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My question comes from this paper (p. 10), where the authors say:

The $\ln(\hat{p_t}) - \ln(p_t) \sim AR(1)$ condition expresses that the LPPLS fitting residuals can be modeled by a mean-reversal Ornstein-Uhlenbeck (O-U) process when the logarithmic price in the bubble regime is attributed to a deterministic LPPLS component. Since the test for the O-U property of LPPLS fitting residuals can be translated into an AR(1) test for the corresponding residuals, both the Phillips-Perron unit-root test and the Dickey-Fuller unit-root test are used to check the O-U property of LPPLS fitting residuals. In this study, the 5% significant level is applied in the tests.

$\ln(\hat{p_t}) - \ln(p_t)$ are the residuals between the log-values of a fitted curve ($\hat{p}_t$) and the observed log-values ($p_t$).

From my understanding the authors use the Phillips-Perron (PP) and Dickey-Fuller (DF) unit-root tests to test if the residuals $\ln(\hat{p_t}) - \ln(p_t)$ satisfy an AR(1) or O-U process.

I know that the unit-root test checks, if for a given AR(1) process:

$$ y_t = \theta y_{t-1} + \epsilon_t $$ one of the following conditions hold:

$$ \begin{align} H_0: \theta=1 \\ H_A: \theta<1 \end{align} $$ If $H_0$ is accepted, the process is non-stationary.

I'm also aware, that an AR(1) process is the time-discrete version of an Ornstein-Uhlenbeck process.

The actual question: I can't see how the two tests (PP and DF) can tell me if the residuals $\ln(\hat{p_t}) - \ln(p_t)$ satisfy an AR(1) or O-U process, since from my understanding they just check if a specific property (non-stationarity) of the processes is satisfied.

In other words: For what purpose are the authors using the PP and DF tests? It seems to me they are using both tests to test wether the residuals can be modeled with a AR(1) / O-U process. But from my understanding that is not what can be tested using the PP / DK tests.

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  • $\begingroup$ Quick answer: Yes, you are correct these test are unit root tests. $\endgroup$ – Harto Saarinen Jun 29 at 18:54
  • $\begingroup$ @HartoSaarinen I know that they are unit root tests, but why should they tell me if the data is actually an AR(1) or O-U process? $\endgroup$ – WolfgangP Jun 29 at 19:03
  • $\begingroup$ They don’t tell you that. They assume the AR model. Of course you could use a different model and then you have some different tests. But AR (and/or MA ) models are the most basic and usually the place to start. $\endgroup$ – Harto Saarinen Jun 29 at 19:20
  • $\begingroup$ @HartoSaarinen OK, that is also my line of reasoning. But for that purpose are the authors then using the PP and ADF tests? $\endgroup$ – WolfgangP Jun 29 at 20:19
  • $\begingroup$ Well usually the OU process is assumed to be mean reverting. So I guess they want to verify that their AR1 process really is also. $\endgroup$ – Harto Saarinen Jun 30 at 8:33
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For a stochastic process $X$ of the form:

$$ X_t = \alpha_0 + \psi X_{t-1} + \epsilon_t $$

where $\epsilon_t$ is white noise, the null hypothesis for the Dickey-Fuller test is

$$ H_0 : \psi = 1 $$

whereas

$$ H_A : \psi < 1 \, . $$

In case $H_0$ cannot be rejected, the process is non-stationary and a random walk with drift. In case $H_0$ is rejected, the process is stationary and an AR(1)-process and also a time-discrete O-U process.

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  • $\begingroup$ You might want to correct your terminology "$H_{0}$ is accepted". $\endgroup$ – StatsPlease Jul 1 at 23:17

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