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I'm writing a little program to explore the relationship/correlation between a stock's current price and its historical seasonality. I'm using the correlation to compare the daily close % returns of each for 60 days, and sometimes the resulting correlation figure is a little confounding. I'm certainly no statistics expert and I could very well be doing something wrong, but I've checked and double-checked my numbers in Excel, and also a free online stats site (www.socscistatistics.com) and my data seem fine. Anyway, I'll submit to you here please, one stock example that has me scratching my head.

Here is a screenshot of a line-chart of the last 60 closes of a stock (TTWO, in green), compared to its 21-year historical average seasonal moves for that same trading day (in yellow).

Charted comparison of stock and its seasonality

To be sure, they are certainly not 100% in synch with each other, but the correlation is -0.22. Given the general upward bias of both lines, I would have expected at least a positive number, albeit weak. This confounds me. I had thought correlation would be the metric to use for this test, but maybe I just need to use another statistic (like regression analysis?) to better show the relationship. If there is a better metric to use, please recommend!

Also, I would have liked to attach a small file of my data (60 numbers for each of the stock and its seasonality), but can't see how to do that.

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Update Feb. 26: Clicking on "Add a comment" seems to do nothing (just auto-scrolls the page up), so it seems like editing my question was the only way i could reply. Anyway, thanks to Dave and Nick for the replies. I'm ideally looking for a statistical measure to quickly and roughly enumerate the relationship between each stock and its seasonality. I can see correlation is not perfect and there will always be "head-scratchers" like the example I gave - that's just the nature of the beast. Instead of trying to compare DAILY % change with correlation, I tried using WEEKLY % changes - figuring the "broader" outlook would at least reveal the common upward positive bias. However, it yielded an even larger negative value of -25.5, so that was surprising. Would there be a "better" metric for me to use for what I'm doing? Thanks.

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  • $\begingroup$ Correlations between time series are often enigmatic. To understand a correlation draw the corresponding scatter plot, noting that when any pair of values occurs in time is irrelevant to the correlation.That said, connecting points in time order can be instructive. Loosely, I see one period when green is rising and yellow is constant, and one where green is constant and yellow is rising. Put those two together, and noting a lot of scatter about each pattern, a weak correlation is indicated. The correlation between one realisation and its historical average is difficult to analyse. $\endgroup$ – Nick Cox Feb 25 at 1:51
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What you are seeing is not likely to be seasonality. Evgeny Slutsky, I think in 1922, showed that "additive errors" produce a sinusoidal pattern, when the shocks are independent. In other words, if you have a model such as $w_{t+1}=Rw_t+\epsilon_{t+1}$ then, even though this is a linear equation, you will get a sine wave effect. Using Excel, you can induce this yourself by adding a normal random variable.

Second, a time series like this will have autocorrelation. Your top line is an autoregressive process, most likely, and your bottom line is a moving average process and both will have autocorrelation.

Third, let us assume that the top line is $w_{t+1}=Rw_t+\epsilon_{t+1},R>1.$ If that is the case, then no standard solution to solve for $R$ exists using anything that minimizes squared loss. You cannot use Excel to solve this. Because of that observation, it is probable that any correlation would be spurious. Note that if you did not believe that your wealth would increase over time, then you would not invest, so $R>1$ must be true. Somebody may be able to show, by theorem, that any correlations would be spurious, but as we have a pretty good proof that no squares minimizing solution exists that is well-behaved, it isn't unlikely that any Pearson product-moment correlation coefficient would be spurious as well.

Finally, if you are running a procedure on many stocks, some stocks will have you scratching your head because they will represent natural, but extreme, behavior. That does not imply that there is anything unusual or abnormal about the security. A purely average security, where every investment has an identical rate of return, will be at extremes due to a random pattern of independent shocks.

You can test that in Excel too. Create a hundred similar time series, each using random, normally-distributed shocks. A few will look very weird in both directions. Also, if you would look at them with some metric, any metric will do, some of them will be extremes as measured by that metric. That is true even though they use identical growth rates in the software.

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