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I'm studying unit root tests and therefore the Dickey Fuller test and I can't seem to figure the following out.

Is it correctly understood that the Dickey Fuller test (with drift and constant) is designed such that under the alternative the data generating process is

$$y_t=c+\lambda t+\alpha y_{t-1}+\varepsilon_t?$$

If yes then i get we want the augmented Dickey Fuller to have a richer class of data generating processes as alternative so we consider the auxilliary regression

$$y_t=c+\lambda t+\alpha y_{t-1}+\sum_{i=1}^k \beta_i \Delta y_{t-1} +\varepsilon_t$$

where $k$ is chosen such that the $\varepsilon$'s are white noise. Which now allows for alternatives $AR(k)$ models (is that correct?). Okay, now the way the test was presented to me the alternative is that the process is stationary, but how does the test perform if i.e. the actual data generating process is say $MA(q)$ or $ARMA(p,q)$? Do we somehow beforehand narrow $y$ down to being either $AR$ or a unit root process?

Thanks in advance.

To the editor: Actually my question was not so much intended to alternatives to Dickey-Fuller as much as to understand the whole "the alternative in the hypothesis is not a general stationary process yet it seems to me that we have that as hypothesis". I feel though my question was answered and of course it is always nice to know generalizations exist. But then again, maybe you just changed the title so that people will find the question if seeking those references.

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To your first question, yes this is the data generating process under the alternative. As for the second question, the "Said-Dickey extension", Said, S. E., & Dickey, D. A. (1984). Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika, 71(3), 599-607. covers the case of an ARMA model (by approximating it with an autoregression).

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For completeness, the reference to Said & Dickey (1984), who converted an ARMA to a long AR prior to the test, is not a full story. Hall (1989) criticized Said & Dickey (1984) because the order of the long AR model will increase with the sample size, leading to loss of degrees of freedom. He used an instrumental variable technique to improve on Said & Dickey (1984) and Phillips & Perron (1988), who earlier proposed an alternative modification. Li (1995) studied the power of the instrumental variable tests. This is from section 11.2 of Howell Tong's Personal Journey through Time Series in Biometrika, where modifications and extensions of the Dickey-Fuller test developed in Biometrika are reviewed.

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