Not Robust

By Steven J. Grisafi, PhD.

Perhaps it was coincidence, we have no way of knowing for certain, but on Tuesday morning September 24, 2019, when I included the previous evening’s U.S. Treasury interest rate yields into my computations the relaxation time calculation reached absolute zero. This is, of course, absolute zero with regard to sixty-four bit floating point calculations. Seven days before, during the overnight period culminating on Tuesday morning September 17th, the New York branch of the Federal Reserve Bank of the United States lost control on their overnight lending rate. One of the more difficult decisions an empirical scientist must make when evaluating non-stationary time series data is choosing the length of the period for the time series data. If the period is too short, extreme volatility would mask any trends that may exist within the data. If the period chosen is too long, historical bias can also mask any current trends. From experience I know that an occurrence which unfolds in real time is not immediately reflected in the performance of its time series data. The time and strength of any effect an occurrence may register within its time series data depends strongly upon the length of the period chosen for the time series computations. Although the real occurrence during the overnight period culminating Tuesday morning September 17th may not have had any bearing upon the computational phenomenon occurring Tuesday morning September 24th, both were a floor breach.

I had intended to collect interest rate yield data for one year before truncating the time series data by beginning an annual time series rolling period. But the floor breach on September 24th compelled me to begin a rolling time period with the next data point taken on September 25th. When using a rolling period for time series computations the addition of each new data point added to the data set requires that the oldest data point within the set be disregarded. By the 24th of September I had accumulated 22 weeks of data so that on the 25th of September I could begin a 22 week time series rolling period. The floor breach within the calculation occurs mathematically when the argument of an exponential function changes sign from negative to positive. The argument of this exponential function is the slope of a line. With the addition of a new data point on the morning of Wednesday, September 25th our estimate of the relaxation time for the United States Treasury yield curve jumped to infinity. Upon beginning the 22 week rolling time period the relaxation time fell to approximately seventeen days. It is known mathematically that such abrupt jumps can occur within a function as its domain changes from one permissible sheet to another. In our case that jump would be across the boundary from one quadrant into another on the unit circle. Finance Rheology utilizes calculations on the unit circle so as to provide periodic boundary conditions for a Dirichlet computation.

Having a firm grasp upon what is happening mathematically with our time series calculations provides a basis for postulating what could be happening within the dynamical system that the calculations seek to model, but it does not guarantee our explanation to be valid. The strange behavior that I have observed over the twenty-five week period, that I have been studying the dynamics of the treasury interest rate yield curve, leads me toward an opinion similar to that which the economist, and central banker, Stephen Williamson has expressed: The floor system of the Federal Reserve Bank is no good. I tend to agree with the central banker Mark Carney of the Bank of England that long term debenture interest rates are determined globally. The Federal Reserve Bank of the United States has no control over the long term interest rates of United States Treasury bonds. At best, all that I believe one can say is, that the Federal Reserve Bank may have some control over short term interest rates. But that control is certainly not robust and definitely flawed when using a floor system. The failure of the floor system has significant repercussions. In my opinion, it invalidates the use of negative interest rates, which implies an interest rate to be charged, rather than paid, on excess reserves commercial banks may hold at the central bank.