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Returns are demonstrably not serially correlated in most financial time series (Day 1 returns are uncorrelated to Day 2 returns etc.) . Since this is the case, how can momentum trading strategies work? What is the mathematical basis that can explain why momentum trading strategies work if there is not, in fact, a correlation between today's price movement and tomorrow's?

EDIT: As an example, take Bitcoin (it's what I'm looking at). I think it is intuitively/visually/conceptually obvious that there are strong momentum patterns. However, I have looked at autocorrelation in returns on timeframes ranging from 1 day to 30 days, and there is no autocorrelation. Perhaps my question should be the following: How can this "subjectively obvious" momentum be expressed, if not in autocorrelation of returns (which is absent)?

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    $\begingroup$ Time series momentum is about exploiting autocorrelation in returns. Historically there has been some serial correlation for individual stock returns for some horizons, though not so much for stock market indices. Also for fixed income and commodities returns. Cross-sectional momentum strategies can additionally exploit some cross-asset predictability patterns other than serial correlation. $\endgroup$
    – fes
    Commented Sep 12, 2021 at 18:29
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    $\begingroup$ You wrote: Day 1 returns are uncorrelated to Day 2 returns. Yes, but momentum strategies work at longer frequencies (6 months, for ex.) where there may be some correlation. But as fesman wrote it depends on the asset, the time period, etc. $\endgroup$
    – nbbo2
    Commented Sep 12, 2021 at 18:38
  • $\begingroup$ @fesman I am not very familiar with cross-sectional momentum, but I have a conceptual grasp on time series momentum. Even though you say "Time series momentum is about exploiting autocorrelation in returns", that's exactly the issue I'm highlighting - for most financial instruments, there is no autocorrelation in returns. Take Bitcoin, for example. I think we can agree that it is very momentum-driven , but if you run some basic analysis, you'll find that last day's/week's/month's returns are not correlated to next day's/week's/month's returns. How can this be? $\endgroup$ Commented Sep 12, 2021 at 21:04
  • $\begingroup$ @noob2 There are momentum strategies that work on the timeframe of 1-2 weeks, but still no autocorrelation in returns. As I mentioned in my response to fesman, take Bitcoin (or any crypto). I think we can agree that it has strong momentum patterns on timeframes smaller than 6 months, yet I cannot statistically detect any autocorrelation in returns on timeframes from 1 to 30 days. How can this be? $\endgroup$ Commented Sep 12, 2021 at 21:05
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    $\begingroup$ @VladimirBelik Take a long panel of individual stock returns. Look at the correlation between one month stock returns and the return of the same stock for the past year. You should be able to detect some significant correlation between the two. $\endgroup$
    – fes
    Commented Sep 13, 2021 at 6:31

2 Answers 2

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Let $R_d$ be the return for day $d$. Mean reversion means that $cor(R_d,R_{d+1})$, $cor(R_{d+1},R_{d+2})$, etc. are all negative--which is actually close to truth. Under conditions not too hard to come up with, you can have $cor(R_d,R_{d+2})>0$ and $cor(R_d+R_{d+1},R_{d+2}+R_{d+3})>0$ simultaneously with mean reversion. The last inequality is a two-day momentum. As correctly stated in the comments, momentum is usually observed over longer horizons (and occasionally also intraday).

Correlation inequalities are useful and demonstrated by a quant interview question (among others in this book): If the correlation between A and B is 90% and the correlation between B and C is 90%, what is the correlation between A and C?

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  • $\begingroup$ I understand the information you presented in your response. However, what I cannot understand is how momentum strategies can work when there is no detectable autocorrelation in returns. I am working with crypto, so I ran analysis on it. Even though there is (subjectively/intuitively) a clear momentum-driven aspect, I cannot detect any autocorrelation of returns on timeframes from 1 to 30 days. How can momentum strategies work while simultaneously not being any detectable autocorrelation in returns? $\endgroup$ Commented Sep 12, 2021 at 21:08
  • $\begingroup$ The idea is that subpopulation of returns may exhibit the autocorrelation you need, the question is how to know what the subpopulation is and how to detect that you are in it. $\endgroup$ Commented Sep 12, 2021 at 22:12
  • $\begingroup$ @HarryCrimmins Hm. Alright, so you're saying sometimes the autocorrelation exists, and sometimes not. But so then indeed, how would you detect this in an unbiased way? Of course if I look at a graph and just analyze the part where there is a trend, naturally there will be a correlation. But how could you independently detect this? $\endgroup$ Commented Sep 12, 2021 at 22:27
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    $\begingroup$ My data are in equities, not in crypto. Serial autocorrelations do exist in equities, but the the signal-to-noise ratio is small, not least due to quant competition keeping tradable signals close to balancing trading costs. People trade by using multiple signals combined, but even then the market is very competitive and mostly arbed out. Crypto is not arbed out this much yet, but you can probably guess the direction it goes. $\endgroup$ Commented Sep 13, 2021 at 0:32
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    $\begingroup$ @Vladimir Belik: If a momentum strategy did generate positive returns over some period $[T_1, T_2]$, then by the definition of auto-correlation, there had to be positive auto-correlation over that period. Of course, whether this exists in general and whether it exists for all stocks or just a subset is a different issue. My point is that, the following is not possible: a momentum strategy generated significant + returns over some period and + autocorrelation did not exist during the same period. Note also that testing for it statistically is not synonymous with the existence of it. $\endgroup$
    – mark leeds
    Commented Sep 13, 2021 at 0:46
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The crux of the answer is that a non-significant serial correlation will produce significant price action. Granger and Newbold (1973) showed that completely unbiased random walks of returns generated statistically significant "trends" in price more often than not :-)

[The rising variance over time violates the stationarity of price, if you want to get formal]

That is the essence of the problem here ;-) DEM

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  • $\begingroup$ So, are you saying that even small serial correlation will "add up" to produce significant price action? Or are you saying that even with non-existent serial correlation, sometimes there will just happen to be big price movements randomly (by chance)? Or am I missing the point completely? :-P $\endgroup$ Commented Sep 12, 2021 at 22:16
  • $\begingroup$ The former. Completely random walks WILL generate (fake) trends, as a matter of course. $\endgroup$
    – demully
    Commented Sep 12, 2021 at 22:49
  • $\begingroup$ So okay, I understand that completely random walks WILL (by chance) generate trends. However, then the second part of my central question remains - if there is no true underlying mechanism/driver of the trends that (randomly) occur in a financial series, why do momentum strategies work? That would suggest momentum strategies pick up on spurious price changes, meaning there is not a tendency for existing trends to continue, meaning these strategies should (on average) fail to outperform the underlying asset, but this is not the case. Do you see my dilemma? $\endgroup$ Commented Sep 12, 2021 at 22:57
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    $\begingroup$ Then we are in agreement :-) the point then being that a statistically significant momentum effect in price need not have a significant autocorrelation! If randomness can generate trends, then how distinguish those from insignificant momentum drift??? That's the essence of the problem here, in a nutshell. $\endgroup$
    – demully
    Commented Sep 12, 2021 at 23:20
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    $\begingroup$ @demully: I get your point about a random walk sometimes generating a trend so that the price can have a trend in it but it's spurious. What I don't get is why the autocorrelation calculated from the price trend can't be significant also? The sample acf can also show significant autocorrelations. In fact, in Stephen Taylor's text, "Modelling Financial Time Series", this is exactly what he tries to exploit except that he doesn't claim it's spurious. (unfortunately, his book is out of print). If you're saying that the spurious nature makes their future existence less reliable, I agree. $\endgroup$
    – mark leeds
    Commented Sep 14, 2021 at 2:47

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