Essays on Exchange Rates Deterministic Chaos and Technical Analysis

2.3.1 Statistical framework

The statistical framework proposed in Paper [ii] is based on a previous work by Gencay (1996a), and utilizes a moving blocks bootstrap procedure to test for the presence of a positive Lyapunov exponent in an observed stochastic time series. One weakness with the proposed framework is that the dynamics are non-chaotic under the null hypothesis.

The main idea with bootstrapping is to use the available observations to design a sort of Monte Carlo experiment in which the observations themselves are used to approximate the distribution of the error terms or other random quantities (Efron, 1979). Kunsch (1989) and Liu and Singh (1992) extended the idea of bootstrapping to the case where the observations form a stationary sequence.

Specifically, consider a sequence {X1 , ..., XN } of weakly dependent stationary random variables. According to Kunsch (1989) and Liu and Singh (1992), the distribution of certain estimators can be consistently constructed by blockwise bootstrap."

Therefore, let Bt denote a moving block of b consecutive observations, i.e.

If k satisfy , then resample with replacement, k blocks from the sequence

Denote the resulting sampled blocks by

and concatenate these blocks into one vector

which constitutesthe bootstrap sample. Let be the unknown parameter of interest, e.g. the largest Lyapunov exponent,
its considered estimator and the statistic computed from the bootstrap sample. By obtaining a large number of bootstrap values

one can estimate the distribution of

More precisely, the bootstrap values form an empirical distribution which can be utilized in statistical hypothesis testing. The idea behind the test schemes below is simply to let the states on the reconstructed trajectory to be the moving blocks.

Test schemes

The null hypothesis H_o and the alternative hypothesis H₁ are formulcted as

where is the unknown parameter, i. e. the largest Lyapunov exponent. Under the null hypothesis, the dynamics are non-chaotic.
Two test schemes are presented; one is general and the other is simplified. The general test scheme allows a reconstruction delay that is different from one, i. e.

in eq. (2.4), whereas the simplified test scheme restricts the reconstruction
delay to one, i. e. J = 1 in eq. (2.4). The latter test scheme is presented because in practice, most reconstructions of the dynamics use a reconstruction delay that is equal to one. The general test scheme consists of the following steps:

(i)

(ii)

and estimate the largest Lyapunov exponent . The embedding dimension m' should satisfy Takens' (1981) embedding criterion, i.e.

where D_A is the dimension of the attractor.

(iii)

The selected block size satisfies

and should also satisfy Takens' (1981) embedding criterion, i.e.

Denote the states on the reconstructed trajectory, i. e. the delay vectors, by T1 , ..., T,,,, where the number of states are

N_ts = N - b + 1.

(iv)

Resample with replacement, k blocks from the sequence {T1 , ..., T,11} where k = N mod b. Denote the resulting sampled blocks by

(v)

where

R".

The sequence

constitutes the bootstrap sample.

(vi)

(vii)

(viii)

Construct a one-sided confidence interval, e.g. a 95% confidence interval,by calculating the critical value as

, following from

is the quantile for the distribution in step (vii).

(ix)

(i) Reconstruct the attractor from the observed scalar N-point time series where m = b and J = 1, and estimate the largest Lyapunov exponent . The embedding dimension b, i. e. the selected block size, should satisfy Takens' (1981) embedding criterion, i. e. b > 2DA where DA is the dimension of the attractor. Denote the states on the reconstructed trajectory, i. e. the delay vectors, by

where the number of states are

(ii) Resample with replacement, k blocks from the sequence

where k = N mod b. Denote the resulting sampled blocks by

The sequence

constitutes the bootstrap sample.

(iii) Estimate the bootstrap value of the largest Lyapunov exponent from the bootstrap sample and calculate

(iv) Repeat steps (ii)-(iii) a large number of times in order to construct an empirical distribution for

(v) Construct a one-sided confidence interval, e.g. a 95% confidence interval, by calculating the critical value as

following from

where q(95%) is the quantile for the distribution in step (iv).

(vi) If

then the null hypothesis is rejected.

Differential equations or discrete time maps as the sources of the observed
dynamics? The proposed framework above can also be utilized to discriminate
between differential equations, i.e. a continuous dynamical system, and discrete time maps, i. e. a discrete dynamical system, as the sources of observed dynamics.

If the dynamics are generated by differential equations, i. e. a flow, then at least one of the Lyapunov exponents is equal to zero (Abarbanel, 1996, and Eckmann and Ruelle, 1985). The reason is that if we choose to make a perturbation to a trajectory in the same direction as the trajectory is going then that perturbation will simply move us along the same trajectory on which we started. Thus, there is no divergence between the "new" trajectory and the "old" trajectory, i. e.

for that particular direction. In the case of finite time maps, there are no flow directions. Thus, if one of the Lyapunov exponents is zero we can be confident that we have a flow. In particular, a two-sided test statistic can be designed in the proposed framework:

where A is the unknown parameter. Under the null hypothesis, the dynamics are generated by differential equations. Thus, if the null hypothesis is rejected for each Lyapunov exponent in the entire Lyapunov exponent spectrum, then the dynamics cannot be generated by differential equations. This information may be useful when modelling the observed dynamics.

Prof. Mikael Bask

Next: Short summaries of Papers [i]

Summary: Index