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K-State Physics



Michael J. O’Shea


Small quasi-periodic signals in very noisy time series - application to financial time series.


Forecasting Ability of a Periodic Component Extracted from Large-Cap Index Time Series,  M.J. O’Shea, Journal of Forecasting, M. J. O’Shea, 5 April 2016 (2016)

The ‘Sell in May and buy at Halloween’ adage has been around for many years and quoted by stock market pundits.  If there is any truth to this adage, then there must be a periodic component embedded in the price of indices.  A visual inspection indicates that large cap indices have stochastic (noisy) and other non-periodic contributions.  If such a periodic contribution is present it is hidden in these much larger contributions.  It’s important to note that this contribution does not have to be perfectly periodic – it can have variations in period and variations in amplitude.  Two properties of this quasi-periodic component must be present:

·   it’s repeat time must vary about a certain average value with the variations about the average period being significantly smaller than this average period

·   the variations in amplitude must not average to zero.

It is very difficult to use standard methods such as Fourier/spectral analysis to extract such a component.  In general, it is not possible to uniquely separate such a quasi-periodic component from these other contributions mathematically.  It is however possible to determine the average contribution of this quasi-periodic term.  This can be done by folding the time series to average out the large non-periodic contributions leaving behind the average of the quasi-periodic contribution.  This folding leads to a so-called repetition function. 


This repetition function is shown above for the Dow Jones Industrial Average. A clear periodicity is present with a maximum in May/June and a minimum in October.  Scaling the Dow Jones Industrial Average with the cost of living index removes some of the gradient but the averaged periodic term is still present.  We looked at eight large cap indices and similar periodic terms were present in six of them. 

It’s important to note that the presence of a quasi-periodic term does not mean that an investor can take advantage of it.  To see if this average periodic term has any predictive power out-of-sample tests were done and indicate some limited predictive power for each of the indices exhibiting a quasi-periodic term.

Caution: None of this is investment advice; it is simply work looking at the predictive power of a quasi-periodic term extracted from a time series.

To give insight into the method, test data was generated and analyzed here using this method.  This folding method, the autocorrelation function method and spectral/Fourier Analysis are compared. 

The effect of outliers is considered here.

Finally the repetition functions of all the DJIA components are given here.