By Steven J. Grisafi, PhD.
Our previous blog post Impulse Stabilized indicated that we cannot assume that a relaxation time response to an impulse is Gaussian if the time series data indicative of the dynamical system is not stationary. Gaussian statistics pertain only to stationary dynamical systems. So one may become curious as to the actual functional form we do observe for the decay of an impulse response from our dynamic model of the United States Treasury interest rates yield curve. For the perturbation beginning during the overnight period on September 16, 2019 at the New York Federal Reserve Bank the data points are insufficient for a concise determination of what functional form the relaxation time decay may have possessed. However, prior to the September 16th perturbation we had observed, in exquisite detail, the relaxation time decay to absolute zero. The data for this case is presented here in the figure below.
Our first indication that this decay does not obey Gaussian statistics is the obvious asymmetry of its functional form. We do observe a long tail decay beyond the peak of the curve, which could be indicative of the Gamma distribution. In a previous theoretical study, which I had published as the book Market Dynamics, I found that the Gamma distribution ought be the most likely functional form for the dynamics of financial markets. At first glance it would seem that the relaxation time is supportive of this proposition, but we need to look more closely. Notice that the long tail decay, on the right hand side of the peak in the figure, is not monotonic. There is a noticeable inflection point occurring just prior to day 200. The presence of an inflection point indicates a much more complex functional form than the Gamma distribution possesses. Most likely, we should find that every relaxation time decay would have unique features adding complexity to its functional form. This means that there would be numerous arbitrary curve-fits to every impulse decay we may observe. Consequently, we ought not expect to have a priori knowledge of any relaxation time of our financial dynamical system with greater specificity than we could have by just assuming the decay to be Gaussian. Hence, as a general rule of thumb, we would use three times an observed relaxation time as the maximum extent of any perturbation to a financial system.