The empirical work based on the BDS statistics presents does not deny the presence of non-linearity and, on the contrary, suggests the relevance of studying the eventual presence of chaos in some economic series. The computation of low dimensions, the general rejection of the null of IID and the detection either of positive Lyapunov exponents in some of the macroeconomic series
(for example, Brock, 1986; Scheinkman and LeBaron, 1989b; Brock, 1990a; Dechert and Gencay, 1992) or of self-similar scaling properties (Mantegna and Stanley, 1995) provide a good case for studying chaos. Yet the general consensus is that the existence of chaos could not be proved: 'Hence the weight of the evidence appears to be against the hypothesis that there is chaos in economics and finance' (Brock, 1990a, p. 430), or 'The direct evidence for deterministic chaos in many economic series remains weak' (LeBaron, 1992b, p. 1).10
In fact, in spite of the indicative empirical results obtained so far and as a consequence of the difficulties of the proof, a large number of authors tend to present a dismissive conclusion about chaos: for example, Brock and Malliaris (1988), Baumol and Benhabib (1989). The only authors strongly claiming to have provided a conclusive demonstration of the existence of chaos are Barnett and Chen (1988), who used Divisia monetary aggregates, but their conclusion was challenged almost immediately by Ramsey et al. (1990), who transformed the data in order to obtain stationarity and then rejected the conclusion of chaos. DeCoster and Mitchell (1991) obtained similar conclusions to those of Barnett and Chen.
Inspecting a series of exchange rates and obtaining a fractal dimension and a positive largest Lyapunov exponent, Scheinkman and LeBaron (1989b) applied the reshuffling diagnostics, getting a larger dimension measure, and consequently did not reject the hypothesis of chaos. Likewise Frank and Stengos (1989) for the gold and silver returns in the London market. In other cases, Brock (1986) excluded the existence of chaos in spite of the positive Lyapunov value and the fractal dimension, since the computed values were not invariant to the magnitude of the radius e, measuring the local distance. Frank et al. (1988), Sayers (1988a, b, 1989) and Granger (1991, p. 263) reached the same conclusion, accepting the non-linear hypothesis but rejecting chaos.
One of the interesting features of the debate between Barnett, Chen and Hinich and their critics, Ramsey, Sayers and Rothman, was the topic of the definition of non-linear dynamical systems as opposed to chaotic systems. In fact, the strict distinction between the two types of systems - in the sense of considering chaotic systems a very peculiar and well identified subset, clearly distinguishable from the other members of the general class of non-linear dynamics - seems to be a unique characteristic of economists involved in complexity theory. This is striking, since the available methods for identifying dynamics and discriminating between a general non-linear structure and a peculiar chaotic process are still so rough and underdeveloped.
In the study of moderately complicated dynamical processes in physics, the generally accepted attitude is to put the burden of proof on those denying chaos, and to accept chaos while no refutation is presented. Yet, in econom ics, the current attitude is the opposite one. The reason for the difference is obvious: physics is not constrained by the constitutive concept of equilibrium, and therefore can dispense with the notion of an intrinsic, well-defined and unique order.
The acceptance of the null in the test of hypotheses is therefore eased by this general option. It is true that current technical limitations impose that form for the BDS test, since the distribution under the alternative hypothesis is not known; but it is also remarkably adjusted to the questionable a priori view dominating mainstream economics. Therefore the BDS does not test for chaos and not even for non-linearity: a rejection of the null may arise from any sort of dependence in the process. The crux of the matter is that common practice, in the face of a situation in which both an autoregressive process of low order n and a non-linear alternative may explain the variability in data, is to accept the linear specification.
It is true that it is virtually impossible to progress in the detection of chaos in economic series while we have no definition of a statistical procedure or a test of chaos defined as the null, but it is also arguable that the current choice of the basic assumptions constrains the development of the theory, imposing a bias against chaos.
Furthermore, some general philosophical problems cannot be avoided in this context. A sophisticated inductive inference, such as the Popperian infirmationist strategy, is also incompatible with the use of the Neyman-Pearson framework, since there is no prediction derived from the alternative hypothesis under inspection, which is accepted simply if the null is rejected and is not by itself submitted to any sort of test; in other words, it is not the basis for any refutable prediction.
However, the crucial question about the use of probability theory in this framework concerns the role of randomness and determinism in models of the economy. The lack of controlled experimental protocols in economics imposes a clear limitation on the use of classical inductive inference and on the definition of the size and power of the tests. Yet it is unreasonable to accept that purely deterministic models can ever explain the working of an economy (Scheinkman, 1990, p. 35). The traditional trade-off between purely deterministic systems and purely exogenous small, random and unimportant shocks is a response to this difficulty, which has been traditionally solved by the postulate of the juxtaposition of both concepts of evolution.
But chaos theory introduced a major shift from this point of view, since it suggested and proved for a specific class of models that deterministic systems could account for time paths virtually indistinguishable from those of traditional systems driven by stochastic impulses: 'white chaos' is not statistically distinguishable from stochastic series (Liu and Granger, 1992, S27). However, one cannot conceive of a deterministic model accounting for all the possible social interactions, as Scheinkman noted, and some noise is always present. Therefore the problem can be redescribed as interpreting the complex generation of intrinsic and extrinsic noise. Moreover, the question arises whether they are separable, and whether their distinction can indeed be established in a non-linear system: 'There is indeed a deep philosophical question concerning the difference between determinism and stochasticity or "randomness"' (Brock and Sayers, 1988, p. 74). This difficulty was noticed in the early discussions about the detection of chaos in economics (Ruelle, 1994, p. 27). In the same sense, Barnett and Hinich argued that this 'deep philosophical question' is virtually unsolvable:
It is well known that solution paths produced from chaotic systems look very much like stochastic processes. This produces a virtually unsolvable problem: should we view chaos as a potential explanation for stochastic appearing data (Barnett, Chen), or should we view stochastic processes as a potential explanation for chaotic appearing data (Ramsey, Sayers, Rothman). (Barnett and Hinich, 1993, p. 255)
Or, paraphrasing Boldrin (1988, p. 250), the question remains whether we should take Laplace or Poincare's point of view. Granger suggests keeping to the tradition:
As economic data include measurement error that is almost certainly stochastic, it seems unlikely that chaos can be observed with the length of series currently available. It seems that econometricians are well advised to continue using the techniques of classical probability theory. (...) The inherent shocks to the economy plus measurement errors will effectively mask any true chaotic signal. Thus, it follows that it will be a sound, pragmatic strategy to continue to use stochastic models and statistical inference as has been developed in the last two decades. (Granger, 1991, p. 268)
Poincare's alternative points in another direction: randomness ceases to be considered either as a perturbation or as a meaningless encapsulation of all other factors, and is defined as the very substrate of the economic evolution, as a constitutive part of the evolutionary process itself. In that sense, the economies are defined as complex processes with sensitive dependence on initial conditions. Consequently, the distinctions between intrinsic and extrinsic randomness, as well as between deterministic and stochastic processes, are overruled. In economic series, the holistic framework that accounts for the creation of both order and disorder blurs these antinomies.
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