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I am looking for literature that suggests the augmented Dickey-Fuller (ADF) test is not completely accurate, or in general criticizing the test.

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    $\begingroup$ A natural source would be papers proposing new unit root tests that appeared after the Dickey-Fuller test was proposed. The authors will normally try to address the weakness of the established tests (ADF test being a prime example) to set the scene for the new test that should hopefully be better than the existing competition. $\endgroup$ Mar 14 '16 at 16:12
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There are various criticisms, some of which, in my opinion, are more pertinent than others:

  1. Lack of power: When testing $\rho=1$ against $|\rho|<1$, it is argued that many macroeconomic time series may be expected to be well-described by a $\rho$ close to, but less than 1. The test lacks power to detect this. To some extent, that criticism is invalid, because all tests have low power when the actual parameter value is close to the null value. To some extent, it is valid because there are tests that apply under broader sets of assumptions, or are simply more powerful than the ADF test.
  2. Size distortion: as for most tests, the null distribution is only available asymptotically, and the finite-sample distribution is often argued to differ substantially from the asymptotic one, leading to actual rejection rates that differ substantially from the nominal level $\alpha$.
  3. General scepticism towards the usefulness of testing point null hypotheses, fairly common among Bayesian statisticians.
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  • $\begingroup$ (+1) But isn't ADF a one-sided test (i.e. it has a composite null) as usually applied to test against a stationary alternative? - then objection 3 wouldn't apply. $\endgroup$ Mar 14 '16 at 17:27
  • $\begingroup$ I do not disagree with you - as I said this is a list of criticisms I remember to have come across, that I do not necessarily endorse (that said, I should provide proper references). But at least when it comes to interpreting the test, my impression is that people mostly see it as a point null $\rho=1$, not as $\rho\geq1$, as explosive processes are typically considered irrelevant in economics (although there is a burgeoning literature on bubbles that are studied with explosive processes). Of course, a left-tailed ADF test indeed has no power statistically against explosive processes. $\endgroup$ Mar 14 '16 at 19:03
  • $\begingroup$ Size distortion is a thing one encounters frequently in statistical tests. Is ADF test particularly bad or notorious in this respect (at least compared to its competitors), or is it just a general criticism? $\endgroup$ Mar 14 '16 at 19:49
  • $\begingroup$ There is probably a point in the nuisance parameter space of most tests where empirical size is particularly bad, so in that sense it's probably general criticism. For adf, such cases have been identified through large negative ma components, which are not implausible. I would not dare to say if these points in the nuisance parameter space are generally as plausible for other testing problems. $\endgroup$ Mar 14 '16 at 19:59

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