Random walk analysis of a univariate timeseries

As the title describes, I want to conduct a random walk analysis of a univariate time series $$Y_t$$. What are the tests and steps that you guys would suggest for this purpose?

My current thinking:

1. Analyse the autocorrelation of the changes $$u_t=Y_t-Y_{t-1}$$ with Ljung-Box test. Here, for different $$q$$'s (amount of lags considered), rejection and non-rejection of the autocorrelation changes and I am not quite sure how to interpret this.

2. Second, I want to conduct an Augmented Dickey-Fuller test to test for a unit root of $$Y_t$$. In this context, I want to do a separrate analysis for the optimal amount of lags considered in the ADF equation. In this separrate analysis I determine the amount of lags that induces whiteness in the regression residuals. For this purpose, I am using Breusch-Godfrey's test. The reason for this additional analysis is because I want to argue that if BG test suggests that further lags are needed for residual whiteness, it is not just a random walk, even though we are not able to reject the null hypothesis of a unit root because the power of the ADF is very small if the coefficient is close to 1. Here, I have the same problem of an appropriate choice of the number of lages residuals considered in the auxiliary regression of the BG test. The resulting decision about residual whiteness is influenced by it a lot.

3. Third, I want to conduct a variance ratio test. A friend of mine told me that the variance ratio test is an analysis that tests the hypothesis that the elements of $$u_t$$ are not only serially uncorrelated but also independent. I don't really understand where this hypothesis comes from. Shouldn't uncorrelation be enough for the linear variance increase that is analysed with a variance ratio test?

Edit: Maybe I should give more background: In the context of testing market efficiency I want to test whether a certain serious fulfills the martingale property. However, the martingale property of mean independence is not testable straight forward (Not sure about that, but most literature switches to random walk testing in context of weak market efficiency). As a random walk with i.i.d innovations also implies a martingale, I switched to testing for that as a "sufficient condition" for the series to exhibit the martingale property.

Any opinions or suggestions would be great.

• The variance ratio test seemingly tests if the variances over 2 periods are equal. So for independence it is necessary that the variance is also equal across the board. Jun 23, 2021 at 18:43
• Could you explain what a "random walk analysis" is?
– whuber
Jun 23, 2021 at 18:52
• Testing whether the underlying time series exhibits random walk properties. Jun 23, 2021 at 21:08
• Right--one would presume that. But exactly which properties do you want to test? What precisely is your null hypothesis and what alternative hypothesis do you intend to use?
– whuber
Jun 23, 2021 at 22:22
• @whuber, That is kind of my question to you. Maybe I should give more background: In the context of testing market efficiency I want to test whether a certain serious fulfills the martingale property. However, the martingale property of mean independence is not testable straight forward. As a random walk with i.i.d innovations also implies a martingale, I switched to testing for that as a "sufficient condition" for the series to exhibit the martingale property. Jun 24, 2021 at 5:39