<p>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).</p>
<p>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.</p>
<blockquote>
<p>How can I use the Dickey-Fuller test statistic in this case to interpret stationarity?</p>
</blockquote>
<p>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.</p>
<p>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.</p>

CC BY-SA 4.0
2021-07-29T15:22:58.107Z
Thank you! What exactly does the test statistic tell me? Does it give me information on if to reject $H_0$?
Hyperbolic PDE friend
CC BY-SA 4.0
2021-07-29T15:30:35.150Z
@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
1muflon1
CC BY-SA 4.0
2021-07-29T15:32:22.117Z
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 $\alpha=1$
1muflon1
CC BY-SA 4.0
2021-07-29T20:43:43.410Z
Thank you! So I would have to read the test statistic in a table?
Hyperbolic PDE friend
CC BY-SA 4.0
2021-07-29T20:48:46.680Z
@Meowdog well yes how else would you know that the test result is? I guess you can also look at p-value (which is just test statistic translated directly to probability) but in professional writing people report test statistics not p-values (those might be indicated by stars)
1muflon1
CC BY-SA 4.0
2021-07-29T20:54:54.203Z
Great, thank you. I am completely new to this stuff and statistics in general :(
Hyperbolic PDE friend
CC BY-SA 4.0
2021-07-29T20:55:49.743Z
@Meowdog no problem good luck with your studies
1muflon1
CC BY-SA 4.0
2021-07-29T21:00:43.590Z
Thank you very much! You too, if you have to do any
Hyperbolic PDE friend
CC BY-SA 4.0
2021-07-29T15:20:10.817Z
2021-07-29T15:20:10.817Z
1muflon1
<p>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 <span class="math-container">$\texttt{adf.test}$</span> function that is provided by the <span class="math-container">$\texttt{tseries}$</span> package. So in my case I coded
<span class="math-container">$$
\texttt{adf.test(nasdaq\\\$nasdaq_adj)}
$$</span>
and received
<span class="math-container">$$
\texttt{Dickey-Fuller = -1.3079, Lag order = 21, p-value = 0.8717
}
$$</span>
as output. My conclusion was that I can not reject the Null-Hypothesis because the <span class="math-container">$p$</span>-value far exceeds <span class="math-container">$0.05$</span>. Therefore the time series is non-stationary.
<br />
Is this correct? Furthermore: How can I use the <span class="math-container">$\texttt{Dickey-Fuller}$</span> test statistic in this case to interpret stationarity?</p>

<econometrics><time-series><r><hypothesis-testing>
Interpreting an ADF test in R
Hyperbolic PDE friend
# Interpreting an ADF test in R

(Asked by: username_0 on 2021-07-29)

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?

**Tags:** econometrics, time-series, r, hypothesis-testing

---

## 1 answers

### Answer by username_1 on 2021-07-29. Score: 2 | Accepted answer

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.

> **Comments:**
> **username_0**: Thank you! What exactly does the test statistic tell me? Does it give me information on if to reject $H_0$?  
> **username_1**: @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  
> **username_1**: 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 $\alpha=1$  
> **username_0**: Thank you! So I would have to read the test statistic in a table?  
> **username_1**: @Meowdog well yes how else would you know that the test result is? I guess you can also look at p-value (which is just test statistic translated directly to probability) but in professional writing people report test statistics not p-values (those might be indicated by stars)  
> **username_0**: Great, thank you. I am completely new to this stuff and statistics in general :(  
> **username_1**: @Meowdog no problem good luck with your studies  
> **username_0**: Thank you very much! You too, if you have to do any  

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