected $\endgroup$– 1muflon1 ♦Commented Jul 29, 2021 at 15:30
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$\begingroup$ in your case the test tells you that the test statistics is -1.309 which is then compared to critical values (where appropriate critical values will depend on sample size and so on), in your case $H_0$ cannot be rejected so you cannot reject the hypothesis that $(1-\alpha)=0$ and consequently that $\alpha=1$ $\endgroup$– 1muflon1 ♦Commented Jul 29, 2021 at 15:32
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$\begingroup$ Thank you! So I would have to read the test statistic in a table? $\endgroup$ Commented Jul 29, 2021 at 20:43
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1$\begingroup$ @Meowdog no problem good luck with your studies $\endgroup$– 1muflon1 ♦Commented Jul 29, 2021 at 20:55