How to use the Phillips-Perron and Dickey-Fuller unit root test to test for AR(1) or Ornstein-Uhlenbeck process

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.

• Quick answer: Yes, you are correct these test are unit root tests. – Harto Saarinen Jun 29 at 18:54
• @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? – WolfgangP Jun 29 at 19:03
• 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. – Harto Saarinen Jun 29 at 19:20
• @HartoSaarinen OK, that is also my line of reasoning. But for that purpose are the authors then using the PP and ADF tests? – WolfgangP Jun 29 at 20:19
• 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. – Harto Saarinen Jun 30 at 8:33

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.

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