Periodicity

Distinguishing Periodicity from Regularity

By Steven J. Grisafi, PhD.

Long time viewers of the central bank computations posted here at Grisafi Finance Rheology may have noticed a similarity between the Fourier spectrum computed for the Bank of Japan with that for the European Central Bank. The observation of this similarity is correct and there is a reason for it. There is an inherent impediment to evaluating the Fourier spectra for the central banks’ money supply periodicity due to the infrequent data available on an annual basis. This impediment vanishes for Fourier spectra evaluations on other analyses whose data collection occurs more frequently. However the nature of the problem addressed for Fourier spectra analysis applied to any of our analyses will always show some similarity existing between various analyses.

This apparent regularity is due to the necessity of including within the Fourier spectrum analysis the schedule upon which the data is evaluated. We have chosen a rolling annual basis. The reporting of any financial data always reflects the periodicity of the business week. Only for the cryptocurrency analyses do we have continuous data availability. All of our Fourier spectra analyses are based upon the time schedule, recorded as the number of the day of the year upon which it became available, and the ratio expressed as the difference between the sum of the minor eigenvalues divided by the dominant eigenvalue for the square of the raw data matrix. Thus, every Fourier spectra analysis is based upon two column vectors, the daily schedule and the eigenvalue ratio. Consequently, one will always observe the effect of the data submission time schedule in any of our Fourier spectra.

As a result of the inclusion of the data time schedule one must search for periodicity within any particular Fourier spectra by comparison with another evaluated on the same data time schedule. Essentially, one must subtract from both the regularity they both possess. The remainder that persists is whatever true periodicity exists within each particular data set. With infrequent data sets, such as the central bank money supply, the regularity of the time schedule overwhelms any actual periodicity within the financial data such that no periodicity is observed. If the reader is curious as to why, the money supply data of the Bank of England has not (yet) show the same regularity as the data for the Bank of Japan and the European Central Bank, it is because our Bank of England data set has not yet reached the same steady state as our data set for the other two banks who also issue monthly data. A data loss caused us to lose several months of data for the Bank of England. The Federal Reserve Bank of the United States issues data weekly and therefore cannot be compared directly with the other three central banks.

The fifty-two weekly data sets available annually for the Federal Reserve Bank show sufficient similarity to the six day per week daily data sets such that one might suggest extending the monthly issuance central banks to a decade long (ten year) rolling basis. This would make available 120 data sets on a rolling basis. If there should exist any true periodicity within the central bank money supply it might become visible over the time schedule regularity. However, then there would arise the difficulty of historical basis within the data set such that Grisafi Finance Rheology considers the data extension unwise. Consequently, we can surmise no periodicity with the monthly data issuance central banks whether or not any such periodicity does exist. The finance profession is unlikely to be concerned about this. Their time horizons are often considerably shorter than one year. Even government officials would be uninterested in any central bank periodicity that extends more than a few years. Therefore we turn our attention to the data sets available on a daily basis.

Three of our six day per week daily data sets have achieved a steady state rolling basis. They are the Treasury Debentures, Dollar Exchange, and Precious Metals. Shown below are graphs of the Fourier spectra for the three analyses:

Their similarity is apparent. But so are differences between them. All three have an upward peak at approximately eight weeks and a down peak at approximately 16 weeks. But even these peaks are slightly different. These two peaks can be seen in the central bank graphs as well. They are a consequence of the regularity of the data time schedules. They are not an indication of any underlying periodicity within the financial data. Since all three share the same time schedule, the characteristics that all three have in common must be subtracted from all three to show any existing periodicity within the financial data. This task could be achieved in several different fashions, either numerically or graphically. Merely superimposing the three graphs upon one another would provide some indication. The most obvious conclusion one can draw is that the Precious Metals spectrum exhibits beats. Notice that there are two strong, approximately one week long, beats at the left side of the graph. Smaller, weaker beats persist throughout the entire spectrum. The Treasury Debentures and Dollar Exchange graphs do not show any beats.

Grisafi Finance Rheology has devised a procedure for quick comparison of any Fourier spectrum. Any numerical comparison should be based upon the Fourier coefficients. For quick comparison, we sum the complex coefficients of the expansion. Then, using the real and imaginary parts of the sum, we evaluate the phase angle. For the examples given today, the phase angle for the Treasury Debentures is 1.861474529 radians; for the Dollar Exchange the phase angle is 1.863834287 radians; and for the Precious Metals the phase angle is 1.863391751 radians. The average phase angle is 1.862900189 radians. Only the phase angle for Treasury Debentures is below the average. The Treasury Debentures have the least activity in their Fourier spectrum.