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My goal is to test the weak-form efficient market hypothesis using time-series on prices of various stocks listed on S&P 500. According to theory, a particular stock is said to be weak-form efficient if it follows a random walk. I would like to ask if using tests such as ADF, KPSS and Elliott-Rothenberg-Stock (which all test for unit roots in the data) is the same as testing whether the series follows a random walk. Confusion arises because I have read on various websites that "not all non-stationary time-series are random walks". However, in some papers I also see people using these unit-root tests to verify the efficient market hypothesis...

To recap, is it fine to use ADF, KPSS, ERS, and other unit root tests to test whether a series exhibits a random walk? Suggestions on other possible weak-form efficiency tests are more than welcome.

Thank you

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[I]s it fine to use ADF, KPSS, ERS, and other unit root tests to test whether a series exhibits a random walk?

No. It is possible that a series has a unit root, yet it is not a random walk. An example would be ARIMA(p,d,q) with $d=1$ and $p>0$ or $q>0$ or both. This process has a unit root (since $d=1$) but it is not a random walk since $p>0$ or $q>0$ or both.

ADF, KPSS and ERS tests assess presence of a unit root (the $d$ parameter), but not for presence of autocorrelation beyond that (characterized by the $p$ and $q$ parameters).

However, in some papers I also see people using these unit-root tests to verify the efficient market hypothesis...

These tests may be a part of the procedure of testing the random-walk hypothesis, but they cannot constitute the whole procedure if it is to be valid.

Suggestions on other possible weak-form efficiency tests are more than welcome.

One way to assess whether a time series $x_t$ is a random walk is first to determine that $d=1$ and then to reject nonzero autocorrelations of the first-differenced process $\Delta x_t$. The latter can be done by referring to the autocorrelation function of $\Delta x_t$ or by other methods. See Chapter 2 of Campbell et al. "The Econometrics of Financial Markets" (1996) for a detailed and pedagogical treatment of precisely the topic you are interested in.

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  • $\begingroup$ Thank you Richard, your answer is extremely useful. Thank you !! $\endgroup$ Feb 15, 2022 at 14:11
  • $\begingroup$ @Hatori_Hanzo, you are welcome! The real use hides in Campbell et al. (1996) that I reference. $\endgroup$ Feb 15, 2022 at 14:15
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    $\begingroup$ Through some research I have come across the 'Generalised Hurst Exponent' as a method to assess whether a series exhibits a RW. Kindly, may I have your opinion on this methodology? $\endgroup$ Feb 15, 2022 at 14:17
  • $\begingroup$ @Hatori_Hanzo, I am not familiar with it, so I cannot comment. $\endgroup$ Feb 15, 2022 at 15:16
  • $\begingroup$ Alright no problem. Thank you again for your help Richard! :) $\endgroup$ Feb 15, 2022 at 15:32

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