Phenomenological Modeling

The Serenity of Empirical Practice

By Steven J. Grisafi, PhD.

I read something, published somewhere, long ago, which asserted that Modern Man is far more neurotic than was Medieval Man. It was the certainty of his faith that gave Medieval Man his serenity. When one attempts to solve a problem of pressing need, instead of propounding the methods by which the problem could be solved, one usually acquires the insight needed to solve the problem. When I tried to model the dynamics of asset price motions within a financial market, using linear force laws in the manner of classical mechanics, I found them to follow the Gamma Distribution. While this is not evidence that financial markets actually obey the Gamma Distribution, it does provide some insight. There are two things regarding the Gamma Distribution that are significant for our purpose. First, is that the mean and variance of the distribution are not independent. One cannot be changed without changing the other. Second, is that the long tail end of the distribution, when plotted on log-log graphs, is reminiscent of power law correlations such as Cobb-Douglas functions.

When I tried to apply the theory to the price motions of various assets that I placed within toy markets which I created, I found that the markets were open systems. The explanation one would provide as to what constitutes an open system varies with regard to the identity of the audience. Practitioners of the natural sciences would give a different answer than would the practitioners of the social sciences. But, for a practitioner of any empirical endeavor, the explanation is always the same: The zeroth moment is not constant. When one recognizes that the zeroth moment of a probability distribution is the normalization constraint, then one understands the problem. We cannot use normalized probability distributions for open systems. This does not imply that the methods of probability theory are not applicable. It only means that we cannot work with probability distributions in their normal form.

The author of one of the textbooks on control theory that I used as a graduate student insisted upon avoiding the use of matrix notation, and writing in complete form all of the summation formulae. His insistence was based upon his belief that abbreviated notation disguised the true computational burden required by the theory. This seems to me to be a problem with the use of the expectation operator by economists. It can often disguise the true nature of the computational burden that a theorist wishes to communicate to his colleagues. Since most development of theory is the progression of manipulated symbols signifying mathematical operations that need to be applied, any ambiguity within the nature of the computation that needs to be fulfilled can lead to confusion. While it may seem like an onerous burden to write in complete form the summation, or integration, required by an expectation operation, doing so would focus attention upon the fact that there must be a probability distribution which the phenomenon of interest obeys. Otherwise, there can be no expectation, and economists would need to describe their thoughts in some other manner.