Scheinkman and LeBaron Test for Predictability
Another interesting use of correlation integral was presented by Scheinkman and Lebaron (1989). As before, Cm(e) stands for the correlation integral for M as a phase dimension of reconstructed space, and threshold e. It is proven that
gives an estimate of conditional probability that
sup jyi+i - У2+І |< e,
sup |yi+i - У2+І |< e,
0 <i <M-1
Scheinkman-LeBaron function behavior i. e. the conditional probability that two states of the system are close, given that their past M histories are close.
This result can be implemented to define the measure of predictability and determinism of the data. If SM(e) does not saturate as M grows, then states of the system depend on the information about its history. Otherwise the dynamics are affected by some random factor unrelated to the system itself, which can be interpreted as stochastic behavior of the process. As a result, Scheinkman and Lebaron’s function gives the following criteria:
• If states are independent, then SM(e) does not depend on M;
• If past values of the series help predict future values, SM(e) will tend to increase with M.
Figure 4 demonstrates the behavior of SM(e) for spread and return series and four different threshold levels. Growth in case of spread indicates its stochastic nature.
Results for all six microstructure variables are given in Table 1. Predictability is observed for all except spread and price series, which is consistent with previous results.