Nonlinear Time Series, Complexity Theory, and Finance

This article describes statistical aspects of a line of recent work in finance that is associated with the words “nonlinearity,” “long-term dependence,” “fat tails,” “chaos theory,” and “complexity theory.” We shall give a rather lengthy introduction in order to give the reader a road map, we shall indicate section headings where each issue is discussed in detail or give references if the issue is not dealt with in this article. Before we begin, let us give a brief overview of some recent “trendy” topics which shall play a role in this article. Centers of research in complexity theory such as the Brussels School, the Stuttgart School, the Santa Fe Institute, and hosts of other related centers and institutes springing up around the world are turning to computer-based methods as well as analytical methods to study phenomena that lie within the rubric of “complex systems.” We shall devote part of this article to an argument for a style of research in statistical finance where models inspired by direct theoretical arguments are estimated by computer-assisted methods such as MSM and where model adequacy (specification testing) is done by bootstrapping financially relevant quantities under the null. That is to say, the quantities that are inputted into the specification tests are themselves motivated by the type of economic and financial behavior one is trying to study. For example, distributions of statistics gleaned off of trading strategies are bootstrapped under the null model being tested in Brock, Lakonishok, LeBaron (1992), and Levich and Thomas (1993). This approach to specification testing is described in Section 4.