estimate the dimension of any chaotic attractor by embedding the time series in m-dimensional space using a time-delay embedding. Each point x, in the time series is mapped to a point in m-dimensional space ffim as follows:

X( = (xt, xtx, xt 2x, ., r-(m i)t)> (13-1)

where the lag x is some positive integer (most commonly x = 1, as in Figure 13.6b) and m is chosen to be larger than the dimension of the attractor. In this way the univariate time series is mapped to a set of points in m-dimensional space, where the time ordering is not apparent but if there is a strange attractor it will be revealed by the pattern generated by these points.

If the system dynamics are chaotic then they will be governed by a strange attractor that has a finite, positive fractal dimension. Among the many definitions of fractal dimension one of the most common is the correlation dimension v (Liu et al., 1992; Grassberger and Procaccia, 1993a, 1993b). The correlation dimension is related to the correlation integral C(e) of a time series {xt} of length N. The correlation integral is defined by embedding the time series in m-dimensional space iJim, where the embedding dimension m is chosen to be suitably large, so that m > v. Then C(e) is given by

C(e) = lim{ number of pairs (i,j) such that x,- - x, < e}/N2 as / -» oo.

Thus the correlation integral C(e) measures the number of points in a ball of radius 8 in 9tm as the length of the time series increases. The correlation dimension v measures the rate of growth of C(e) with 8. In other words, C(e) ev for small 8. The correlation dimension v is normally estimated by finding a finite approximation to each of the correlation integrals for different e and then estimating dlnC(e)/dlne.

Another standard test for chaotic dynamics is based on the largest Lyapunov exponent X. A Lyapunov exponent measures the average rate of exponential divergence of neighbouring trajectories, so it is a measure of sensitivity to initial conditions. The greater the Lyapunov exponent, the faster the divergence in trajectories. If all Lyapunov exponents are negative then all trajectories converge, so a statistical test for chaos is to estimate the largest Lyapunov exponent, and if it is negative there is no evidence for chaos. If xl+i = f{x,) then the Lyapunov exponent is defined as

X - lim(ln\dffdx\/r) as r -» oo,

where the trajectory of x is {x, f(x), f2(x), f3(x), . . .}. To see that X measures the rate of divergence of neighbouring points, note that \df/dx \ exp(rX), and by Taylor linearization this is approximately equal to the difference between x and x + e after r iterations.

Since the function / is unknown (all we observe are the realizations of a time series) algorithms must be designed to estimate the Lyapunov exponents

The greater the Lyapunov exponent, the faster the divergence in trajectories

(Wolff, 1992). Estimates of the Lyapunov exponent by Abhyankar et al. (1997) for the S&P 500, the Nikkei 225, the DAX 30 and the FTSE 100 equity indices confirm the presence of non-linear dependence in the returns but provide no evidence of low-dimensional chaotic processes.

There are several important points to note about algorithms such as these that are designed to detect chaotic dynamics:

» The sample size should be upwards of 100 000 points if the algorithms are to work with any degree of accuracy. Erroneous conclusions might be drawn even with quite long data sets (Ruelle,1990; Vassilicos et al, 1992).

>- Data must not contain much noise because these methods cannot detect underlying chaos when mixed with even a small amount of noise (Smith, 1992).

» Correlation dimension algorithms depend on counting the number of points inside a small ball in m-dimensional space, and results can be quite sensitive to the choice of diameter for this ball (Liu et al., 1992).

» The time series must be stationary if results are to be correctly interpreted (Casdagli, 1992).

Taken together, it should be clear that a large amount of cleaned and pre-whitened high-frequency data is necessary in order to apply these algorithms. However, not all the research that has been published for and against chaos in capital markets has been based on empirical methods that meet these stringent criteria. Therefore it is not surprising that opinions are very mixed. Those that claim to have found chaos in financial markets include Peters (1991), Medio (1992) and de Grauwe et al. (1993). But many more papers find no conclusive evidence of chaos, among them Scheinkman and LeBaron (1989), Hseih (1991), Tata and Vassilicos (1991), Liu et al. (1992), Alexander and Giblin (1994), Drunat et al. (1996), and Abhyankar et al. (1997).

Not all the research that has been published for and against chaos in capital markets has been based on empirical methods that meet these stringent criteria

13.3.2 Nearest Neighbour Algorithms

Now let us consider which concepts in the chaos detection algorithms just described could be used as a basis for high-frequency prediction models. First, since Lyapunov exponents will be positive if the series is chaotic, points on neighbouring trajectories will diverge. Chaos-based prediction algorithms can only be designed for very short-term predictions in high-frequency data. Secondly, note that the correlation dimension required embedding the time series in a higher-dimensional space and then taking points in a small ball around the current point. Such nearest neighbour methods have formed the basis of some quite successful prediction algorithms for high-frequency data.

Figure 13.6 shows two different representations of the hourly returns data from a high-frequency price series. The first representation in Figure 13.6a is simply

0.01-

Figure 13.6 (a) Traditional time series representation; (b) two-dimensional time-delay embedding of the same series.

a time series graph, which is useful for investigating properties such as autocorrelation and conditional heteroscedasticity. The second representation in Figure 13.6b is a two-dimensional time-delay embedding, a scatter plot of current against lagged returns. This is just another view of the same data. The dimension of this embedding is only 2, far too small to reveal any strange attractor even if it were to exist; it is only used here to illustrate the concept of a time-delay embedding.

In standard nearest neighbour prediction methods, first each historic point of the time series is mapped into w-dimensional space Stm using the time-delay embedding (13.1). This creates a library of historic data in the form of a pattern of points in W\ Then, to make a prediction for the current point in time x„ it is mapped into the library using the same embedding. The nearest neighbours of x, in 9im are found and then the first coordinates of these points are used as data on the explanatory variable in a univariate prediction model.