Chaos Theory Research Paper

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Chaos theory is the study of deterministic difference (differential) equations that display sensitive dependence upon initial conditions (SDIC) in such a way as to generate time paths that look random.

This research paper (a) explains what ‘chaos’ is in mathematics, (b) explains why economists became interested in this concept, (c) sketches economic forces that can ‘smooth’ out dynamical irregularities that can lead to chaos, (d) very briefly discusses theoretical models that generate chaotic dynamical systems as equilibria, and (e) discusses empirical testing for chaos. We shall spend most of the time on (e) because even though the vast bulk of studies has not found convincing evidence for chaotic dynamics that are short term predictable (out-of-sample) using nonlinear methods with high ability to detect chaos, the quest for chaos has generated useful statistical methods.

1. Chaos

The mathematical apparatus for rigorous treatment of chaos can be quite intimidating to the nonmathematician. Therefore we shall attempt to explain the concepts verbally as much as possible. This expositional strategy will allow the reader to decide whether it is worth their while to investigate this area further. Reference to wide-ranging surveys are given.

We follow Brock (1986) for a very brief treatment of some mathematics of chaos below. A deterministic discrete time dynamical system on n-dimensional space is a system,

of n difference equations in n variables. Here F is a map from n-dimensional real space to itself, x (t ) and x (t +1) are n-dimensional vectors at date t and t +1 respectively, x₀ denotes the initial condition vector, and t is time.

A useful pedagogical example is the scalar ‘tent map’ T from the closed interval [0, 1] to itself defined by

The hallmark of chaos is SDIC, i.e. for two nearby initial conditions, x (0), y (0), the map F magnifies the distance between them:

for most pairs of nearby initial conditions. Here denotes a distance measure. The tent map (2) illustrates this idea nicely because |x (0) -y (0)| is magnified by a factor of two for small enough |x (0) – y (0)| except for the rare case where x (0), y (0) straddle the maximizer, x = 0.5.

A popular way to precisely capture the idea of SDIC and to measure it is the largest Lyapunov exponent, L, which is defined by the following limit as h tends to infinity,

Here DG denotes the n by n derivative matrix of map G, F^h (x (0)) denotes the application of map F to the n-vector x (0) h times, v denotes a nonzero n-dimensional direction vector, ‘ e denotes dot product, |·| denotes the Euclidean norm on n-dimensional space, and ln denotes natural logarithm.

Sufficient conditions are available for L (x (0), v ) to exist and to be independent of x (0), v for most initial conditions x (0) and most vectors v. A popular definition of chaos is:

Definition (Chaos) The map F is said to be chaotic if L > 0.

This is not the only definition of chaotic map in the literature, but it is a popular one and we shall use it here.

The value of L for the tent map T is L = ln (2) > 0 so T is chaotic by this definition of chaos. As we saw before, T displays SDIC. The quantity L measures how fast (on average per iteration) a tiny measurement error (captured by y) in the initial condition x(0) is magnified by the map. If the iteration h in (2) is thought of as a forecast then |DF ^h (x (0))·v| represents the error in an h-horizon forecast caused by measurement error at date zero. If L > 0, this error is growing exponentially as h increases by a factor exp (L). This kind of behavior is associated with deterministic maps generating random-looking time series output. Clearly the tent map example generates random-looking time series output. Another example of a type of chaotic map is a psuedo-random number generator for computers.

If the solution {x(t), t = 1, 2, …} to (1) is bounded for each initial condition, x₀ , under regularity conditions the limiting behavior of chaotic dynamical systems is contained in a set A that is invariant under application of F which is called a ‘strange attractor.’ It is called ‘strange’ because it is not a rest point or a p-cycle. Here a p-cycle is a collection of p vectors, x (1), x (2), … , x ( p) , such that

A rest point is a p-cycle where p = 1.

It is useful to briefly explain the notion of bifurcations and ‘routes to chaos.’ Consider a dynamical system with ‘fast’ and ‘slow’ variables, x(t), and a(t) where the fast variables x(t) have a rate of change that is much faster than the slow variables. Write

where ‘ ≤ 1’ means ‘much less than one’ to capture the idea that the second difference equation in (6) moves much more slowly than the first. Hence, it is useful sometimes to assume the fast variables have already converged to an attractor conditional on the value of the slow variables. A bifurcation value is a value of the slow variables such that passing through it leads to an abrupt change of the attractor of the fast variables. For example suppose the attractor is a rest point which abruptly changes to a p-cycle, p > 1; or a p-cycle which abruptly changes a more complex attractor. There is a classification theory for bifurcations (Kuznetsov (1995)).

A well-studied route to chaos in one dimensional discrete dynamical systems is the Feigenbaum cascade (and closely related Sharkovsky ordering). Here a rest point (i.e., a one-cycle) bifurcates into a two-cycle, followed by bifurcation into a four-cycle, followed by bifurcation into an eight-cycle, … , followed by bifurcation into a 2m-cycle, … to fully developed chaos as a slow ‘tuning’ parameter increases. This particular route to chaos is used a lot in economics (Benhabib 1992). Some economists argue that fully developed chaos is not as useful to economic science as the analysis and classification of bifurcations themselves, because the evidence from data is stronger for the existence of bifurcations.

2. Chaotic Economic Dynamics?

Even though predictability of a chaotic dynamics is futile in the long term, nonlinear prediction methods can do a good job on short term prediction when chaos is present. Consider a general stylized dynamic economic model represented by the following stochastic dynamical system

where x(t) is a vector of economic quantities that represent the state vector of the economy at date t, R(x(t)) represents the response of economic agents to the state of the economy at date t, and the function f produces the new state vector x(t +1) at date t + 1. Here n(t) denotes a stochastic process, called ‘forcing noise,’ (often chosen to be an independent and identically distributed sequence of mean zero, finite variance random variables) which represents outside shocks to the economic system and s denotes standard deviation of each random variable n(t).

Many economic models (e.g., many of the models treated in Benhabib 1992) can be put into the mathematical form of Eqn.(7). If we define

Then Eqn.(1) is a difference equation of the form (1)when s = 0. Since economic theory generates nonlinear dynamics it is theoretically easy to produce economic models of the form (8) that generate chaotic dynamics when external forcing noise is set equal to zero (e.g., Benhabib 1992, Boldrin’s chapter in Anderson et al. 1988, Dechert 1996).

An important issue is whether, in such models, the parameter values needed to obtain chaos (especially a chaos where prediction in the near term can be improved by exploiting it) are consistent with empirical measurements in economics. For example, when real interest rates are low (as they typically are), intertemporal smoothing operations such as intertemporal arbitrage (e.g., the trading of assets across different points of time in order to profit after adjustment for interest costs and for risk bearing) tend to squash cycles and chaos in economic systems with a rich enough variety of market instruments.

Brock’s chapter in Anderson et al. (1988) goes through a list of empirical plausibility checks for the magnitude of ‘frictions’ needed to obtain short term forecast able deterministic cycles and chaos at high to usuably (usuably from a policy-relevant predictive point of view) low frequencies in macroeconomic and financial data. Brock’s general conclusion is not encouraging for the presence of persistent deterministic cycles and chaos for economic dynamics of macro variables and financial asset returns in countries with well-developed asset markets and financial markets but maybe better for countries with poorly-developed markets. It is argued that well-developed market economies just have too many instruments through which self-interested intertemporal smoothing behavior can operate to be consistent with persistent deterministic chaos or cycles at high to medium frequencies.

After all, the usual arguments behind the efficient markets hypothesis suggests that there is money lying on the table if there are potentially predictable patterns such as deterministic chaos or deterministic cycles in asset returns. That is, arbitrage opportunities are available unless interest rates are high, market instruments for arbitrage are few, risk adjustments are high, or constraints on borrowing and lending are high. Of course, none of these arguments are germane to the presence of chaos and cycles at very low frequencies. For example, Day’s chapter in Pesaran and Potter (1992) is more encouraging for the presence of persistent ‘complex’ economic dynamics such as chaos, especially at long-term historical frequencies. However, while theory is suggestive, there is no substitute for rigorous statistical testing.

More interesting to empirical economists is whether there is evidence in economic and financial time series data for the presence of chaos. While the evidence is weak for financial data and for macroeconomic data for the presence of chaos which can be short term predicted, there does seem to be useful evidence in favor of nonlinear structure which can be predicted in the short term, conditional on appropriate information sets. The evidence for extra unconditional out-of-sample predictability (i.e., prediction of data that was not in the sample used to fit the model) using nonlinear methods appears to be weak for asset returns data. The evidence for extra conditional out-of-sample predictability of asset returns is better (LeBaron 1994). See the review of Brock et al. (1991) with especial attention to the references to Diebold, Nason, and LeBaron. However evidence for bifurcations, complex dynamics, abrupt changes, and other nonlinear phenomena seems quite strong (Dechert and Hommes 2000, see especially Chavas for animal dynamics; Carpenter et al. 1999, for ecological dynamics; LeBaron 1994, for nonlinear patterns in financial data; Dechert 1996, for an overall review of evidence).

3. New Statistical Methods

The quest for evidence of deterministic chaos, deterministic cycles, and other complex deterministic persistent patterns in economic and financial data, has inspired the development of new statistical methods. These methods have turned out to be useful in areas having nothing to do with deterministic chaos. We briefly explain two of them here. The first method is a specification test called the ‘BDS test’ by many writers (Bollerslev et al.’s chapter in Engle and McFadden 1994, De Grauwe et al. 1993). The method emerged out of the work that culminated in Brock et al. (1996). Here is a very brief explanation.

Suppose one formulates a model that relates a set of variables to be predicted to another set of variables called predictor variables, and fits this model to data and saves the residuals. If one has done a proper job of theorizing, model formulation, and model fitting, then the residuals should be unforecastible using histories based upon observables. The BDS test is used to formulate and carry out tests for unpredictability of residuals of fitted models. (See Brock et al. 1991 for extensive discussion and Monte Carlo work.) The BDS test has been used to test the adequancy of fitted models to data (Dechert 1996, De Grauwe et al. 1993, Pesaran and Potter 1992). The BDS test plays a similar role for nonlinear and general models as the Box and Jenkins Q-test does for auto-regressive integrated moving average (ARIMA) models (Box and Jenkins 1976). Tests like the Q-test and the BDS test are useful for testing the adequacy of fitted models and evaluating whether the evidence warrants a more costly exploration of alternatives to the null hypothesis model. See Dechert (1996) for a collection of studies where the BDS test is used in this manner.

The bootstrap-based specification test is a more sophisticated specification testing method. This method is especially relevant for detecting subtle nonlinearities in settings like finance where the economic logic dictates that any patterns are likely to be hard to detect (e.g. Maddala and Li’s chapter in Maddala and Rao 1996). The idea here is to use a version of bootstrap to compute the null-model distribution of statistics gleaned from various trading strategies, and to use the data values of these statistics to suggest refinements of the null model in financially relevant directions. For example, in financial applications, the null model is sometimes taken to be a random walk. This procedure emphasizes development of statistical quantities to evaluate the null model which are motivated by the behavior that one is modeling. This method was made possible by recent technical increases in computing speed, and reduction in cost, as well as advances in computationally based inference methods such as bootstrap.

4. Conclusion And Future Directions

The study of chaos and general complex dynamics as well as general complexity theory in economics, finance, and social studies has already been fruitful in generating new methods and new evidence. This kind of work is likely to grow in prominence as computational costs fall and advances continue in computational-based methods of theory and statistical inference.

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