Mathematical Investor (MAFFIA) I’ll give you the benefit of the doubt and assume you misread this post, and that you didn’t choose to defraud readers of your blog. Here are a few points to clarify your misunderstanding of the arguments above.

I) STATIONARITY CANNOT BE TESTED: Throughout this post, I make the case that stationarity, as a property of a time series, simply cannot be tested from a single realization, WITHOUT MAKING ANY ADDITIONAL ASSUMPTIONS, and I thoroughly explain why. The aim here is to draw the reader’s attention to the fact that, existing so-called ‘stationarity tests’ do not test whether the underlying process is stationary, but rather test whether, ASSUMING THE MATHEMATICAL SETUP UNDERPINNING THE DERIVATION OF THE TEST STATISTIC and ASSUMING ONE HAS ENOUGH DATA TO CHARACTERIZE THE UNDERLYING PROCESS, empirical evidence of stationarity can be found from the data.

Contrary to your understanding (as explained in the blog post above), this does not mean all existing ‘stationarity tests’ are useless, or that one should not assume that one has enough data to characterize the underlying process. This means that the operator needs to be mindful of the interpretation he/she makes of a ‘stationarity test’ success or failure and, in particular, the operator should bear in mind that the mathematical setup underpinning the statistical test cannot be invalidated by the test itself. This does not mean that the test is useless. This means that any result from the test should always be used IN CONJUNCTION WITH the foregoing mathematical setup and underlying assumptions, and one should bear in mind that derived results are only valid to the extent that mathematical setup relied upon is valid. For example, one cannot conclude from a unit root test that “a time series is stationary”, but one can conclude that “assuming that the underlying process is an auto-regressive process, it is stationary with confidence level x”, where x is related to the p-value of the test used.

It is simply impossible to test whether a time series is stationary from a single realization; any data scientist attempting to do so should not trust the tools he/she is using, as he/she is probably misusing such tools. Making data scientists understand that they always test stationarity under specific assumptions empowers them to properly understand the tools they use, and allows them to question underlying assumptions if/when needed.

If you still don’t get why stationarity can’t be tested in isolation, and/or disagree, why don’t you suggest a statistical test that tests whether a time series is stationary without making any additional assumptions?! Considering the hyperbolic claims you make in your blog post, you owe as much to your readers and mine!

II) USING (NON)STATIONARITY AS AN AXIOM: Given that neither stationarity nor nonstationarity can be proved or disproved statistically in isolation, either one can be used as an axiom — the same way unit root tests use as axiom that the underlying diffusion is an AR process. Such an axiom can be guided by expert knowledge in certain cases — contrary to what you claim in your blog post, I have never said otherwise.

In the absence of expert knowledge, one can choose to go with the axiom that is the MOST USEFUL. At Pit.AI Technologies, we are humble enough to believe that all models are wrong, but some are useful. In our Yellow Paper, we assume that time series of returns are stationary because it is a MORE USEFUL/GENERAL alternative axiom to the traditional Gaussian i.i.d. assumption! This relaxation allows one to empirically see evidence of memory and/or nonlinearities in time series of returns, and to better capture financial risk.

III) THE ITERATED DIFFERENTIATION FALLACY: You either do not have the right mathematical background to understand this, in which case you should refrain from talking about what you do not understand, or you are displaying intellectual dishonesty. In this very post, we give an example of a time series that cannot be made stationary, no matter how many times it is differentiated in the backshift operator sense. Instead of relying on misguided arguments from authority, WHY DON’T YOU TELL US HOW MANY TIMES YOU NEED TO DIFFERENTIATE THE TIME SERIES {W_{exp(t)}} ABOVE TO MAKE IT STATIONARY?! Any decent graduate student in maths who too stochastic analysis 101 can tell you that the reasoning above is solid.

IV) THE MEMORY VS. STATIONARITY AND SKEWNESS/KURTOSIS VS. STATIONARITY FALLACIES: We do not engage in unsubstantiated narratives and arguments from authority, we deal in logic, maths, machine learning, and data. In this post, I gave a specific example of a time series that has maximum memory AND is neither skewed, NOR leptokurtic, and an example of a time series that is stationary AND has maximum memory. WHY DON’T YOU FOCUS ON THOSE EXAMPLES AND PROVE ME WRONG?

V) OVERFITTING: Our readers will not be fooled by your unsubstantiated claims that our Yellow Paper is a case study of overfitting, and that our technology was not built with an understanding of how markets work. From our very first medium post, we have been promoting a finance-first approach to machine learning research, one that recognizes that, for a ML revolution to arise in finance, we need to acknowledge that ML methodologies that worked in other domains might not necessarily be transposable to finance. You simply have no understanding of what we do to make such outrageous claims, and quite frankly, given the aforementioned blog post, I question the depth of your command of probability theory and your understanding of modern machine learning methodologies.

It is very easy to sit on the sideline and write blog posts about why most start-up hedge funds fail that only consist of basic arguments and naive Monte-Carlo simulations. The field doesn’t need more storytellers wannabe-ML-experts, who rely on pseudo arguments from authority to attack small hedge funds in order to suck-up to and get favors from bigger ones while lacking a basic understanding of modern machine learning. The field needs more capable researchers, pushing the boundaries of ML methodologies applied to finance, and with a #FinanceFirst mindset.

If you want to criticize our research, we welcome precise logic/maths/ML-based arguments. Unlike your group and your blog, we do not post articles and comments anonymously, we take responsibility for all our posts, and we have an open comments section to facilitate public discussion. Our readers are smart enough to determine who is the fraud between you and I.