Time-Series: Testing for stationarity and random walks

Question

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

Answer 1

[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.