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I am new to statistics and I have been struggling to understand the three different versions of (A)DF test. For simplicity I will use the naming conventions from https://en.wikipedia.org/wiki/Dickey%E2%80%93Fuller_test.

My understanding is that we if generate a time series under the null hypothesis, our t-statistics would follow a well-defined distribution. Our null hypothesis is $H_0 :\delta = 0$.

For example let's take version 2 from the wiki page, i.e. $\Delta y_t = a_0+\delta y_{t-1}+u_t$. We use this formula under the null hypothesis ($\Delta y_t = a_0+u_t$) as the data generating process. Intuitively I would assume that fitting a constant term ($\Delta y_t = a_0+\delta y_{t-1}+u_t$) would pick up the real $a_0$ used in the data generating process (which it did), and that should be enough to remove the effects of $a_0$ on the regression. If we compute the t-statistic for $\delta y_{t-1}$ for each of the time series generated, they would follow a well-defined distribution regardless of what value $a_0$ takes, but it turned out to be wrong.

I did this simulation and it seems the percentiles of this distribution is dependent on $a_0$. However, by adding the trend term ($a_1t$ from version 3), the distribution of out test statistic seems to be independent of the $a_0$ used in data generation. What really confused me was that this trend term implies the $y_t$ process is somewhat quadratic. Is there any intuition behind why we have to include this term?

That brings me to my questions

  • What exactly are the $H_0$ and $H_1$ of the three versions of the (A)DF tests? Is $\delta = 0$ the only hypothesis (other than the data generating process itself) ?

  • Why does adding in the $a_1t$ trend term make the test statistic independent of $a_0$ (of the data generating process)? Intuition / proof both works.

  • (How) does this extend to higher orders of t? E.g. if the data generating process is $\Delta y_t = a_0+u_t + kt$, how should I set up my ADF regression? Perhaps something like $\Delta y_t = a_0+a_1t+a_2t^2+\delta y_{t-1}+u_t$?

**update

On page 3, last paragraph of https://www.econstor.eu/bitstream/10419/67744/1/616664753.pdf. The author mentioned the same trick. Is there a proof / some intuition behind this?

The simulation code,

  • If I set c to be nonzero, the distribution changes
  • c = 0 gives me the same ADF percentiles as the statsmodels adf functions
  • using regression = 'ct' this problem goes away
from statsmodels.tsa.stattools import adfuller as adf
import numpy as np
adf_distribution = []
c = 0
for j in range(5000):
    x = 0
    xs = []
    for i in range(1000):
        xs.append(x)
        x = x + np.random.normal(0,1) + c
    statistic = adf(xs,regression = 'c',maxlag = 0,autolag = None)[0]
    adf_distribution.append(statistic)
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  • $\begingroup$ See e.g. stats.stackexchange.com/questions/210885/… and stats.stackexchange.com/questions/213551/… $\endgroup$
    – Christoph Hanck
    Commented Dec 1, 2022 at 15:53
  • $\begingroup$ @ChristophHanck Thank you for your reply, but I don't think these two posts answer my question. Under H0, our time series has a unit root so assuming it has a constant term it is a random walk with drift, and we do not know the value of that drift. We want to construct a linear regression whose t-stat follows a distribution that is well defined regardless of the drift so that we can define pvalues. What I do not understand is why this linear regression has to include a t term, since this implies y_t is quadratic. $\endgroup$
    – Tobi
    Commented Dec 1, 2022 at 17:00
  • $\begingroup$ It is indeed a bit subtle, and I'm not sure I interpret your question correctly, but I believe that the first link does relate to your question. In particular, it shows which DGP we need to have in mind in the trend case (and the same idea applies in the constant case): under the null, the process becomes a unit root with drift, while under the alternative, it is a linear time trend. In particular, the DGP is such that the process is not a unit root process with linear time trend (which would indeed amount to a quadratic trend when recursively substituting for $y_{t-1}$) under the null. $\endgroup$
    – Christoph Hanck
    Commented Dec 2, 2022 at 7:35
  • $\begingroup$ @ChristophHanck Thank you for your response. I think my question focused on why the t-statistic (of the regression with constant term + linear trend) is independent of the drift in the DGP, when the DGP is a random walk with constant drift. However, when the regression is of the same form as the DGP (constant term + no linear trend), our t-statistic becomes dependent on the drift. I think the answer in the first link shows that under H0, the difference between yt and a random walk with no drift is linear in t, but I failed to see how that relates to the t-statistic. $\endgroup$
    – Tobi
    Commented Dec 7, 2022 at 14:05
  • $\begingroup$ @ChristophHanck I think the answer in the second link is quite similar to what I am looking for. In particular, I am looking for a similar proof with constant and trend in the test regression, and the DGP is a random walk with constant drift. Do you know where I can find a detailed proof? I have looked at the original Dickey-Fuller Test paper and their proof was too difficult for me to read. $\endgroup$
    – Tobi
    Commented Dec 7, 2022 at 14:10

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